Global pseudo-differential calculus on Euclidean spaces

Contents Preface Introduction ix 1 Background material 0.1 Basic Facts and Notation . . . . . . . . . . . . . . . . ...

Author: Nicola F. | Rodino L.

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Contents Preface Introduction

ix 1

Background material 0.1 Basic Facts and Notation . . . . . . . . . . . . . . . . . . . . . . . 0.2 Function Spaces and Fourier Transform . . . . . . . . . . . . . . . 0.3 Identities and Inequalities for Factorials and Binomial Coefficients

9 9 11 13

1

15 15 19 23 23 29 31 34 35 38 40 41 45 46 50 51 52 54

Global Pseudo-Differential Calculus Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Symbol Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Action on S . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Adjoint and Transposed Operator. Action on S . . . . . . . 1.2.3 Composition of Operators . . . . . . . . . . . . . . . . . . . 1.3 Global Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Hypoellipticity and Construction of the Parametrix . . . . . 1.3.2 Slow Variation and Construction of Elliptic Symbols . . . . 1.4 Boundedness on L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Fredholm Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Abstract Theory . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Pseudo-Differential Operators . . . . . . . . . . . . . . . . . 1.7 Anti-Wick Quantization . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Short-Time Fourier Transform and Anti-Wick Operators . . 1.7.2 Relationship with the Weyl Quantization . . . . . . . . . . 1.7.3 Applications to Boundedness on L2 and Almost Positivity of Pseudo-Differential Operators . . . . . . . . . . . . . . . 1.7.4 Sobolev Spaces Revisited . . . . . . . . . . . . . . . . . . . 1.8 Quantizations of Polynomial Symbols . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58 60 62 64

vi

Contents Γ-Pseudo-Differential Operators and H-Polynomials Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Γ-Pseudo-Differential Operators . . . . . . . . . . . . . . . . 2.2 Γ-Elliptic Differential Operators; the Harmonic Oscillator . 2.3 Asymptotic Integration and Solutions of Exponential Type 2.4 H-Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Quasi-Elliptic Polynomials . . . . . . . . . . . . . . . . . . . 2.6 Multi-Quasi-Elliptic Polynomials . . . . . . . . . . . . . . . 2.7 ΓP -Pseudo-Differential Operators . . . . . . . . . . . . . . . 2.8 Lp -Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

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67 . 67 . 70 . 77 . 82 . 88 . 94 . 97 . 106 . 116 . 126

G-Pseudo-Differential Operators Summary . . . . . . . . . . . . . . . . . . . . . . . 3.1 G-Pseudo-Differential Calculus . . . . . . . . 3.2 Polyhomogeneous G-Operators . . . . . . . . 3.3 G-Elliptic Ordinary Differential Operators . . 3.4 Other Classes of Globally Regular Operators Notes . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

129 129 132 137 145 148 150

. . . . . . . . . . . . Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

153 153 155 158 164 166 173 177 193 195 199

5 Non-Commutative Residue and Dixmier Trace Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Non-Commutative Residue for Γ-Operators . . . . . . . . . . . 5.2 Trace Functionals for G-Operators . . . . . . . . . . . . . . . . 5.3 Dixmier Traceability for General Pseudo-Differential Operators Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

203 203 206 213 221 224

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. . . .

227 227 229 241 256

2

3

4

6

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. . . . . .

. . . . . .

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Spectral Theory Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Unbounded Operators in Hilbert spaces . . . . . . . 4.2 Pseudo-Differential Operators in L2 : Realization and 4.3 Complex Powers . . . . . . . . . . . . . . . . . . . . 4.3.1 The Resolvent Operator . . . . . . . . . . . . 4.3.2 Proof of Theorem 4.3.6 . . . . . . . . . . . . 4.4 Hilbert-Schmidt and Trace-Class Operators . . . . . 4.5 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . 4.6 Weyl Asymptotics . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

Exponential Decay and Holomorphic Extension of Solutions Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Function Spaces Sνμ (Rd ) . . . . . . . . . . . . . . 6.2 Γ-Operators and Semilinear Harmonic Oscillators . . . 6.3 G-Pseudo-Differential Operators on Sνμ (Rd ) . . . . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . . . .

. . . .

. . . .

. . . .

Contents

vii

6.4 A Short Survey on Travelling Waves . . . . . . . . . . . . . . . . . 271 6.5 Semilinear G-Equations . . . . . . . . . . . . . . . . . . . . . . . . 275 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Bibliography

287

Index

301

Index of Notation

305

Preface The modern theory of general linear PDEs was largely addressed to local problems, i.e., to the study of solutions in a suitably small neighbourhood of x0 ∈ Rd . The analysis of global solutions, in fixed domains, subsets of Rd , or even manifolds, appears as a second step, reconnecting with classical results. A simple but important global situation, where interest goes back to Quantum Mechanics, Signal Analysis and other applications in Physics and Engineering, is represented by the study of solutions in the whole Euclidean space Rd . To such study we devote the present book, making systematic use of the techniques of pseudo-differential operators. In fact, the Fourier transform and pseudo-differential operators play an essential role in the modern theory of PDEs, both from the local and global points of view. Actually, the pseudo-differential calculus was initially introduced by KohnNirenberg and Hörmander and then developed by other authors, mainly in a local context, to study local regularity and local solvability of PDEs. On the other hand, the Fourier transform and pseudo-differential calculus find in Rd their natural setting; for example, the more recent treatment of Hörmander, the so-called Weyl-Hörmander calculus, is formulated in symplectic vector spaces. However, genuine global settings in Rd , providing regularity of solutions in the Schwartz space S(Rd ) and compactness in Rd of resolvents of globally-elliptic operators, require symbols with a precise asymptotic control when the variable x goes to infinity. It is exactly these symbols that we address in the present book. Let us list some topics that we discuss in the volume, and mention several others which we neglect for the sake of brevity. First, our attention is restricted to globally elliptic equations, whereas the important study of the corresponding evolution case, in particular the hyperbolic case, is omitted. Moreover, concerning classes of pseudo-differential operators, we mainly treat symbols possessing a homogeneous structure, namely the so-called classic Γ and G symbols. Nevertheless in the first part of the book, addressed to non-experts, we propose a relatively general calculus, very easy to handle and involving only elementary computations. With respect to the Weyl-Hörmander calculus, we make here a restrictive assumption, namely our weights are bounded from below by positive constants; such a condition is satisfied indeed by Γ and G classes. Also, we omit the study of the symplectic invariance of our pseudo-differential calculus, limiting ourselves

x

Preface

to an emphasis on its behaviour under Fourier conjugation, which is of relevance in some applications. Similarly, extensions to more general types of non-compact manifolds, in particular the so-called manifolds with exits, are omitted, as well as general index formulas. The arguments which are treated in the book reflect the research interests of the authors and their collaborators, and collect results obtained by them in the last 10 years. Our first main line of discourse is devoted to Spectral Theory. We pay particular attention to complex powers and asymptotics for the counting function (without sharp remainder). In this regards, the present volume owes much to the preceding monograph of Boggiatto, Buzano and Rodino. We also discuss the non-commutative residue in Rd and, in strict connection with Spectral Theory, the Dixmier trace. The second main line of discourse is devoted to the study of exponential decay and holomorphic extension of the solutions. We refer here to the Gelfand-Shilov spaces Sνμ (Rd ), replacing successfully the Schwartz space in Applied Mathematics; a self-contained presentation is given in this book. The results, coming from a series of papers by Cappiello, Gramchev and Rodino, refer to semi-linear perturbations of Γ and G equations. Applications, besides to non-linear Quantum Mechanics, are also to travelling waves. We wish finally to express our gratitude to Ernesto Buzano, of the University of Torino. In addition to reading a large part of the manuscript and suggesting many improvements, he discussed with us the structure of the book and took part in the choice of the contents. We wish also to express our warmest thanks to Claudia Garetto (University of Innsbruck), Alessandro Morando (University of Brescia), Ubertino Battisti, Paolo Boggiatto, Marco Cappiello, Elena Cordero, Sandro Coriasco, Giuseppe De Donno, Gianluca Garello, Alessandro Oliaro, Patrik Wahlberg (University of Torino) for contributing scientific material, reading the manuscript and suggesting improvements.

Torino, December 2009

Fabio Nicola, Luigi Rodino

Introduction To give an introduction to the contents of the book, let us consider initially the basic models to which our pseudo-differential calculus will apply, namely the linear partial differential operators with polynomial coefficients in Rd : (I.1) P = cαβ xβ Dα , x ∈ Rd , cαβ ∈ C, where in the sum (α, β) ∈ Nd × Nd runs over a finite subset of indices. A natural setting for P is given by the Schwartz space S(Rd ) and its dual S (Rd ). These spaces are invariant under the action of the Fourier transform d Fu(ξ) = u (ξ) = e−ixξ u(x) dx, ¯ with dx ¯ = (2π)− 2 dx. (I.2) Note also that the conjugation FP F −1 gives still an operator of the form (I.1). The preliminary problem in our analysis will be to establish the global regularity of P by means of the construction of parametrices in the pseudo-differential form, with suitable symbols a(x, ξ) ∈ C ∞ (Rd × Rd ): Au(x) = a(x, D)u(x) = eixξ a(x, ξ) u(ξ) dξ. ¯ (I.3) Namely, a parametrix A of P is a linear map S(Rd ) → S(Rd ), S (Rd ) → S (Rd ) such that (I.4) P A = I + R1 , AP = I + R2 where R1 , R2 are regularizing, i.e., R1 , R2 : S (Rd ) → S(Rd ). Starting from the equation P u = f with u ∈ S (Rd ), f ∈ S(Rd ), the second identity in (I.4) implies AP u = u + R2 u, hence we conclude u ∈ S(Rd ), i.e., in our terminology P is globally regular. In particular all the solutions u ∈ S (Rd ) of P u = 0 belong to S(Rd ). From the existence of a pseudo-differential parametrix A, we may also deduce the Fredholm property of P in Sobolev spaces with suitable weights. Define the symbol of P as standard: (I.5) p(x, ξ) = cαβ xβ ξ α .

2

Introduction

Which properties of the polynomial p(x, ξ) warrant the construction of the parametrix? The standard (local) ellipticity can be read for p(x, ξ) in (I.5), where we assume |α| ≤ m, |β| ≤ n, as well as the condition

cαβ xβ ξ α = 0

for ξ = 0.

(I.6)

|α|=m |β|≤n

This provides the existence of local pseudo-differential parametrices and local regularity, but does not give the required control on the asymptotic behaviour of the solutions for x → ∞. The basic idea is then to add to (I.6) conditions involving the x-variables in the homogeneous structure. As suggested by the invariance of the class of the operators (I.1) under Fourier conjugation, we expect in these conditions a joint or symmetrical role for the x and ξ variables. We may identify two different approaches, somewhat in competition with each other from the historical point of view, both based on model equations of Quantum Mechanics. Γ-classes. Our starting model is the harmonic oscillator −Δ + |x|2 − λ.

(I.7)

We generalize it by considering

P =

cαβ xβ Dα

(I.8)

|α|+|β|≤m

satisfying the Γ-ellipticity assumption cαβ xβ ξ α = 0 pm (x, ξ) =

for (x, ξ) = (0, 0),

(I.9)

|α|+|β|=m

which implies local ellipticity. The operator in (I.7) is Γ-elliptic, independently of λ ∈ C. A parametrix for P in (I.8) is constructed as a pseudo-differential operator with symbol having principal part 1/p(x, ξ), which we cut off in the bounded region where possibly p(x, ξ) = 0. The natural class containing the symbol of the parametrix is defined by considering z = (x, ξ) ∈ R2d and imposing the estimates |∂zγ a(z)| zm−|γ| ,

z ∈ R2d ,

(I.10)

where now m ∈ R, γ ∈ N2d and z = (1 + |z|2 )1/2 = (1 + |x|2 + |ξ|2 )1/2 . The corresponding pseudo-differential calculus was first given in Shubin [182], 1971. G-classes. The basic model is the free particle operator in Rd −Δ − λ.

(I.11)

Introduction

3

As a generalization, consider the operator P = cαβ xβ Dα

(I.12)

|α|≤m |β|≤n

satisfying the following G-ellipticity condition. First, considering the so-called bi-homogeneous principal symbol, we impose cαβ xβ ξ α = 0 for x = 0, ξ = 0. (I.13) pm,n (x, ξ) = |α|=m |β|=n

Then we assume the standard local ellipticity (I.6) and the dual property, obtained from (I.6) by interchanging the role of x and ξ:

cαβ xβ ξ α = 0

for x = 0.

(I.14)

|α|≤m |β|=n

In the case n = 0, i.e., when P = p(D) is an operator with constant coefficients, the local ellipticity (I.6) implies (I.13), whereas (I.14) is satisfied if and only if p(ξ) = 0 for all ξ ∈ Rd . So the operator in (I.11) is G-elliptic when λ ∈ R+ ∪ {0}. Under the assumptions (I.6), (I.13), (I.14), a parametrix can be constructed with symbol in the G-classes defined by the estimates |∂ξα ∂xβ a(x, ξ)| ξm−|α| xn−|β| ,

(x, ξ) ∈ R2d ,

(I.15)

for m ∈ R, n ∈ R. The corresponding pseudo-differential operators were introduced by Parenti [156], 1972, and then studied in detail by Cordes [59], 1995. In short: the first aim of this book is to present a simple pseudo-differential calculus, containing both Γ-classes and G-classes as particular cases, and to give in this framework some main results, concerning construction of parametrices, weighted Sobolev spaces, Fredholm property and global regularity. Peculiarities of Γ and G operators, depending on the respective homogeneous structures, are then emphasized. Attention is given to recent results, concerning in particular Lp -boundedness, non-commutative residues, exponential decay and holomorphic extension of solutions of semi-linear Γ and G equations. A more detailed description of the contents can be found in the Summary preceding each chapter. In the following we illustrate general ideas, list models having importance in the applications and provide some references. A detailed bibliography will be found in the Notes at the end of each chapter. As for the pseudo-differential calculus in Chapter 1, the symbols are defined by the estimates |∂ξα ∂xβ a(x, ξ)| M (x, ξ)Ψ(x, ξ)−|α| Φ(x, ξ)−|β| ,

(x, ξ) ∈ R2d ,

(I.16)

4

Introduction

where M , Φ, Ψ are positive weight functions in R2d satisfying suitable conditions. In particular, the effectiveness of the symbolic calculus is granted by the strong uncertainty principle (1 + |x|2 + |ξ|2 )/2 Φ(x, ξ)Ψ(x, ξ),

(x, ξ) ∈ R2d ,

(I.17)

for some > 0. The Γ-classes in (I.10) are recaptured by setting Φ(x, ξ) = Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )1/2 , so that (I.17) is valid with = 2. The choice Φ(x, ξ) = x, Ψ(x, ξ) = ξ gives the G-classes. The strong uncertainty principle is satisfied with = 1 since (1 + |x|2 + |ξ|2 )1/2 ≤ xξ = Φ(x, ξ)Ψ(x, ξ). In Chapter 2 we fix attention on Γ-classes and their generalizations, considering the case when p(x, ξ) in (I.5) is a H-polynomial in R2d , or in particular a multi-quasi-elliptic polynomial as in Boggiatto, Buzano, Rodino [19]: p(x, ξ) = cαβ xβ ξ α , (I.18) (β,α)∈P

where the Newton polyhedron P of p(x, ξ) is assumed to be complete (in short: the normals to the faces have strictly positive components), with lower bound |p(x, ξ)| ΛP (x, ξ) =

x2β ξ 2α

1/2 (I.19)

(β,α)∈P

for large (x, ξ). Taking Ψ(x, ξ) = Φ(x, ξ) = ΛP (x, ξ)1/μ , where μ is the so-called formal order of P, we obtain a symbolic calculus which satisfies the strong uncertainty principle. Following the results of Morando [148], Garello and Morando [85], we prove that the corresponding pseudo-differential operators are Lp -bounded, with 1 < p < ∞. For the operator P with symbol (I.18) we deduce in particular the a priori estimates

xβ Dα u Lp P u Lp + u Lp (I.20) (β,α)∈P

and related Fredholm properties. The results apply for example to the following generalizations of the harmonic oscillator P = −Δ + V (x)

(I.21)

where the potential V (x) is a positive multi-quasi-elliptic polynomial with respect to the x-variables. In Chapter 3, besides G-pseudo-differential operators, we consider extensions of (I.21) to more general potentials V (x), by constructing the parametrix in the classes with weights Φ(x, ξ) = 1, Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )ρ/2 , ρ > 0, for which (I.17) is still satisfied, cf. Buzano [27]. Chapter 4 is devoted to Spectral Theory for pseudo-differential operators with symbol in the classes defined by (I.16), (I.17). For generic weights Φ, Ψ, we

Introduction

5

study the complex powers following Buzano and Nicola [30], and compute the trace of the heat kernel. Precise asymptotic expansions are deduced for the counting functions N (λ), λ → +∞, of self-adjoint operators in the case of Γ and G classes. Namely, for a Γ-operator with principal symbol pm (x, ξ) as in (I.9) we prove the formula of Shubin [182] 2d −d m dx dξ. (I.22) N (λ) ∼ λ (2π) pm (x,ξ)≤1

This gives in particular for the eigenvalues of the harmonic oscillator (I.7) the well-known formula λd N (λ) ∼ d . (I.23) 2 d! For a G-operator with bi-homogeneous principal symbol pm,n (x, ξ), m > 0, n > 0, as in (I.13), we obtain according to Maniccia and Panarese [138]: ⎧ d ⎪ ⎨Cλ m log λ, for m = n, d N (λ) ∼ C λ m , (I.24) for m < n, ⎪ ⎩ nd for m > n, C λ , for constants C, C , C which can be computed in terms of pm,n , of the symbol in (I.6) and the symbol in (I.14), respectively. In Chapter 5 we present results on non-commutative residue and Dixmier’s trace, from Boggiatto and Nicola [20], Nicola [152], Nicola and Rodino [154]. Let us recall that a linear map on an algebra over C is called a trace if it vanishes on commutators. For the algebra of the classical pseudo-differential operators on a compact manifold, a trace is given by the so-called Wodzicki’s non-commutative residue. For classical Γ-operators in Rd , namely with symbol having asymptotic expansion in homogeneous terms a(z) ∼ ∞ j=0 am−j (z), m ∈ Z, we define a−2d (Θ) dΘ, (I.25) Res a(x, D) = S2d−1

2d

where z = (x, ξ) ∈ R , and dΘ is the usual surface measure on S2d−1 . The map Res in (I.25) turns out to be the unique trace on the algebra of the classical Γoperators which vanishes on regularizing operators, up to a multiplicative constant. Similarly to the result of Connes in the case of a compact manifold, even in Rd the map Res coincides with the Dixmier trace Trω , when applied to a Γ-operator of order −2d. Namely Res a(x, D) = 2d(2π)d Trω (a(x, D))

(I.26)

where, limiting for simplicity to a positive self-adjoint operator a(x, D) with eigenvalues λj → 0+ , j = 1, 2, . . ., N 1 λj . N →∞ log N j=1

Trω (a(x, D)) = lim

(I.27)

6

Introduction

We may test (I.26), (I.27) on the inverse of the d-th power of the harmonic oscillator in (I.7): h(x, D) = (−Δ + |x|2 )−d (I.28) for which we have Res h(x, D) = S2d−1

dΘ = 2d(2π)d Trω (h(x, D)),

(I.29)

as we easily deduce from (I.23). We obtain similar results for the algebra of Goperators. Concerning operators defined by general weights Φ and Ψ, we content ourselves with giving a sufficient condition for Dixmier traceability. Because of the lack of homogeneity, the problem of the definition of the non-commutative residue is in this case difficult, and challenging. In the conclusive Chapter 6 we replace, as a basic space, the Schwartz space S(Rd ) with the subspaces Sνμ (Rd ), μ > 0, ν > 0, which are better adapted to the study of the problems of Applied Mathematics. We have f ∈ Sνμ (Rd ) if and only if |f (x)| e−|x| |f(ξ)| e

1/ν

1/μ

−|ξ|

,

x ∈ Rd ,

(I.30)

,

ξ∈R ,

(I.31)

d

for some > 0. If μ ≤ 1, the function f extends to the complex domain, and the estimates (I.31) imply precise bounds for this extension. After a detailed discussion of the properties of the classes Sνμ (Rd ) we present the results of Cappiello, Gramchev and Rodino [41], [44], Cappiello and Rodino [45], concerning exponential decay and holomorphic extension of the solutions of Γ-elliptic and G-elliptic equations. In short, the conclusions are the following. As suggested by the behaviour of the eigenfunctions of the harmonic oscillator, i.e., the Hermite functions, all the 1/2 solutions u ∈ S (Rd ) of a generic Γ-elliptic equation P u = 0 belong to S1/2 (Rd ), that implies super-exponential decay and holomorphic extension in Cd . Instead, eigenfunctions of G-elliptic equations are in S11 (Rd ); this gives exponential decay and holomorphic extension limited to a strip {x + iy ∈ Cd : |y| < T }. Particular attention in the second part of Chapter 6 is reserved for semi-linear equations, because of their importance in applications. Consider for example the semi-linear harmonic oscillator of the Quantum Mechanics: −Δ + |x|2 u − λu = G[u],

(I.32)

where the non-linear term is G[u] = uk , k ≥ 2, or more generally G[u] = L(uk ) with L a first-order Γ-operator. For the eigenfunctions of (I.32), i.e., homoclinics, we still have super-exponential decay; however, with respect to the linear Γ-case, the entire extension is lost, and analyticity in the complex domain is limited to a strip.

Introduction

7

A basic example in the G-case is given by a linear part with constant coefficients cα Dα u = F [u]. (I.33) |α|≤m

As observed before, we have G-ellipticity if and only if the symbol p(ξ) = |α|≤m cα ξ α is elliptic in the standard sense and p(ξ) = 0 for all ξ ∈ Rd . The non-linear term F [u] is an arbitrary polynomial in u and lower order derivatives. The result for (I.33) is the same as in the linear case, namely homoclinics belong to S11 (Rd ). Relevant examples are (I.34) u − Cu + u2 = 0 in R for the solitary wave v(t, x) = u(x−Ct), C > 0, of the Korteweg-de Vries equation, and higher order travelling waves equations; we recapture for all of them the results of exponential decay, expected by the physical intuition. Our result also applies to the d-dimensional extension of (I.34) −Δu + u = uk ,

k ≥ 2,

(I.35)

appearing in plasma physics and non-linear optics. Finally, as a non-local example, the intermediate-long-wave equation in R eD + e−D Du + γu = u2 , eD − e−D

γ > −1,

(I.36)

is contained in our theory, because the symbol ξCtgh ξ +γ of the Fourier multiplier in the left-hand side of (I.36) is G-elliptic, and its homoclinics belong to S11 (R).

Background Material In this chapter we fix the notation used in the present book and we collect some results from Real Analysis which will be useful in the sequel. For a comprehensive account we refer the reader to Hörmander [119, Vol I]; see also Grubb [102].

0.1 Basic Facts and Notation We employ standard set-theoretic notation, with one peculiarity: set-theoretic inclusion A ⊂ B does not exclude equality. Thus proper inclusion has to be stated explicitly: A ⊂ B and A = B. We set N = {0, 1, 2, ...}, whereas Z, R stand for the set of all integer and real numbers, respectively. We define R+ = {x ∈ R : x > 0}, R− = {x ∈ R : x < 0}. Given a real number x, we set x+ = max{x, 0} = 12 (x + |x|) and x− = min{x, 0} = 12 (x − |x|). Hence x+ and x− are the positive and the negative part of x respectively. The integer part of a real number x is denoted by [x]. Then [x] is the unique integer such that [x] ≤ x < [x] + 1. If X is a non-empty set and f, g : X → [0, +∞), we set f (x) g(x),

x ∈ X,

if there exists C > 0 such that f (x) ≤ Cg(x), for all x ∈ X. Moreover, if f and g depend on a further variable z ∈ Z, the statement that, for all z ∈ Z, f (x, z) g(x, z),

x ∈ X,

means that for every z ∈ Z there exists a real number Cz > 0 such that f (x, z) ≤ Cz g(x, z) for every x ∈ X. Of course, it may happen that supz∈Z Cz = ∞. Also, we set f (x) g(x), x ∈ X, if f (x) g(x) and g(x) f (x), x ∈ X, and similarly as above if the functions depend on a further parameter.

10

Background Material We will employ the multi-index notation. Given α, β ∈ Nd and x ∈ Rd , we

set α! = α1 ! · · · αd !, |α| = α1 + · · · + αd , α ≤ β ⇐⇒ αj ≤ βj , for j = 1, . . . , d, α < β ⇐⇒ α = β and α ≤ β,

d αj α! α = , β ≤ α, = β β!(α − β)! β j j=1 αd 1 xα = xα 1 · · · xd .

Functions are always understood complex-valued, if not stated otherwise. Partial derivatives are denoted by ∂j = ∂xj =

∂ , ∂xj

Dj = Dxj = −i∂j ,

j = 1, . . . , d,

where i is the imaginary unit. More generally, we set ∂ α = ∂1α1 · · · ∂dαd = ∂xα = ∂xα11 · · · ∂xαdd , Dα = D1α1 · · · Ddαd = Dxα = Dxα11 · · · Dxαdd . If x, ξ ∈ Rd , we set xξ = x · ξ = x, ξ =

d

xj ξ j ,

j=1

|x| =

d j=1

x2j

1/2 ,

1/2 2 . x = 1 + |x| Observe that x is a smooth function satisfying ∂xj x = xj /x. In general one verifies that (0.1.1) |∂xα x| ≤ 2|α|+1 |α|!x1−|α| , for all x ∈ Rd and all α ∈ Nd . The following elementary inequality, sometimes called Peetre’s inequality, will be used throughout the book: x + ys ≤ cs xs y|s| ,

x, y ∈ Rd , s ∈ R,

with a constant cs > 0. The power of multi-index notation is well explained by the following las. For an open subset X of Rd , consider the spaces C n (X) of functions continuous partial derivatives of order ≤ n, and C ∞ (X) = ∩n∈N C n (X). f, g ∈ C n (X), we have Leibniz’ formula: α ∂ β f ∂ α−β g, |α| ≤ n. ∂ α (f g) = β β≤α

(0.1.2) formuhaving Given

(0.1.3)

0.2. Function Spaces and Fourier Transform

11

If we assume furthermore that X is convex, we have Taylor’s formula: f (y) =

1 ∂ α f (x)(y − x)α α! |α| d/2.

(0.2.6)

Finally, we will also use the fact that the space H s (Rd ), for s > d/2, is an algebra with respect to pointwise multiplication, and the following Schauder estimates are satisfied: (0.2.7)

f g H s (Rd ) ≤ Cs f H s (Rd ) g H s (Rd ) , f, g ∈ H s (Rd ).

0.3 Identities and Inequalities for Factorials and Binomial Coefficients We collect in the sequel some identities and inequalities for factorials and binomial coefficients. First we recall the generalized Newton formula:

(t1 + . . . + td )N =

|α|=N

N! d tα1 . . . tα d , α 1 ! . . . αd ! 1

(0.3.1)

where N ∈ N and t1 , ..., td are real numbers. Fixing t1 = . . . = td = 1 in (0.3.1) we deduce N! . (0.3.2) dN = α! |α|=N

This implies in particular

|α|! ≤ d|α| α!.

(0.3.3)

When d = 2, we obtain from (0.3.2) and (0.3.3), respectively 2N =

k+j=N

N! k! j!

(0.3.4)

and for any k, j ∈ N. Hence

(k + j)! ≤ 2k+j k! j!

(0.3.5)

(α + β)! ≤ 2|α|+|β| α! β!

(0.3.6)

for any α, β ∈ Nd , whereas obviously α! β! ≤ (α + β)!.

(0.3.7)

From (0.3.4) we have then α β≤α

β

= 2|α| ,

α ∈ Nd ,

(0.3.8)

14

Background Material

and in particular

α ≤ 2|α| , β ≤ α. β Recall now the definition of the Euler Gamma function +∞ e−s st−1 ds, t > 0. Γ(t) =

(0.3.9)

0

Observing that Γ(t + 1) = tΓ(t) and Γ(1) = 1, one obtains n! = Γ(n + 1),

n ∈ N.

(0.3.10)

The asymptotic behaviour of the Gamma function is given by the well-known Stirling formula: √ Γ(t + 1) = tt e−t 2πt eθ(t)/(12t) , t ≥ 1, where 0 < θ(t) < 1. In particular we have tt ≤ et Γ(t + 1), and also Γ(t + 1) ≤ tt for large t. Taking into account (0.3.10), we then obtain for N = 1, 2, . . . , √ (0.3.11) N ! = N N e−N 2πN eθN /(12N ) , where 0 < θN < 1; in particular we have N N ≤ eN N !,

(0.3.12)

whereas obviously N ! ≤ N for all N = 1, 2, . . . . We have from the Taylor expansion of et for t > 0, N

tN ≤ N !et , N

which implies t

N = 0, 1, . . . ,

(0.3.13)

≤ N e . This last inequality is improved by N t

tA ≤ AA et−A ,

A > 0, t > 0,

(0.3.14)

as one gets by taking logarithms. In particular we have tA e−t ≤ AA . Finally it will be useful to observe that

#{α = (α1 , . . . , αd ) ∈ Nd : |α| ≤ m} = whereas

m+d , m

m+d−1 . d−1

(0.3.15)

#{α = (α1 , . . . , αd ) ∈ Nd : |α| = m} =

(0.3.16)

Chapter 1

Global Pseudo-Differential Calculus Summary Pseudo-differential operators, in a broad sense, are linear operators of the type u(ξ) dξ, ¯ u ∈ S(Rd ). (1.0.1) a(x, D)u(x) = eixξ a(x, ξ) The function a(x, ξ), satisfying suitable estimates in R2d , is called the symbol of a(x, D). In fact, (1.0.1) makes sense as a temperate distribution even for a ∈ S (R2d ). A linear partial differential operator a(x, D) = cα (x)Dα , (1.0.2) |α|≤m

with slowly increasing coefficients can be regarded as a pseudo-differential operator with symbol cα (x)ξ α , a(x, ξ) = |α|≤m

as one sees by using (0.2.1), (0.2.2). The way of associating to a symbol a the operator a(x, D) in (1.0.1) is called standard, or left, quantization. The terminology is justified by the special case of the partial differential operators, where this quantization amounts to replacing ξ by D and to putting the coefficients to the left of the derivatives. Other important examples of pseudo-differential operators are given by convolution operators u → T ∗ u; in fact, assuming the Fourier transform of the distribution T is well defined, we have u(ξ) dξ, ¯ u ∈ S(Rd ), (1.0.3) (T ∗ u)(x) = eixξ a(ξ)

16

Chapter 1. Global Pseudo-Differential Calculus

where a(ξ) = (2π)d/2 T(ξ), see (0.2.4). As a particular case we quote the linear partial differential operators with constant coefficients and the convolution with fundamental solutions whose Fourier transform is defined in some sense. The first task of the theory is to express functional operations between pseudo-differential operators, as transposition and compositions, in terms of analytic operations on the corresponding symbols. For this program to work, the symbols need to satisfy some growth estimates at infinity, together with their derivatives. In Sections 1.1, 1.2, we will consider symbols a(x, ξ) satisfying, for every α, β ∈ Nd , estimates of the type |∂ξα ∂xβ a(x, ξ)| M (x, ξ)Ψ(x, ξ)−|α| Φ(x, ξ)−|β| ,

for (x, ξ) ∈ R2d ,

(1.0.4)

where Φ, Ψ, M are positive weight functions in R2d satisfying some technical conditions. We denote by S(M ; Φ, Ψ) the space of functions a(x, ξ) satisfying estimates (1.0.4). The choice of the functions Φ, Ψ, M depends on the problem under consideration; we will clarify this point later. Now, the symbol of the composition of two operators of the form (1.0.3) is exactly the product of the corresponding symbols. This is generally not true when dealing with two operators of the form (1.0.1), say a(x, D) and b(x, D); nevertheless, if a ∈ S(M1 ; Φ, Ψ) and b ∈ S(M2 ; Φ, Ψ), the composition of a(x, D) and b(x, D) is still a pseudo-differential operator with symbol in S(M1 M2 ; Φ, Ψ) which can be written as a(x, ξ)b(x, ξ) + r(x, ξ), (1.0.5) with a remainder r ∈ S(M1 M2 h; Φ, Ψ), where h := Φ−1 Ψ−1 is the so-called Planck function. It is assumed h(x, ξ) 1, so that r(x, ξ) also belongs to S(M1 M2 ; Φ, Ψ). Precisely, we have for r(x, ξ) the asymptotic expansion r(x, ξ) ∼

1 ∂ α a(x, ξ)Dxα b(x, ξ), α! ξ

(1.0.6)

α=0

which means that, for every N ≥ 2, 1 α r(x, ξ) − ∂ a(x, ξ)Dxα b(x, ξ) ∈ S(M1 M2 hN ; Φ, Ψ). α! ξ

(1.0.7)

0 0, which guarantees, for example, that the expression in (1.0.7) has a strong decay at infinity provided N is large enough. Similarly, we will prove formulas for the symbol of the transpose and the adjoint of a pseudo-differential operator. Here we only remark that the transpose of the differential operator a(x, D) = |α|≤m cα (x)D α is given by t a(x, D)u(x) = (−D)α (cα (x)u(x)) , |α|≤m

Summary

17

which suggests the study of pseudo-differential operators of the form Op1 (b)u(x) = ei(x−y)ξ b(y, ξ)u(y) dy ¯ dξ. ¯

(1.0.8)

α In fact, we have t a(x, D) = Op1 (b), with b(x, ξ) = |α|≤m cα (x)(−ξ) . The correspondence b → Op1 (b) is called right quantization. More generally, we will develop the whole theory for the so-called τ -quantization: Opτ (a)u(x) = ei(x−y)ξ a((1 − τ )x + τ y, ξ)u(y) dy ¯ dξ, ¯ (1.0.9) where τ ∈ R. When τ = 0 we recapture the operator a(x, D) in (1.0.1), whereas for τ = 1 we obtain the quantization in (1.0.8). The case τ = 1/2 is also of special interest and yields the Weyl quantization

x+y aw u(x) = ei(x−y)ξ a , ξ u(y) dy ¯ dξ. ¯ (1.0.10) 2 For example, in dimension d = 1, if a(x, ξ) = xξ, aw =

1 (xDx + Dx x). 2

This quantization has the nice property that real-valued symbols give rise to (formally) self-adjoint operators, which is one reason for its usefulness in Quantum Mechanics. We will prove in Section 1.2 that every operator of the form (1.0.10), with symbol a in some class S(M ; Φ, Ψ), is continuous on S(Rd ) and extends to a continuous operator on S (Rd ). Moreover symbols in S(1; Φ, Ψ) yield bounded operators in L2 (Rd ), cf. Section 1.4. This is easy to see by the aid of the symbolic calculus, assuming the strong uncertainty principle. A more general result will be discussed later as an application of Anti-Wick techniques. Let us now come to the main applications of the symbolic calculus in Section 1.3. The role of (1.0.5) and (1.0.6) is fundamental in this connection. Suppose, for example, that a symbol a(x, ξ) satisfies the estimate |a(x, ξ)| M (x, ξ), for |x| + |ξ| large; we say that a is elliptic. Then (1.0.5) and (1.0.6) show that the operator B with symbol a(x, ξ)−1 fulfills Ba(x, D) = I + S1 ,

a(x, D)B = I + S2 ,

(1.0.11)

where S1 and S2 have symbols in S(h; Φ, Ψ). Under the strong uncertainty principle, and by a slightly more refined argument based on (1.0.7), one can in fact construct B as a pseudo-differential operator verifying (1.0.11) with S1 and S2 regularizing operators, i.e., continuous as maps S (Rd ) → S(Rd ). We will refer to B as the parametrix of a(x, D). The existence of a parametrix gives at once a global regularity result for any operator with elliptic symbol; namely, if u ∈ S (Rd )

18


and a(x, D)u ∈ S(Rd ), then u ∈ S(Rd ). This also explains the advantage of considering the above quite general symbol classes: dealing with a given symbol a(x, ξ) (typically, of a particular differential operator), one looks for weights Φ, Ψ such that a satisfies (1.0.4) with M (x, ξ) = |a(x, ξ)| and, if there exist any, a(x, D) is then globally regular. Many examples and applications of this general principle will be detailed in the subsequent chapters. Another application of the parametrix is the theory of Sobolev spaces tailored to the above symbol classes. In Section 1.5 we will define the Sobolev space H(M ) as A−1 (L2 (Rd )), endowed with a convenient norm, where A is any pseudodifferential operator with an elliptic symbol in S(M ; Φ, Ψ). Any pseudo-differential operator P with symbol in S(M ; Φ, Ψ) will define a bounded operator from H(M ) to H(M/M ). These spaces measure the regularity and decay of functions and temperate distributions in Rd and allow us to give precise global regularity results for pseudodifferential operators with elliptic symbols. For example, if P has an elliptic symbol in S(M ; Φ, Ψ) and u ∈ S (Rd ) with P u ∈ H(M/M ), then u ∈ H(M ). It is worth noticing that when regarded as operators on such tailored Sobolev spaces, elliptic operators become Fredholm (they have kernel and cokernel of finite dimension) and an index theory can be developed, cf. Section 1.6. In Section 1.7 we will consider yet another type of quantization, the so-called Anti-Wick quantization. To this end, let 1

2

Gy,η (x) = π −d/4 eixη e− 2 |x−y| , where y, η ∈ Rd are regarded as parameters. These functions satisfy Gy,η L2 (Rd ) = 1 for every (y, η) ∈ R2d , and have the property to be localized near y and to have the Fourier transform localized near η, i.e., by denoting the open ball in Rd of center x0 and radius R by B(x0 , R), Gy,η L2 (B(y,1)) ≥ c, and G y,η L2 (B(η,1)) ≥ c, for some c > 0, uniformly with respect to y, η. By this reason, the behaviour near y of a given function u(x), and the behaviour near η of its Fourier transform u (ξ), are captured together by the short-time Fourier transform V u of u, defined by V u(y, η) = (u, Gy,η ). It is easy to see that V acts on S(Rd ) → S(R2d ) continuously and extends to a continuous map S (Rd ) → S (R2d ). Moreover V : L2 (Rd ) → L2 (R2d ) is an isometry, up to a constant factor. Then, the Anti-Wick operator Aa with symbol a ∈ S (R2d ) reads Aa u = (2π)−d V ∗ (aV u), u ∈ S(Rd ), namely as a Fourier multiplier but with the Fourier transform replaced by the short-time Fourier transform. It turns out that Aa : S(Rd ) → S (Rd ) continuously, and that Aa is in fact a pseudo-differential operator whose Weyl symbol b(x, ξ) is obtained by convolving a(x, ξ) with a Gaussian function in R2d . The main feature of the Anti-Wick

1.1. Symbol Classes

19

quantization is that it preserves positivity, i.e., non-negative symbols yield nonnegative operators. This is not true for the Weyl quantization, but the result for Anti-Wick operators can be fruitfully used to prove lower bounds for Weyl operators. Anti-Wick techniques also yield a very elementary proof of the L2 -boundedness of pseudo-differential operators whose symbols a(x, ξ) are bounded together with their derivatives, i.e., satisfying for every α, β ∈ Nd , |∂ξα ∂xβ a(x, ξ)| ≤ Cαβ ,

(x, ξ) ∈ R2d .

The functions Gy,η can in fact be used to almost diagonalize such operators, in the sense that, after conjugating with the short-time Fourier transform, the operator V a(x, D)V ∗ has an integral kernel K(y , η ; y, η) dominated by a convolution kernel in L1 . Young’s inequality then gives the desired boundedness on L2 .

1.1 Symbol Classes A positive continuous function Φ(x, ξ), (x, ξ) ∈ R2d , is called a sub-linear weight if (1.1.1) 1 ≤ Φ(x, ξ) 1 + |x| + |ξ|, for x, ξ ∈ Rd . It is called a temperate weight if, for some s > 0, Φ(x + y, ξ + η) Φ(x, ξ)(1 + |y| + |η|)s ,

for x, ξ, y, η ∈ Rd .

(1.1.2)

Observe that (1.1.2) with x−y and ξ−η in place of x and ξ gives, for any temperate weight Φ, the lower bound Φ(x + y, ξ + η) Φ(x, ξ)(1 + |y| + |η|)−s ,

for x, ξ, y, η ∈ Rd ,

(1.1.3)

i.e., Φ−1 is temperate as well. Moreover, the product and real powers of temperate weights are still temperate weights. Also, (1.1.2) and (1.1.3) with x = ξ = 0 yield (1 + |y| + |η|)−s Φ(y, η) (1 + |y| + |η|)s . Basic examples of temperate weights are the functions (1 + |x| + |ξ|)m , and (1 + |x|2 + |ξ|2 )m/2 , m ∈ R (by Peetre’s inequality (0.1.2) in R2d ). Similarly, xm and ξm , m ∈ R, regarded as functions in R2d are temperate weights. Definition 1.1.1. (Symbol classes) Let Φ(x, ξ), Ψ(x, ξ) be sub-linear and temperate weights. Let M (x, ξ) be a temperate weight. We denote by S(M ; Φ, Ψ), or for short S(M ), the space of all smooth functions a(x, ξ), (x, ξ) ∈ R2d , such that for every α, β ∈ Nd , |∂ξα ∂xβ a(x, ξ)| M (x, ξ)Ψ(x, ξ)−|α| Φ(x, ξ)−|β| ,

for (x, ξ) ∈ R2d .

(1.1.4)

20


The following remarks follow at once from Definition 1.1.1. If a ∈ S(M ), then ∂ξα ∂xβ a ∈ S(M Ψ−|α| Φ−|β| ). By Leibniz’ rule, if a ∈ S(M1 ) and b ∈ S(M2 ), then ab ∈ S(M1 M2 ). Moreover, if M1 M2 , then S(M1 ) ⊂ S(M2 ). The following family of seminorms

a k,S(M ;Φ,Ψ) =

sup

sup

|α|+|β|≤k

(x,ξ)∈R2d

|∂ξα ∂xβ a(x, ξ)|M (x, ξ)−1 Ψ(x, ξ)|α| Φ(x, ξ)|β| , (1.1.5)

with k ∈ N, defines a Fréchet topology on S(M ; Φ, Ψ) (notice that these seminorms are actually norms). When M (x, ξ) = ξm , m ∈ R, Φ(x, ξ) = 1, Ψ(x, ξ) = ξρ , 0 ≤ ρ ≤ 1, the m in the literature. class S(M ; Φ, Ψ) is usually denoted by Sρ,0 Now, according to the general theory of Fréchet spaces, a subset B ⊂ S(M ; Φ, Ψ) is bounded if supa∈B a k,S(M ;Φ,Ψ) < ∞ for every k ∈ N. Hence, a bounded subset of S(M ; Φ, Ψ) is also bounded in C ∞ (R2d ). Since every bounded sequence in C ∞ (R2d ) has a convergent subsequence in C ∞ (R2d ) (see, e.g., Treves [190, Theorem 14.4, page 146]), we deduce the following fact. Proposition 1.1.2. For any bounded sequence an ∈ S(M ; Φ, Ψ) the following conditions are equivalent: 1) an converges in S (R2d ); 2) an converges pointwise; 3) an converges in C ∞ (R2d ). Notice that the Schwartz space S(R2d ) is contained in every symbol class S(M ; Φ, Ψ). This is due to the fact that Φ and Ψ are sub-linear, and M satisfies a lower bound of the type (1.1.3). Example 1.1.3. Let p(x, ξ) be a polynomial of degree N . Then |p(x, ξ)| (1 + |x| + |ξ|)N . Since the derivatives ∂ξα ∂xβ p(x, ξ), |α| + |β| ≤ N , are polynomials of degree N − |α| − |β|, we see that p ∈ S(M ; Φ, Ψ), with M (x, ξ) = (1 + |x| + |ξ|)N , and Φ(x, ξ) = Ψ(x, ξ) = 1 + |x| + |ξ|. Actually, since all the weights Φ and Ψ we are considering have sub-linear growth, for any couple of such weights we have p ∈ S(M ; Φ, Ψ) with M (x, ξ) = (1 + |x| + |ξ|)N . The following proposition shows a way to approximate symbols in S(M ; Φ, Ψ) by Schwartz functions. Lemma 1.1.4. Let χ ∈ C0∞ (R2d ) and let Φ and Ψ be any sub-linear weights. Then χ (x, ξ) := χ( x, ξ), 0 ≤ ≤ 1, is bounded in S(1; Φ, Ψ). If moreover χ(x, ξ) = 1 for (x, ξ) in a neighbourhood of the origin, χ tends to 1 in S(M0 ; Φ, Ψ) as → 0, where M0 (x, ξ) is any (temperate) weight which tends to inf inity at infinity.

1.1. Symbol Classes Proof. We have

21

∂ξα ∂xβ χ (x, ξ) = |α|+|β| (∂ξα ∂xβ χ)( x, ξ).

Now there exists a constant C > 0, independent of , such that on the support of χ we have (|x| + |ξ|) ≤ C, and therefore ≤ (C + 1)(1 + |x| + |ξ|)−1 . Since Φ and Ψ are sub-linear weights, it follows that, for every α, β ∈ Nd , |∂ξα ∂xβ χ (x, ξ)| Ψ(x, ξ)−|α| Φ(x, ξ)−|β| ,

(x, ξ) ∈ R2d , 0 ≤ ≤ 1,

(1.1.6)

which is the first part of the statement. If χ(x, ξ) = 1 for (x, ξ) in a neighbourhood of the origin, we see that ≥ C (1 + |x| + |ξ|)−1 on the support of χ − 1, for a constant C > 0 independent of . Hence, for every L > 0 there exists 0 > 0 such that M0 (x, ξ) > L on the support of χ − 1, if < 0 . As a consequence, |∂ξα ∂xβ (χ (x, ξ) − 1) | L−1 M0 (x, ξ)Ψ(x, ξ)−|α| Φ(x, ξ)−|β| ,

(x, ξ) ∈ R2d ,

if < 0 , which is the second part of the statement.

Proposition 1.1.5. Let a ∈ S(M ; Φ, Ψ) and let M0 be any (temperate) weight which tends to inf inity at infinity. Then there exists a sequence an of Schwartz functions, bounded in S(M ; Φ, Ψ) and convergent to a in S(M M0 ; Φ, Ψ). Proof. Let χ ∈ C0∞ (R2d ), χ = 1 in a neighbourhood of the origin. It suffices to set an (x, ξ) = χ(x/n, ξ/n)a(x, ξ)

and apply Lemma 1.1.4.

A key role in the development of the symbolic calculus will be played by the so-called uncertainty principle, i.e., the lower bound Φ(x, ξ)Ψ(x, ξ) 1,

(x, ξ) ∈ R2d ,

(1.1.7)

which follows from the above hypotheses Φ(x, ξ) ≥ 1 and Ψ(x, ξ) ≥ 1. The function h(x, ξ) := Φ(x, ξ)−1 Ψ(x, ξ)−1

(1.1.8)

will be called the Planck function. As a consequence of the uncertainty principle we have the inclusions S(M hN ; Φ, Ψ) ⊂ S(M ; Φ, Ψ),

N ∈ N.

(1.1.9)

Often in the following we will make use of the strong uncertainty principle, that is Φ(x, ξ)Ψ(x, ξ) (1 + |x| + |ξ|)δ ,

(x, ξ) ∈ R2d ,

(1.1.10)

for some δ > 0. The importance of the strong uncertainty principle is clear when dealing with operators of the form Dξα Dxα , |α| = N , which map S(M ; Φ, Ψ) into

22


S(M hN ; Φ, Ψ). If the strong uncertainty principle is satisfied we see that symbols in S(M hN ; Φ, Ψ) decay at infinity, together with their derivatives, as much as we want if N is large enough, and in fact S(M hN ; Φ, Ψ) = S(R2d ). (1.1.11) N ∈N

For any given sequence of symbols an ∈ S(M hn ; Φ, Ψ), n ∈ N, we write an (x, ξ) (1.1.12) a(x, ξ) ∼ n

if, for every N ≥ 1, a−

N −1

aj ∈ S(M hN ; Φ, Ψ).

(1.1.13)

j=0

The right-hand side of (1.1.12) is called asymptotic expansion of a. Because of (1.1.9), from (1.1.12) we have that a ∈ S(M ; Φ, Ψ) and, generally speaking, nothing more. Instead, if the strong uncertainty principle is satisfied, the above asymptotic expansion determines a modulo Schwartz functions. More precisely, the following result holds. Proposition 1.1.6. Assume the strong uncertainty principle (1.1.10). Let an ∈ S(M hn ; Φ, Ψ), n ∈ N. Then there exists a symbol a(x, ξ) ∈ S(M ; Φ, Ψ) verifying (1.1.12). Moreover a is uniquely determined modulo Schwartz functions. Proof. The uniqueness of a modulo Schwartz functions is a consequence of (1.1.11). Let now χ ∈ C0∞ (R2d ), χ = 1 in a neighbourhood of the origin. Let j be a positive sequence converging to 0, and set Aj (x, ξ) = (1 − χ( j x, j ξ))aj (x, ξ). Hence Aj ∈ S(M hj ). By Lemma 1.1.4, 1 − χ( j x, j ξ) tends to zero in S(M0 ) with, say, M0 (x, ξ) = 1 + |x| + |ξ|. Hence, we can choose j converging to zero so rapidly that

Aj k,S(M M0 hj ) ≤ 2−j , for k ≤ j. ∞ Set a(x, ξ) := j=0 Aj (x, ξ). Since this sum is locally finite, a(x, ξ) is a well defined smooth function. Moreover, by the strong uncertainty principle, there is N ∈ N such that h−N M0 . Hence, if k ≤ N , N −1 Aj − a j=0

k;S(M hN −N )

∞ Aj

≤

j=N ∞

k;S(M M0 hN )

Aj k;S(M M0 hN )

j=N ∞ j=N

Aj k;S(M M0 hj ) < ∞.

1.2. Basic Calculus

23

Then, for every fixed k, N , we choose N such that N − N ≥ N , and we obtain N −1 − Aj a

j=0

k,S(M hN )

N −1 a − Aj

j=0

k,S(M hN −N )

< ∞.

Since, for j ≥ N , Aj ∈ S(M hj ) ⊂ S(M hN ), we see that N −1 Aj a − j=0

k,S(M hN )

< ∞.

The same holds with aj in place of Aj , because Aj − aj ∈ S(R2d ). This concludes the proof.

Remark 1.1.7. It is worth noticing that (1.1.13) could give significant information from a micro-local point of view, even if the strong uncertainty principle is not satisfied (e.g. looking at sub-regions, as cones, where the strong uncertainty principal holds). Hence in the next section we will deal with asymptotic expansions without assuming the strong uncertainty principle.

1.2 Basic Calculus 1.2.1 Action on S Let a ∈ S (R2d ), τ ∈ R, and consider the operator Opτ (a) defined formally by ¯ dξ, ¯ u ∈ S(Rd ). (1.2.1) Opτ (a)u(x) = ei(x−y)ξ a((1 − τ )x + τ y, ξ)u(y) dy We will call a the τ -symbol of the pseudo-differential operator Opτ (a). Important special cases are obtained when τ = 0, τ = 1 and τ = 1/2, which correspond to the so-called left quantization a(x, D) in (1.0.1), the right quantization in (1.0.8) and the Weyl quantization aw in (1.0.10), respectively. The rigorous meaning we give to (1.2.1) is the following one: Opτ (a) is the continuous operator S(Rd ) → S (Rd ) with distribution kernel Kτ ∈ S (R2d ) given by −1 a((1 − τ )x + τ y, ξ). (1.2.2) Kτ (x, y) = (2π)−d/2 Fξ→x−y This means that Opτ (a)u, v = Kτ , v ⊗ u,

u, v ∈ S(Rd ),

(1.2.3)

where ·, · stands for the usual duality between temperate distributions and Schwartz functions. Observe that when a ∈ S(R2d ) the integral in (1.2.1) is absolutely convergent and the action just defined reduces to (1.2.1). Other interpretations are given in Proposition 1.2.3 below.

24


Proposition 1.2.1. The correspondence a → Kτ is an isomorphism of S(R2d ), of S (R2d ), and also of L2 (R2d ). The inverse map is given by a(x, ξ) = (2π)d/2 Fy→ξ Kτ (x + τ y, x − (1 − τ )y).

(1.2.4)

Proof. The partial Fourier transform as well as the composition with the change of variable T(x, y) = ((1 − τ )x + τ y, x − y) are isomorphisms of S(R2d ), of S (R2d ), and also of L2 (R2d ). This gives the first part of the statement. The second part is just an easy computation. In particular, any pseudo-differential operator corresponds to only one τ symbol. Operators with Schwartz symbols, namely Schwartz kernels, are called (globally) regularizing operators. They are characterized by the property that they extend to continuous operators S (Rd ) → S(Rd ). The following simple result will be used often in the following. Proposition 1.2.2. For any given sequence an ∈ S (R2d ), convergent to a ∈ S (R2d ), it turns out that Opτ (an )u → Opτ (a)u in S (Rd ) for every u ∈ S(Rd ). Proof. The kernels of Opτ (an ) converge to that of Opτ (a) in S (R2d ), since the map a → Kτ is continuous on S (R2d ). Proposition 1.2.3. Let a ∈ S(M ; Φ, Ψ). Then the integral in (1.2.1) is well defined as an iterated integral. Moreover, Opτ (a)u, u ∈ S(Rd ), defined as a distribution in (1.2.3), coincides with the function expressed by that integral. Proof. Since u ∈ S(Rd ) and all the derivatives of a are dominated by the same function M (x, ξ), having polynomial growth, we see by repeated integrations by parts that the integral ei(x−y)ξ a((1 − τ )x + τ y, ξ)u(y) dy defines a function b(x, ξ) satisfying the estimates |b(x, ξ)| xs ξ−N , for some s ∈ R and every N . This gives the first part of the statement. This estimate also implies that we can apply Fubini’s theorem and interchange the integrals with respect to x and ξ in ei(x−y)ξ a((1 − τ )x + τ y, ξ)u(y)v(x) dy ¯ dξ ¯ dx. The change of variable )x+τ y, x−y) then reduces that expression T(x, y) = ((1−τ to (2π)−d/2 a, F2−1 (v ⊗ u) ◦ T −1 , (F2 being the partial Fourier transform in the second variable), which is exactly the meaning of the right-hand side of (1.2.3).

1.2. Basic Calculus

25

Now we show that every pseudo-differential operator Opτ1 (a) can be written as Opτ2 (b) for another symbol b, with a precise dependence of a. To study this problem we introduce the following family of Fourier multipliers in R2d : a(y, η) dy ¯ dη, ¯ a ∈ S (R2d ), τ ∈ R. (1.2.5) eiτ Dx ·Dξ a(x, ξ) := ei(xy+ξη) eiτ yη In particular, for τ = 0 we have the identity operator. These operators define isomorphisms of S(R2d ) and S (R2d ), because such are the Fourier transform in R2d and the multiplication by eiτ yη . Moreover, as operators on S(R2d ) and S (R2d ) they satisfy the group property eiτ1 Dx ·Dξ eiτ2 Dx ·Dξ = ei(τ1 +τ2 )Dx ·Dξ ,

τ1 , τ2 ∈ R,

(1.2.6)

as one sees at once from their definition. Also, by Parseval’s formula, eiτ Dx ·Dξ is in fact a unitary operator on L2 (R2d ). Now we show other useful expressions for eiτ Dx ·Dξ a, when a ∈ S(R2d ). Since the Fourier transform of (2π|τ |)−d e−iyη/τ , τ = 0, is (2π)−d eiτ xξ (x and ξ being the variables dual to y and η respectively), cf. Hörmander [119, Vol. I, Theorem a b, for a ∈ S(R2d ), b ∈ S (R2d ), cf. 7.6.1], and using the formula a ∗ b = (2π)d (0.2.4), we see that, for τ = 0, iτ Dx ·Dξ −d a = |τ | ¯ dη, ¯ a ∈ S(R2d ). (1.2.7) e e−iyη/τ a(x + y, ξ + η) dy By changes of variables we obtain from this formula the following two expressions, as absolutely convergent integrals (τ = 0): eiτ Dx ·Dξ a = e−iyη a(x + y, ξ + τ η) dy ¯ dη, ¯ a ∈ S(R2d ), (1.2.8) eiτ Dx ·Dξ a =

e−i(y−x)(η−ξ) a((1 − τ )x + τ y, η) dy ¯ dη, ¯

a ∈ S(R2d ).

(1.2.9)

We now study the action of the operator eiτ Dx ·Dξ on the class of symbols S(M ). According to Remark 1.1.7, we do not assume the strong uncertainty principle. Theorem 1.2.4. Let a ∈ S(M ; Φ, Ψ). Then b(x, ξ) := eiτ Dx ·Dξ a ∈ S(M ; Φ, Ψ) and the following asymptotic expansion holds: (α!)−1 τ |α| ∂ξα Dxα a(x, ξ). (1.2.10) b(x, ξ) ∼ α

Moreover the map eiτ Dx ·Dξ is continuous on S(M ; Φ, Ψ). Formula (1.2.10) means that (α!)−1 τ |α| ∂ξα Dxα a(x, ξ) ∈ S(M hN ; Φ, Ψ), b(x, ξ) − |α| 0 for almost every (x, ξ) ∈ R2d , then Aa > 0, that is (Aa u, u)L2 (Rd ) > 0 for u = 0. Proof. We have, from (1.7.8), (Aa u, u) =

a(y, η)|(u, Gy,η )|2 dy ¯ dη. ¯

54


Under the assumption a(y, η) ≥ 0 we conclude (Aa u, u) ≥ 0. If a(y, η) > 0 for almost every (y, η) ∈ R2d , then (Aa u, u) = 0 would imply (u, Gy,η ) = 0 for almost every (y, η) ∈ R2d . Since |x−y|2 − d4 d/2 (u, Gy,η ) = π (2π) Fx→η u(x)e− 2 (η),

this gives u = 0. ∞

2d

Proposition 1.7.7. Let a ∈ L (R ). Then Aa extends to a bounded operator on L2 (Rd ), with

Aa B(L2 (Rd )) ≤ a L∞ (R2d ) . Proof. The result is clear from the definition in (1.7.7), taking into account (1.7.3) and (1.7.5). Alternatively, one can use (1.7.3), (1.7.8) and duality.

1.7.2 Relationship with the Weyl Quantization Anti-Wick operators, as continuous operators S(Rd ) → S (Rd ), have a distribution kernel in S (R2d ), and therefore can be regarded as pseudo-differential operators with distribution symbols. We now study in detail the relationship between the Anti-Wick symbol and the Weyl one. Observe, first of all, that the integral kernel of Py,η in (1.7.10) is the Schwartz function (1.7.12) Ky,η (x, t) = Gy,η (x)Gy,η (t). As a consequence, Py,η can be regarded as a pseudo-differential operator with Schwartz symbol. For example, considering the Weyl quantization, we can write

x+t Py,η u(x) = ei(x−t)ξ σy,η , ξ u(t) dt ¯ dξ, ¯ (1.7.13) 2 where the Weyl symbol σy,η of Py,η can be computed from the above expression for its kernel by means of (1.2.4) with τ = 1/2. Namely,

2 2 1 1 σy,η (x, ξ) = (2π)d/2 Ft→ξ Ky,η x + t, x − t = 2d e−(|x−y| +|ξ−η| ) . (1.7.14) 2 2 We have therefore the following basic result. Proposition 1.7.8. The orthogonal projection operator Py,η on the unitary vector (1.7.1) can be written as a formally self-adjoint pseudo-differential operator, with real Weyl symbol σy,η ∈ S(R2d ) given by (1.7.14). Consider now a ∈ S(R2d ). Inserting (1.7.13) into (1.7.9) we obtain, for u ∈ S(R ),

x+t Aa u(x) = ei(x−t)ξ b , ξ u(t) dt ¯ dξ, ¯ (1.7.15) 2 d

1.7. Anti-Wick Quantization

55

with

d

b(x, ξ) = 2

2

a(y, η)e−(|x−y|

+|ξ−η|2 )

dy ¯ dη ¯

= (2π)−d (a ∗ σ0,0 )(x, ξ).

(1.7.16)

Proposition 1.7.9. Let a ∈ S (R2d ). Then Aa = bw , where b ∈ S (R2d ) is given in (1.7.16). Proof. The result was proved above for Schwartz symbols. The general case follows by a limiting argument, taking into account Propositions 1.7.4 and 1.2.2 and the fact that the map a → a ∗ σ0,0 is continuous on S (R2d ). We now take symbols a in the classes S(M ; Φ, Ψ). We consider, for simplicity, only the case in which Φ ≡ Ψ. This implies Ψ ≡ h−1/2 (h being the Planck function in (1.1.8)). Theorem 1.7.10. Let a ∈ S(M ; Ψ, Ψ). Then Aa is a pseudo-differential operator with symbol in the same class. More precisely, its Weyl symbol b(x, ξ) defined in (1.7.16) belongs to S(M ; Ψ, Ψ) and has asymptotic expansion cαβ b(x, ξ) ∼ ∂ α ∂ β a(x, ξ), (1.7.17) α!β! ξ x α,β

in the sense that, for every N ≥ 1, cαβ ∂ α ∂ β a ∈ S(M hN/2 ; Ψ, Ψ), RN = b − α!β! ξ x

(1.7.18)

|α+β| d in (1.7.28), we see that the kernel K(y , η ; y, η) is dominated by a convolution kernel in L1 (R2d ). Therefore (1.7.26) follows by Young’s inequality. We now come to an application to the almost positivity of pseudo-differential operators. Contrary to the result in Proposition 1.7.6 for the Anti-Wick quantization, pseudo-differential operators with positive Weyl symbol are not positive in general. Consider for example in dimension d = 1 the symbol a(x, ξ) = x2 ξ 2 ≥ 0. The operator A with Weyl symbol a can be written as A= and therefore

1 1 (xDx + Dx x)2 − , 4 4

1 1

(xDx + Dx x)u 2L2 − u 2L2 , 4 4 where the right-hand side is negative for suitable u ∈ S(R). We have however the following lower bound, “with gain of one derivative”, for pseudo-differential operators with positive Weyl symbol. It represents one of the basic applications of the Anti-Wick theory. (Au, u)L2 =

60


Theorem 1.7.15. (Sharp Gårding inequality) Let a ∈ S(h−1 ; Ψ, Ψ), a(x, ξ) ≥ 0, (x, ξ) ∈ R2d . There exists a constant C > 0 such that (aw u, u)L2 (Rd ) ≥ −C u L2 (Rd ) ,

u ∈ S(Rd ).

(1.7.29)

Proof. By Proposition 1.7.11 we have aw = Aa + sw , where s ∈ S(1; Ψ, Ψ). Since Aa satisfies (1.7.11) and sw is bounded on L2 (Rd ) by Theorem 1.7.14, the desired result follows. Much more refined techniques allow one to prove (1.7.29) for symbols in S(h−2 ; Φ, Ψ) (under additional assumptions on Φ and Ψ), which is the so-called Fefferman-Phong inequality.

1.7.4 Sobolev Spaces Revisited As another application of Anti-Wick techniques we now show, as promised in Section 1.5, that the Sobolev spaces H(M ) introduced there are in fact independent of the choice of the weights Φ and Ψ. Let M be a temperate weight, cf. (1.1.2). Let L2M (R2d ) be the weighted Lebesgue space of measurable functions F in R2d such that M F ∈ L2 (R2d ), with the norm F L2M (R2d ) = M F L2 (R2d ) . ˜ We define the space H(M ) as the completion of the Schwartz class S(Rd ) with respect to the norm

u H(M ˜ ) = V u L2M (R2d ) =

2

2

M (y, η) |V u(y, η)| dy dη

1/2 ,

u ∈ S(Rd ),

where V u is the short-time Fourier transform of u in (1.7.2). Notice that the above norm is well defined for u ∈ S(Rd ), because V u ∈ 2d S(R ) and M has polynomial growth. In fact, we have a continuous inclusion ˜ ˜ S(Rd ) → H(M ). Moreover, for M (x, ξ) ≡ 1 we get H(1) = L2 (Rd ) by (1.7.3). ˜ Let us show that the space H(M ) in Definition 1.5.2 coincides with H(M ), and is therefore independent of Φ and Ψ. To this end, we need the following continuity result. Theorem 1.7.16. Let M, M be temperate weights and a ∈ S(M ; 1, 1) (Φ(x, ξ) = ˜ ˜ Ψ(x, ξ) = 1). Then a(x, D) extends to a bounded operator H(M ) → H(M/M ). Proof. We argue as in the proof of Theorem 1.7.14. Namely, using (1.7.6) we write a(x, D) = (2π)−4d V ∗ V a(x, D)V ∗ V on S(Rd ). Observe that the map (2π)−d/2 V ˜ clearly extends to an isometry H(M ) → V (S(Rd )) ⊂ L2M (R2d ) and similarly ˜ (2π)−d/2 V ∗ extends to an isometry V (S(Rd )) ⊂ L2M/M (R2d ) → H(M/M ). Hence ∗ we need only prove that V a(x, D)V extends to a bounded map L2M (R2d ) → L2M/M (R2d ). We know that the integral kernel K(y , η ; y, η) of this map has the expression in (1.7.27).

1.7. Anti-Wick Quantization

61

Repeated integrations by parts in (1.7.27) (as in the proof of Theorem 1.7.14), combined with the symbol estimates for a(x, ξ) and the temperance M (x + y , ξ + η) M (y , η)(1 + |x| + |ξ|)s , show that for every N , |K(y , η ; y, η)| M (y , η)y − y −N η − η −N . As a consequence, for F ∈ S(R2d ), we get M (y , η ) | (V a(x, D)V ∗ F ) (y , η )| M (y , η ) M (y , η ) M (y , η) y − y −N η − η −N M (y, η)|F (y, η)| dy dη. M (y , η ) M (y, η) Again by temperance we have M (y , η ) (1 + |y − y | + |η − η |)s , M (y, η)

M (y , η) (1 + |η − η |)s , M (y , η )

so that we get, for every N , M (y , η ) | (V a(x, D)V ∗ F ) (y , η )| M (y , η ) y − y −N η − η −N M (y, η)|F (y, η)| dy dη. An application of Young’s inequality then gives the estimate

V a(x, D)V ∗ F L2

M/M

(R2d )

F L2M (R2d ) ,

F ∈ S(R2d ),

which concludes the proof.

Corollary 1.7.17. Assume further that the temperate weight M is regular, in the sense of Definition 1.5.1. The Sobolev space H(M ) in Definition 1.5.2 coincides, ˜ for every choice of the weights Φ, Ψ, with the above space H(M ), with equivalent norms. Proof. Let a ∈ S(M ; Φ, Ψ) be an elliptic symbol and set A = a(x, D). Let B = b(x, D), b ∈ S(M −1 ; Φ, Ψ) be a (left) parametrix of A, so that BA = I + R, ˜ ˜ with R regularizing. By Theorem 1.7.16, A and R act H(M ) → H(1) = L2 (Rd ) continuously. Hence we have the estimate

Au L2 (Rd ) + Ru L2 (Rd ) u H(M ˜ ), ˜ which is H(M ) → H(M ). ˜ ˜ Similarly, by Theorem 1.7.16, B acts L2 (Rd ) = H(1) → H(M ) continuously, so that using u = BAu + Ru we get

u H(M ˜ ˜ ) Au L2 (Rd ) + Ru H(M ).

62


˜ Clearly Au L2 (Rd ) u H(M ) . On the other hand, since S(Rd ) → H(M ) and d

u

, and the opposite R is continuous H(M ) → S(R ), we get Ru H(M ˜ H(M ) ) ˜ inclusion H(M ) → H(M ) is therefore proved.

1.8 Quantizations of Polynomial Symbols We conclude this chapter with a few remarks about relationships between several types of quantizations, especially for polynomial symbols. First of all one could wonder about the reason for such a large number of quantization rules. Indeed, the main motivations come from Quantum Mechanics. In Classical Mechanics the state of a system is described by a point in Rd (or in a d-dimensional manifold); for example d = 6N in the case of N particles in the space R3 , the coordinates being the position and momentum of each particle. In Quantum Mechanics, the state of a system is instead described by a function φ in L2 (Rd ) (up to a non-zero multiplicative constant). The (real) observables a(x, ξ) are replaced by densely defined self-adjoint operators A, and (φ, Aφ)L2 represents the mean value of the observable a. The main examples are given by the operators associated with the position observable a(x, ξ) = xj which corresponds to the operator Mj φ = xj φ, and the momentum observable a(x, ξ) = ξj , which corresponds to the derivative Dj = −i∂j . Every reasonable quantization rule should verify these two axioms. The quantization of general observables is formally performed by replacing, in the expression of a(x, ξ), the variables xj and ξj by the above mentioned operators. This unfortunately leads to a non-trivial commutativity problem, as we have of course xj ξj = ξj xj , but Mj Dj = Dj Mj . One preferred quantization rule is certainly the Weyl one since, as we know from Proposition 1.2.10, real-valued symbols a give rise to (formally) self-adjoint operators aw . Another interesting quantization is the so-called Feynmann quantization, which is 1 a → a(x, D) + a(x, D)∗ , 2 where a(x, D) is the standard quantization. Observe that, if cα (x)D α a(x, D) = |α|≤m

is a differential operator with smooth coefficients, then D α cα (x)f (x) . a(x, D)∗ f (x) = |α|≤m

Notice also that, if a(x, ξ) = xj ξj , aw =

1 1 (xj Dj + Dj xj ) = (a(x, D) + a(x, D)∗ ) . 2 2

(1.8.1)

1.8. Quantizations of Polynomial Symbols

63

More generally (1.8.1) still holds for every polynomial α,β cα,β xβ ξ α with real coefficients, if each monomial has degree at most 1 either with respect to x or ξ. This follows by R-linearity from the following result, which shows, among other things, that generally we have aw = 12 (a(x, D) + a(x, D)∗ ) even if a is real-valued. Proposition 1.8.1. Let a be a real-valued polynomial. Then aw = with r(x, ξ) =

1 (a(x, D) + a(x, D)∗ ) + r(x, D), 2

(α!)−1 2−|α| − 2−1 ∂ξα Dxα a(x, ξ),

|α|≥2

where the above sum is finite because a is a polynomial. Proof. The desired result follows by an application of the previous Remark 1.2.6 and Proposition 1.2.9 (or, better, from the proofs of those results). In fact, the implied asymptotic expansions become here finite and the corresponding asymptotic equivalences become equalities, because a(x, ξ) is a polynomial. For example, in dimension d = 1, for a(x, ξ) = x2 ξ 2 , we have 1 (a(x, D) + a(x, D)∗ + I) . 2 Another remark is that, if a(x, ξ) = |β|≤m cβ xβ is a polynomial with complex coefficients, then aw = a(x, D). That is, the Weyl quantization yields multiplication operators for polynomials independent of ξ. Notice that the same trivially happens on the frequency side, namely aw = a(x, D) if a(x, ξ) = |α|≤m cα ξ α , cα ∈ C. aw =

Let us now come to the relationship between Weyl and Anti-Wick quantization. We saw in Section 1.7.2 that any operator Aa with Anti-Wick symbol a(x, ξ) can be written as a Weyl operator with symbol b(x, ξ) given by (1.7.16), i.e., b(z) = (2π)−d (a ∗ σ0,0 )(z), z ∈ R2d , 2

where σ0,0 (z) = 2d e−|z| . Vice-versa, as we already observed, it is not always possible to write an operator with a given Weyl symbol in the Anti-Wick form. In −|ζ|/4 a σ , we have fact, using a ∗ σ0,0 = (2π)d 0,0 , cf. (0.2.4), and σ 0,0 (ζ) = e b = e−|D|

2

/4

a = eΔ/4 a.

Hence one would be tempted to solve this equation as a = e−Δ/4 b, but this does 2 not make sense for general b ∈ S (R2d ), because e|ζ| /4 is not a multiplier for S (R2d ). However this formula holds when b is a polynomial, because its Fourier transform is then supported at the origin. Namely, we have the following result.

64


Proposition 1.8.2. Let b(z) be a polynomial, z ∈ R2d . Then the equation b = eΔ/4 a is satisfied by a=

∞ (−Δ/4)k

k!

k=0

b,

where the sum is finite because b is a polynomial. k ∞ b = b. But the Fourier transform Proof. We have to verify that eΔ/4 k=0 (−Δ/4) k! of the left-hand side is e−|ζ|

2

/4

∞ (|ζ|2 /4)k k=0

k!

b(ζ) = b(ζ).

Notes Pseudo-differential operators were introduced explicitly by Kohn and Nirenberg [127] and Hörmander [115], although early related issues appeared in works by m , Calderón, Seeley, Zygmund and others. The first symbol classes considered, S1,0 were suitable for the construction of a parametrix for elliptic differential operators, but in order to treat similarly non-elliptic differential operators, the more m general classes Sρ,δ , 0 ≤ δ ≤ ρ ≤ 1, defined by the estimates |∂ξα ∂xβ a(x, ξ)| (1 + |ξ|)m−ρ|α|+δ|β| , were soon revealed to be essential. The study of the local solvability problem for operators of principal type required Beals and Fefferman [11] to introduce much more general symbol classes; see also the subsequent works by Beals [9], [10], and Fefferman and Phong [78] for applications to sharp lower bounds for pseudo-differential operators. Meanwhile, in connection with problems in Quantum Mechanics and PDEs in unbounded domains, Berezin and Shubin [14], [15], Cordes [58], Grushin [103], Kumano-go [128], Parenti [156], and Shubin [182] studied pseudo-differential global calculi in Rd , where x and ξ enter mostly in a symmetric way in the corresponding symbol estimates. Eventually, Hörmander [117] gave a systematic treatment of the whole theory, introducing general symbol classes which included all those considered until then. Applications to spectral theory of globally regular operators appeared in Hörmander [118] and to local solvability of pseudo-differential operators in Dencker [72]. The symbol classes considered in this chapter are inspired by [11], [117] and Hörmander [119, Vol III, Section 18.4]. More precisely, the class S(M ; Φ, Ψ) corresponds to Hörmander’s class S(M, g), with the metric gx,ξ = |dx|2 /Φ(x, ξ)2 + |dξ|2 /Ψ(x, ξ)2 . Our calculus is in fact simpler, since we assume the conditions Φ ≥ 1 and Ψ ≥ 1. On the other hand the above metric g, under our assumptions, does not need to be slowly varying or temperate in Hörmander’s sense. The conditions of sub-linear growth Φ(x, ξ), Ψ(x, ξ) 1 + |x| + |ξ| basically play a similar

Notes

65

role to (but do not imply) the slow-variation condition for g (cf. Lemma 1.1.4 and Remark 1.1.7). The conditions in (1.3.8) are instead equivalent to saying that the metric g is slowly varying and the weight M is g-continuous, in Hörmander’s terminology. For the invariance of the Weyl calculus with respect to symplectic changes of variables and the connection with the theory of metaplectic operators, which is not discussed here, we refer the reader to Folland [82] and [119, Vol. III, Theorem 18.5.9]. The proof of the L2 -boundedness in Theorem 1.4.1 is a classical and nice application of the symbolic calculus, but it just works under the assumption of the strong uncertainty principle. Hence, that result does not contain the cele0 brated Calderón-Vaillancourt Theorem [34] for symbols in S0,0 , which is instead recaptured in Section 1.7.3. The theory of the general Sobolev spaces tailored to a given symbol class in Section 1.5 is inspired by Shubin [183], Bony and Chemin [23], but is again much simpler here, because we assume the strong uncertainty principle and we can consequently rely on the existence of a parametrix. Similarly, Fredholm properties in the subsequent section are immediate once one has a full symbolic calculus and the continuity and compactness, on every Sobolev space, of operators “of order 0” and “of order lower than 0”, respectively. The presentation of the abstract theory of Fredholm operators in Hilbert spaces in Section 1.6.1 is inspired by Grubb [102] and [119, Vol. III] (where the case of operators on Banach spaces is considered), but the proof of the stability result for the index (Theorem 1.6.3) is extracted from Atiyah and Singer [6]. The short-time Fourier transform, or windowed transform, is a basic tool in Time-Frequency Analysis (see Gröchenig [97]), and plays a fundamental role in signal analysis. Indeed, contrary to the Fourier transform, its magnitude |V u(y, η)| contains the information of what frequencies of a signal are mostly present for a given time interval. Anti-Wick operators were introduced as a quantization rule in Physics by Berezin [13]; see also Friedrichs [84]. They also appeared under the name of localization operators in Daubechies [67], which proposed them as a mathematical tool to localize a signal on the time-frequency plane. Since then they have been extensively studied in signal analysis and found many other applications (see Cordero and Gröchenig [52], [53], Cordero and Rodino [57], Ramanathan and Topiwala, [167], Wong [198] and the references therein). We also recall their employment as approximation of pseudo-differential operators (wave packets); see Córdoba and Fefferman [60], Folland [82], Lerner [133], Tataru [186]. Anti-Wick operators are also called short-time Fourier transform multipliers, see Feichtinger and Nowak [81] for a survey. The proof of the L2 -boundedness in Theorem 1.7.14, for operators with sym0 bols in S0,0 , is a continuous version of the almost diagonalization via Gabor frames in Rochberg and Tachizawa [171]. It is one of the most elementary proofs we know, since it avoids the use of the Cotlar-Stein lemma (and Schur’s test). Moreover it

66


generalizes to less regular symbols (see Gröchenig [98]) and even to Fourier integral operators (see Cordero, Nicola and Rodino [55] and the references therein). Different proofs can be found in the already quoted paper [34], Coifman, and Meyer [51], [97, Theorem 14.5.2], [119, Vol. III, Theorem 18.6.3], Hwang [121], Stein [185], Buzano and Toft [31]. Finally, the use of Anti-Wick techniques in the proof of the Sharp Gårding inequality (Theorem 1.7.15) appeared in [60] and [119, Vol. III, Theorem 18.1.4]. More general symbol classes were considered by similar techniques in [133]. The Fefferman-Phong inequality mentioned after Theorem 1.7.15 was proved in [78] 2 for symbols in S1,0 and in [119, Vol. III, Theorem 18.6.8] for the classes S(h−2 , g); see also Fefferman and Phong [77], [79]. For recent work about lower bounds for pseudo-differential operators we refer the reader to the references in Nicola [153]. ˜ The spaces H(M ) in Section 1.7.4 are a type of modulation spaces, widely used in Time-Frequency Analysis. They were first introduced by H. Feichtinger in [80]; see [97, Chapter 11] and the references therein for a detailed study. The results in Section 1.8 can also be proved without using the symbolic calculus, by direct computations, see Boggiatto and Rodino [21].

Chapter 2

Γ-Pseudo-Differential Operators and H-Polynomials Summary Let P be a linear operator, P : S(Rd ) → S(Rd ), with extension to a map from S (Rd ) to S (Rd ). According to Definition 1.3.8 we say that P is globally regular if, for any f ∈ S(Rd ), all the solutions u ∈ S (Rd ) of the equation P u = f belong to S(Rd ). In particular then, all the solutions u ∈ S (Rd ) of the equation P u = 0 belong to S(Rd ). An important tool for deducing global regularity, when P is a pseudo-differential operator, is given by Theorem 1.3.6, namely: the existence of a left parametrix P˜ of P , i.e., P˜ P = I + R where R : S (Rd ) → S(Rd ), implies global regularity for P , as well as precise estimates in generalized Sobolev spaces, cf. Proposition 1.5.8. Besides, the simultaneous existence of a right parametrix gives Fredholmness, cf. Theorem 1.6.9. The main subject of this chapter, and of Chapter 3, is the study of the above properties for relevant classes of operators, including as basic examples partial differential operators with polynomial coefficients in Rd : P =

cαβ xβ Dα .

(2.0.1)

|α|+|β|≤m

The key idea is to model on the structure of P , and define a suitable couple of weights Φ, Ψ satisfying the strong uncertainty principle (1.1.10), in order to obtain a pseudo-differential framework where parametrices of P can be constructed. In this chapter attention is limited to Γ-pseudo-differential operators, corresponding to a case when Φ and Ψ coincide. A simple example is Φ(x, ξ) = Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )1/2 .

68

Chapter 2. Γ-Pseudo-Differential Operators and H-Polynomials

We review the corresponding pseudo-differential calculus in Section 2.1. In short, by writing z = (x, ξ) ∈ R2d , the symbols in Γm (Rd ), m ∈ R, are defined by the estimates, for γ ∈ N2d : |∂zγ a(z)| zm−|γ| ,

z ∈ R2d .

(2.0.2)

According to Definition 1.3.1, we call Γ-elliptic the symbols satisfying |z|m |a(z)| m d for large z. A relevant subclass of Γ (R ) is given by all the symbols admitting an asymptotic expansion a ∼ ∞ a , k=0 m−k with am−k (z) positively homogeneous of degree m−k in R2d ; for these symbols Γ-ellipticity amounts to assuming am (z) = 0 for every z = 0. For m ∈ N the corresponding Sobolev spaces HΓm (Rd ) can be explicitly defined by setting

xβ Dα u L2 < ∞. (2.0.3)

u HΓm = |α|+|β|≤m

Section 2.2 and 2.3 are devoted to the study in this framework of the partial differential operators P in (2.0.1). With z = (x, ξ) and γ = (β, α) the symbol of P , in the standard quantization, can be re-written in the form cγ z γ . (2.0.4) p(z) = |γ|≤m

The Γ-ellipticity reads pm (z) = |γ|=m cγ z γ = 0 for z = 0; this provides global regularity and Fredholm property of the map P : HΓm (Rd ) → L2 (Rd ). A motivating example is the harmonic oscillator of Quantum Mechanics: H = −Δ + |x|2

(2.0.5)

with elliptic symbol |z|2 . As appetizer to Chapter 4, devoted to Spectral Theory, we begin here to compute eigenvalues and eigenfunctions of H. We also study Γ-elliptic ordinary differential equations, by using the theory of Asymptotic Integration; we prove here some decay properties of the solutions, which we shall extend later in the book to linear and semi-linear partial differential equations, see Chapter 6. In Section 2.4 we consider Γ-hypoelliptic symbols. Namely, we limit attention to polynomial symbols p(z), as in (2.0.4), and assume |∂ γ p(z)| |p(z)|z−ρ|γ|

(2.0.6)

for some ρ, 0 < ρ ≤ 1 and large z (the case ρ = 1 corresponds to Γ-ellipticity, defined as before). Polynomials p(z) satisfying (2.0.6), called in the sequel Hpolynomials, were first studied by Hörmander in connection with the Schwartz’ problem of the interior regularity of the solutions of equations with constant coefficients. We recall in Section 2.4 some equivalent definitions and basic properties. Following Hörmander’s presentation, we shall argue in arbitrary dimension

Summary

69

n, though in our applications n = 2d is even. If (2.0.6) is satisfied, we can construct parametrices for the operator P in (2.0.1), hence in particular P is globally regular. However, with respect to the Γ-elliptic case, P is not Fredholm on the spaces HΓm (Rd ), cf. (2.0.3), and there is a loss of regularity in the related a priori estimates. It is then convenient to look for more appropriate weight functions, depending on the particular form of p(z), so that p(z) can be regarded as elliptic symbol in the new frame. This is performed in Sections 2.5, 2.6, 2.7 and 2.8 for some special subclasses of the H-polynomials. Namely in Section 2.5 we consider the quasi-elliptic polynomials; representative examples for the corresponding class of operators are given by Dx + rxk ,

x ∈ R, k ≥ 1, Im r = 0,

(2.0.7)

and by the generalized harmonic oscillator −Δ + |x|2k ,

x ∈ Rd , k ≥ 1.

(2.0.8)

In Section 2.6 we study multi-quasi-elliptic polynomials, which contain quasielliptic polynomials as a particular case. Simple examples for the related class of operators are Dx6 + x4 Dx4 + x6 , x ∈ R (2.0.9) and −Δ + V (x1 , x2 ),

(x1 , x2 ) ∈ R2 ,

(2.0.10)

where the potential V is given by V (x1 , x2 ) = x61 + x41 x42 + x62 . In general, an H-polynomial is multi-quasi-elliptic if 1/2 2γ |p(z)| ΛP (z) = z

(2.0.11)

(2.0.12)

γ∈V (P)

for large z, the sum being extended to the set V (P) of all the vertices γ of the Newton polyhedron P of p(z). The corresponding pseudo-differential calculus is developed in Section 2.7. In fact Φ(z) = Ψ(z) = ΛP (z)ρ give a couple of weights for 0 < ρ ≤ 1/μ, where μ ≥ 1 is the so-called formal order of P. The condition (2.0.12) can be read as ellipticity with respect to such a couple of weights; the m Γ-ellipticity is recaptured when ΛP (z) |z| . The results of Chapter 1 apply m d therefore to the class ΓP (R ), m ∈ R, of all the symbols a(z) satisfying |∂zγ a(z)| ΛP (z)m−|γ|/μ ,

z ∈ R2d .

(2.0.13)

Moreover, the weights ΛP (z) in (2.0.12) enjoy peculiar properties, which suggest m d to consider also the subclass M Γm P of all a(z) ∈ ΓP (R ) satisfying d z γ ∂zγ a(z) ∈ Γm P (R )

(2.0.14)

70


for any γ = (γ1 , . . . , γ2d ) ∈ N2d with γj ∈ {0, 1}. In particular, the symbol of the parametrix of a multi-quasi-elliptic operator belongs to this class. This approach allows us to establish precise Lp estimates for multi-quasielliptic operators. Namely, in Section 2.8 we prove that every pseudo-differential operator with symbol in M Γ0P (Rd ) is bounded on Lp , 1 < p < ∞. We may then consider, as generalization of (2.0.3), the Banach space HPp with norm

xβ D α u Lp , (2.0.15)

u p,P = (β,α)∈V (P)

cf. (2.0.12) with z = (x, ξ), γ = (β, α). The operator P corresponding to the polynomial symbol p(z) in (2.0.12) is then a Fredholm map HPp → Lp for 1 < p < ∞.

2.1 Γ-Pseudo-Differential Operators The first basic examples of the general classes of symbols considered in the previous Chapter 1 are the Γ-classes, defined below. Write z = (x, ξ) ∈ R2d and, according to the preceding notation, 1/2 . z = (1 + |z|2 )1/2 = 1 + |x|2 + |ξ|2 Definition 2.1.1. We define Γm (Rd ), m ∈ R, as the set of all functions a(z) ∈ C ∞ (R2d ) satisfying, for all γ ∈ N2d , |∂zγ a(z)| zm−|γ| ,

z ∈ R2d .

(2.1.1)

As we observed before Definition 1.1.1, the weights Φ(z) = Ψ(z) = z are sub-linear and temperate, cf. (1.1.1), (1.1.2). More generally we can define the following classes. d Definition 2.1.2. We define Γm ρ (R ), with m ∈ R, 0 < ρ ≤ 1, as the set of all the functions a(z) ∈ C ∞ (R2d ) satisfying, for all γ ∈ N2d ,

|∂zγ a(z)| zm−ρ|γ| ,

z ∈ R2d .

(2.1.2)

In the above definition we assumed ρ > 0, so the strong uncertainty principle d m d (1.1.10) is satisfied. Note that Γm 1 (R ) = Γ (R ). Also, it is useful to introduce m d the subspace of Γ (R ) of the polyhomogeneous symbols. Let us write Hm (R2d \ {0}), m ∈ R, for the class of functions a(z) ∈ C ∞ (R2d \ {0}) which are positively homogeneous of degree m in R2d , i.e., a(tz) = tm a(z) for t > 0, z ∈ R2d , z = 0. d m d Definition 2.1.3. We define Γm cl (R ) as the subset of Γ (R ) of all the symbols a(z) which admit asymptotic expansion

a(z) ∼

∞ k=0

am−k (z)

(2.1.3)

2.1. Γ-Pseudo-Differential Operators

71

for a sequence of functions am−k ∈ Hm−k (R2d \ {0}), k = 0, 1, . . . . By (2.1.3) we mean that, considering χ(z) ∈ C ∞ (R2d ), with χ(z) = 1 for |z| ≥ 2, χ(z) = 0 for |z| ≤ 1, we have for every integer N ≥ 1: a(z) − χ(z)am−k (z) ∈ Γm−N (Rd ). (2.1.4) 0≤k 0 such that |z|m |a(z)| for |z| ≥ R. (2.1.9) d m d When applying this definition in Γm cl (R ) ⊂ Γ (R ), we shall rather rely on the following equivalent notion of Γ-ellipticity. d Proposition 2.1.5. The symbol a ∈ Γm cl (R ) is Γ-elliptic if and only if the principal part am satisfies (2.1.10) am (z) = 0 for every z = 0.

Proof. Because of the homogeneity of am , the assumption (2.1.10) is equivalent to the estimate m (2.1.11) |z| |am (z)| for z = 0. On the other hand, from (2.1.4) with N = 1 we have, for any fixed R > 0, |a(z) − am (z)| |z|m−1

for |z| ≥ R.

Then, if (2.1.9) is satisfied, we deduce for suitable positive constants C and : |z|m ≤ |a(z)| ≤ |am (z)| + C |z|m−1

for |z| ≥ R.

Since for sufficiently large |z| we have C |z|

m−1

≤

m |z| , 2

m

we conclude 2 |z| ≤ |am (z)|, hence (2.1.11). Symmetrically we may prove that (2.1.11) implies (2.1.9). From Theorem 1.3.6 we have the following result on the existence of parametrices. Let us observe that Γ-ellipticity in (2.1.9), (2.1.10) is invariant under different quantizations. Theorem 2.1.6. Let a ∈ Γm (Rd ) be Γ-elliptic. Then there exists b ∈ Γ−m (Rd ) such that b(x, D) is a parametrix of a(x, D), i.e., a(x, D)b(x, D) = I + S1 ,

b(x, D)a(x, D) = I + S2

where S1 and S2 are regularizing operators. Hence a(x,D) is globally regular, i.e., u ∈ S (Rd ) and a(x, D)u ∈ S(Rd ) imply u ∈ S(Rd ). d The result remains valid for symbols a ∈ Γm cl (R ) satisfying (2.1.10), the −m d symbol of the parametrix belonging then to Γcl (R ), with principal part 1/am . We may now apply the results of Section 1.7, concerning the Anti-Wick quantization, to symbols in the Γ-classes. In particular we shall use the following version of Proposition 1.7.12.


73

Theorem 2.1.7. Let a ∈ Γs (Rd ) be Γ-elliptic, s ∈ R, with a(z) > 0 for every z ∈ R2d . Denote by Aa the operator with Anti-Wick symbol a. The map Aa : S(Rd ) → S(Rd ) is an isomorphism, extending to an isomorphism on S (Rd ). Moreover, the −m inverse A−1 (Rd ). a belongs to OPΓ Take for example, for s ∈ R, 2

2

as (z) = zs = (1 + |x| + |ξ| )s/2 ,

(2.1.12)

satisfying the assumptions of Theorem 2.1.7. We may refer to such symbols in the following definition of Γ-Sobolev spaces. Definition 2.1.8. Let as (z), s ∈ R, be the symbol in (2.1.12) and write Ws for the operator with Anti-Wick symbol as . Then we set (2.1.13) HΓs (Rd ) = Ws−1 L2 (Rd ) = {u ∈ S (Rd ) : Ws u ∈ L2 (Rd )}. All the properties in Sections 1.5 and 1.6 apply. In particular HΓs are Hilbert spaces with respect to the scalar product (u, v)HΓs = (Ws u, Ws v)L2

(2.1.14)

and the corresponding norm

u HΓs = Ws u L2 .

(2.1.15)

In Definition 2.1.8 we may of course replace as (z) with any other strictly positive Γ-elliptic symbol, with equivalence of norms. More generally we can define the HΓs (Rd ) spaces using arbitrary Γ-elliptic symbols, as stated in the following proposition, cf. Definition 1.5.2. Proposition 2.1.9. Let T ∈ OPΓs (Rd ) have Γ-elliptic symbol, cf. Definition 2.1.4. Then HΓs (Rd ) = {u ∈ S (Rd ) : T u ∈ L2 (Rd )} and on HΓs (Rd ) an equivalent Hilbert space structure is given by the scalar product (u, v)HΓs = (T u, T v)L2 + (Ru, Rv)L2

(2.1.16)

where R is a regularizing operator associated to a parametrix T˜ ∈ OPΓ−s (Rd ) of T , namely T˜T = I + R. From Proposition 1.5.5, we have: Theorem 2.1.10. Every A ∈ OPΓm (Rd ) defines, for all s ∈ R, a continuous operator A : HΓs (Rd ) → HΓs−m (Rd ). (2.1.17) Later on in the book we shall also use the following result of boundedness in the standard Sobolev spaces H s (Rd ).


74

Theorem 2.1.11. Every A ∈ OPΓ0 (Rd ) defines, for all s ∈ R, a continuous operator (2.1.18) A : H s (Rd ) → H s (Rd ). Proof. Consider the pseudo-differential operator Dt with standard symbol ξt , t ∈ R. We know that u ∈ H s (Rd ) if and only if Ds u ∈ L2 (Rd ), cf. Section 0.2. Hence, by writing Ds Au = Ds AD−s Ds u, we have to prove the boundedness of the map Ds AD−s : L2 (Rd ) → L2 (Rd ).

(2.1.19)

The operator in (2.1.19) does not belong to OPΓ0 (Rd ), however we may regard Ds , A and D−s as operators with symbols in the classes S(ξs ; 1, 1), S(1; 1, 1) and S(ξ−s ; 1, 1) respectively. Hence Ds AD−s ∈ OPS(1; 1, 1) by Theorem 1.2.16 and (2.1.19) is granted, cf. Theorem 1.7.14. Other properties of the HΓs (Rd ) spaces, following from Section 1.5, are compact immersions j : HΓs (Rd ) → HΓt (Rd ) for s > t, and compactness of the maps A : HΓs (Rd ) → HΓt (Rd ) for A ∈ OPΓm (Rd ) whenever ! s − t > m. In particular if A is regularizing, then A belongs to OPΓ−∞ (Rd ) := m OPΓm (Rd ), hence it is continuous and compact from HΓs (Rd ) to HΓt (Rd ) for any s, t ∈ R. Moreover, for every s ∈ R we have continuous immersions j : S(Rd ) → HΓs (Rd ),

j : HΓs (Rd ) → S (Rd )

and moreover HΓs (Rd ) = S(Rd ),

s∈R

HΓs (Rd ) = S (Rd ).

s∈R

Finally, the spaces HΓs (Rd ), HΓ−s (Rd ) are dual to each other with respect to the bilinear pairing u, v = u(x)v(x) dx. The following equivalent definition when m ∈ N is peculiar for the spaces HΓs (Rd ). Theorem 2.1.12. Let m ∈ N. An equivalent definition of the space HΓm (Rd ) is given by HΓm (Rd ) = {u ∈ S (Rd ) : xβ Dα u ∈ L2 (Rd ) for |α| + |β| ≤ m}

(2.1.20)

with equivalent norm

u ∗HΓm =

|α|+|β|≤m

xβ Dα u L2 .

(2.1.21)


75

Proof. Assume first u ∈ HΓm (Rd ), according to the Definition 2.1.8. Since xβ Dα ∈ OPΓm (Rd ) if |α| + |β| ≤ m, then xβ D α u ∈ HΓ0 (Rd ) = L2 (Rd ) in view of Theorem 2.1.10, and u ∗H m can be estimated in terms of u HΓm . Γ In the opposite direction, it will be convenient to estimate the norm coming from (2.1.16) in terms of u ∗H m . Namely, we shall fix an operator T ∈ OPΓm (Rd ) Γ with Γ-elliptic symbol, see below, and consider

u 2HΓm = T u 2L2 + Ru 2L2 ,

(2.1.22)

where R is a regularizing operator associated to a left parametrix T˜ ∈ OPΓ−m (Rd ) of T . We may then estimate, for some C > 0,

u HΓm ≤ T u L2 + C u L2 .

(2.1.23)

T = A−m P

(2.1.24)

Choose T as follows: d where A−m is any operator in OPΓ−m cl (R ) with elliptic symbol and xβ D α xβ D α . P =

(2.1.25)

|α|+|β|≤m d In fact, using Leibniz’ rule we have that the principal symbol of P ∈ OPΓ2m cl (R ) is given by p2m (x, ξ) = x2β ξ 2α , |α|+|β|=2m

which satisfies the Γ-ellipticity condition in Proposition 2.1.5. Hence T in (2.1.24) d belongs to OPΓm cl (R ) with elliptic symbol, cf. (2.1.10). Combining (2.1.23) with (2.1.24), (2.1.25), and observing that A−m xβ Dα ∈ OPΓ0 (Rd ), we obtain

A−m xβ D α (xβ Dα u) L2 + C u L2

u HΓm ≤ |α|+|β|≤m

≤

C u ∗HΓm

for a new constant C . The theorem is therefore proved.

HΓs (Rd ),

we may now give a more precise Referring to the Sobolev-type spaces frame to Theorem 2.1.6. Namely, particularizing the results of Sections 1.5, 1.6, we have the following two theorems. Theorem 2.1.13. Consider A ∈ OPΓm (Rd ) with Γ-elliptic symbol and assume u ∈ S (Rd ), Au ∈ HΓs (Rd ). Then u ∈ HΓs+m (Rd ) and, for every t < s + m,

u H s+m ≤ C Au HΓs + u HΓt (2.1.26) Γ

for a positive constant C depending on s and t.

76


In particular if m is a positive integer, we may refer to the equivalent norm (2.1.21) and re-write (2.1.26) for s = 0, t = 0:

xβ Dα u L2 ≤ C ( Au L2 + u L2 ) . (2.1.27) |α|+|β|≤m

To be definite, in the following statement we shall denote by As the restriction of A : S (Rd ) → S (Rd ) to HΓs (Rd ), s ∈ R, or equivalently the extension of A : S(Rd ) → S(Rd ) to HΓs (Rd ). Theorem 2.1.14. Consider A ∈ OPΓm (Rd ) with Γ-elliptic symbol. Then: (i) As ∈ Fred(HΓs (Rd ), HΓs−m (Rd )). (ii) ind As = dim Ker A − dim Ker A∗ , ind As = dim Ker A − dim Ker t A, where A∗ is the formal adjoint and t A is the transposed operator. Observe that the index is then independent of s.

(iii) If T ∈ OPΓm (Rd ) with m < m, then As + Ts ∈ Fred (HΓs (Rd ), HΓs−m (Rd )) and ind (As + Ts ) = ind As . (iv) Suppose As : HΓs (Rd ) → HΓs (Rd ) is invertible for some s ∈ R; then it is invertible for all s ∈ R, and the inverse is an operator in OPΓ−m (Rd ). In conclusion, we pass to consider the hypoelliptic case, the natural frame d being now the classes Γm ρ (R ), 0 < ρ ≤ 1. d Definition 2.1.15. We say that a ∈ Γm ρ (R ), 0 < ρ ≤ 1, m ∈ R, is Γρ -hypoelliptic if there exist m ∈ R, m ≤ m, and R > 0 such that m

|z|

|a(z)| ,

for |z| ≥ R

(2.1.28)

and, for every γ ∈ N2d , |∂zγ a(z)| |a(z)| z−ρ|γ| ,

for |z| ≥ R.

(2.1.29)

d Note that the assumption of Γ-ellipticity (2.1.9) for a symbol a ∈ Γm ρ (R ) corresponds to taking m = m in (2.1.28) and gives automatically (2.1.29). So the Γ-ellipticity implies Γρ -hypoellipticity. The new interesting case is then m < m, for which we have the following counterpart of Theorems 2.1.6 and 2.1.13. d Theorem 2.1.16. Let a ∈ Γm ρ (R ), 0 < ρ ≤ 1, be Γρ -hypoelliptic, for some m < m −m d in (2.1.28). Then there exists b ∈ Γρ (R ) such that, denoted A = a(x, D), B = b(x, D), we have

AB = I + S1 ,

BA = I + S2 ,

with S1 and S2 regularizing operators. Hence A is globally regular. Moreover, if we assume u ∈ S (Rd ), Au ∈ HΓs (Rd ), then we have u ∈ HΓs+m (Rd ) and, for every t<s+m,

u H s+m ≤ C Au HΓs + u HΓt (2.1.30) Γ

for a positive constant C depending on s and t.

2.2. Γ-Elliptic Differential Operators; the Harmonic Oscillator

77

The estimate (2.1.30) presents a loss of m − m in the order of the Γ-Sobolev spaces, with respect to the Γ-elliptic case. So, if m < m we cannot regard A as a Fredholm map from HΓs (Rd ) to HΓs−m (Rd ).

2.2 Γ-Elliptic Differential Operators; the Harmonic Oscillator d All differential operators in OPΓm ρ (R ), 0 < ρ ≤ 1, have polynomial coefficients. Namely: d Proposition 2.2.1. Assume p(x, ξ) ∈ Γm ρ (R ) is of the form p(x, ξ) = aα (x)ξ α |α|≤m

for some aα (x) ∈ C ∞ (Rd ). Then aα (x) is a polynomial. Proof. From (2.1.2) we have, for all β ∈ Nd , |∂xβ p(x, ξ)| zm−ρ|β| ,

z = (x, ξ) ∈ R2d .

Hence, taking |β| > m/ρ, we have, for = ρ|β| − m, ∂xβ aα (x)ξ α z− , |α|≤m

which can be satisfied only if ∂xβ aα (x) = 0 for all x ∈ Rd , i.e., aα is a polynomial. Let us then consider P =

cαβ xβ Dα ,

cαβ ∈ C,

(2.2.1)

|α|+|β|≤m

with symbol, according to the standard quantization: p(x, ξ) = cαβ xβ ξ α .

(2.2.2)

|α|+|β|≤m

Setting z = (x, ξ), γ = (β, α) we shall also write for short cγ z γ . p(z) =

(2.2.3)

|γ|≤m d We have p ∈ Γm cl (R ) and the Γ-ellipticity condition (2.1.9) reads cγ z γ , for |z| ≥ R |z|m |γ|≤m

(2.2.4)

78


with R > 0 sufficiently large. The principal part is given by pm (z) = cγ z γ ,

(2.2.5)

|γ|=m

and the equivalent Γ-ellipticity condition (2.1.10) is cγ z γ = 0 for z = 0.

(2.2.6)

|γ|=m

Under the assumption (2.2.6) we may apply Theorems 2.1.6, 2.1.13 and 2.1.14. In the following, we discuss some relevant examples of Γ-elliptic differential operators. Consider first the one-dimensional case. Define for x ∈ R and D = −id/dx as standard: (2.2.7) L = Dx − rx, r ∈ C. The principal symbol is given by ξ + rx and (2.2.6) is satisfied if and only if Im r = 0. The transposed operator is given by t

L = −Dx − rx.

(2.2.8)

The classical solutions of Lu = 0 are the functions u(t) = C exp[irx2 /2],

C ∈ C,

(2.2.9)

which for C = 0 belong to S(R), or S (R), if and only if Im r > 0. We may regard L as a Fredholm operator, setting for example L : HΓ1 (R) → L2 (R), with ind L = dim Ker L − dim Ker t L given by 1 for Im r > 0, ind L = −1 for Im r < 0.

(2.2.10)

(2.2.11)

We may now consider the case of a generic ordinary differential operator with polynomial coefficients. The Γ-ellipticity condition (2.2.6) on the principal symbol reads cαβ xβ ξ α = 0 for (x, ξ) ∈ R2 , (x, ξ) = (0, 0). (2.2.12) pm (x, ξ) = α+β=m

Factorizing we then obtain pm (x, ξ) = c(ξ − r1 x) · · · (ξ − rm x)

(2.2.13)

with Im rj = 0, j = 1, . . . , m, c = 0. Hence we may write after multiplication by c−1 : P = (Dx − r1 x) · · · (Dx − rm x) + aαβ xβ Dα (2.2.14) α+β<m


79

for some constants aαβ ∈ C. We may regard P as a Fredholm operator P : HΓm (R) → L2 (R).

(2.2.15)

The index does not depend on the lower order terms in the sum in (2.2.14), in view of Theorem 2.1.14, (iii), and it is then given by the sum of the indices of the factors Dx − rj x, j = 1, . . . , m, cf. Corollary 1.6.5, which we may compute by (2.2.11). We therefore obtain the following result. Theorem 2.2.2. Consider P in (2.2.14), (2.2.15) and assume Im rj > 0 for j = 1, . . . , m+ , Im rj < 0 for j = m+ + 1, . . . , m, m = m+ + m− (we do not exclude the case m+ = 0 or m− = 0). Then P is a Fredholm operator with ind P = m+ − m− .

(2.2.16)

Concerning the existence of a non-trivial solution u ∈ S (R), hence u ∈ S(R), of P u = 0, we may obtain the following conclusions. If m+ − m− > 0, then ind P > 0, therefore dim Ker P > 0 and a non-trivial solution exists. Moreover, since dim Ker P and dim Ker t P cannot exceed m, if m+ = 0, m− = m, then ind P = −m and a non-trivial solution does not exist. See the next section for further investigation. As an example in dimension d ≥ 1, we then fix attention on the harmonic oscillator of Quantum Mechanics, H = − + |x|2 ,

x ∈ Rd ,

(2.2.17)

which will play an important role in the sequel of the book. Consider for λ ∈ C: 2

P = H − λ = − + |x| − λ, 2

2

P : HΓ2 (Rd ) → L2 (Rd ).

(2.2.18)

2

The principal symbol |z| = |x| +|ξ| satisfies obviously the Γ-ellipticity condition, and P is a Fredholm operator with ind P =ind H = 0 because t H = H. Non-trivial solutions u ∈ S (Rd ) of P u = Hu − λu = 0 belong to S(Rd ) and exist or not, according to the values of λ. Namely, let us return to the one-dimensional case and consider first the equation Hu − λu = −u + x2 u − λu = 0,

x ∈ R.

(2.2.19)

It is convenient to introduce Ψ+ = x −

d , dx

Ψ− = x +

with respective Γ-elliptic symbols x − iξ,

x + iξ.

d dx

(2.2.20)


80

The operators Ψ± are the celebrated creation (Ψ+ ) and annihilation (Ψ− ) operators of Quantum Mechanics. They are of type (2.2.7), and the preceding arguments apply obviously. Since Ψ+ Ψ− = H − 1, then for λ = 1 the equation (2.2.19) admits the solution 2

u0 (x) = e−x

/2

.

(2.2.21)

n = 0, 1, . . . ,

(2.2.22)

un (x) = Ψn+ u0 (x) ∈ S(R).

(2.2.23)

We may then prove that for λ = 2n + 1, a solution is given by In fact, arguing by induction on n, we set un = Ψ+ un−1 and compute Hun = (Ψ+ Ψ− + 1)un = (Ψ+ Ψ− + 1)Ψ+ un−1 = Ψ+ (Ψ− Ψ+ − 1)un−1 + 2Ψ+ un−1 . Since Ψ− Ψ+ = H + 1, and Hun−1 = (2n − 1)un−1 by the inductive hypothesis, we obtain Hun

=

Ψ+ Hun−1 + 2Ψ+ un−1

=

Ψ+ (2n − 1)un−1 + 2Ψ+ un−1 = (2n + 1)Ψ+ un−1 .

Also, let us compute

un 2L2 = (un , un )L2 = (Ψn+ u0 , Ψn+ u0 )L2 . Since the formal adjoint of Ψ+ is Ψ− , we have n−1 n−1

un 2L2 = (Ψ− Ψn+ u0 , Ψn−1 + u0 )L2 = ((H + 1)Ψ+ u0 , Ψ+ u0 )L2 n−1 2 = 2n(Ψn−1 + u0 , Ψ+ u0 )L2 = 2n un−1 L2 .

Hence

√

un 2L2 = 2n n! u0 2L2 = 2n n! π.

(2.2.24)

Since Ψ− u0 = 0, a similar computation also shows that (un , um )L2 = 0 for n = m, that yields the orthogonality of the system {un }, n ∈ N. Let us now define the n-th order Hermite polynomial by

n 2 2 d e−x /2 , Pn (x) = cn ex /2 x − dx

(2.2.25)

(2.2.26)


81

cn = 2−n/2 (n!)−1/2 π −1/4 . We have therefore proved that the Hermite functions, still written by un by abuse, 2

un (x) = Pn (x)e−x

/2

,

n = 0, 1, . . . ,

(2.2.27)

give an orthonormal system of L2 (R). To prove completeness, it will be sufficient to argue on g ∈ S(R) and assume +∞ g(x)un (x)dx = 0 for all n ∈ N. (2.2.28) −∞

Since we may write xn as a linear combination of Pj (x), 0 ≤ j ≤ n, then (2.2.28) implies +∞ 2 g(x)xn e−x /2 dx = 0 for all n ∈ N. −∞

Compute now the Fourier transform +∞ ∞ 2 (−ixξ)n −x2 /2 ¯ Fx→ξ g(x)e g(x)e−x /2 dx = n! −∞ n=0 (−iξ)n +∞ 2 = lim g(x)xn e−x /2 dx ¯ = 0. N →∞ n! −∞ n≤N

A function in S(R) which has zero Fourier transform must be the zero function, 2 hence g(x)e−x /2 = 0 for all x ∈ R, and therefore g(x) ≡ 0. Let us pass to consider the higher dimensional case. It will be sufficient to search for solutions of Hu − λu = 0 of the form u(x) = uk (x) = Πdj=1 Pkj (xj )e−

|x|2 2

,

where k = (k1 , . . . , kd ) ∈ Nd and Pn stands for the n-th Hermite polynomial. Writing Hj = Dx2j + x2j , since Hj uk = (2kj + 1)uk , we obtain Huk =

d

H j uk =

j=1

d

(2kj + 1)uk .

j=1

We may summarize the preceding results as follows. Theorem 2.2.3. The equation 2

P u = Hu − λu = −u + |x| u − λu = 0, admits for λ = λk =

d j=1

(2kj + 1),

u ∈ S (Rd ),

k = (k1 , . . . , kd ) ∈ Nd ,

(2.2.29)

82


the solutions in S(Rd ), uk (x) =

d

|x|2 2

Pkj (xj )e−

,

(2.2.30)

j=1

which form an orthonormal system in L2 (Rd ). Because of the completeness of the Hermite functions uk , k ∈ Nd , we know from Spectral Theory, see Theorem 4.2.9 in the next Chapter 4, that for λ = λk the map P = H − λ : HΓ2 (Rd ) → L2 (Rd ) is an isomorphism, with inverse P −1 = (H − λ)−1 : L2 (Rd ) → HΓ2 (Rd ) belonging to OPΓ−2 (Rd ), cf. Theorem 2.1.14, (iv). Returning to (2.2.29), (2.2.30) we observe that the eigenvalues λ = 2K + d, K ∈ N, appear with multiplicity if d > 1. Precisely, corresponding to λ = 2K + d we have in (2.2.30) " k ∈ Nd ,

d

# kj = K

=

j=1

K +d−1 d−1

different eigenfunctions, cf. (0.3.16). Let us also observe that, taking into account multiplicity, the number of the eigenvalues which do not exceed λ ∈ R+ is given by " N (λ) = k ∈ Nd ,

d

kj ≤ K

j=1

=

K h+d−1 h=0

d−1

#

K +d , d

=

(2.2.31)

cf. (0.3.15). Here we have set K = [(λ − d)/2], the integer part of (λ − d)/2. Hence for λ → +∞, N (λ) =

(K + d)(K + d − 1) · · · (K + 1) Kd λd ∼ ∼ d . d! d! 2 d!

(2.2.32)

2.3 Asymptotic Integration and Solutions of Exponential Type Let us return to consider the generic Γ-elliptic ordinary differential operator, P ∈ OPΓm (R). In view of the discussion in the preceding section, we may write, for

2.3. Asymptotic Integration and Solutions of Exponential Type x ∈ R, P = (Dx − r1 x) . . . (Dx − rm x) +

aαβ xβ Dα ,

83

(2.3.1)

α+β<m

with Im rj = 0, j = 1, . . . , m, aαβ ∈ C. The classical solutions of P u = 0 extend to entire functions of x ∈ C. A precise analysis in the complex domain is given by the theory of Asymptotic Integration. Assuming initially that all the rj , j = 1, . . . , m, are distinct, we formally solve P u = 0 in C by ∞ x2 uj (x) = xsj exp irj + ηj x β−p,j x−p , 2 p=0

j = 1, . . . , m.

(2.3.2)

We may first determine ηj and sj by computing x2 x2 − ηx P exp irj + ηx xs exp − irj 2 2 = g(η)xs+m−1 + f (s)xs+m−2 + O(xs+m−3 ). The function g(η) is linear in η. Namely, setting g(η) = 0 we obtain

η = ηj = −i

aαβ rjα . h=j (rj − rh )

$

α+β=m−1

The expression of f (s), involving aαβ with α + β ≥ m − 2, is linear in s and imposing f (s) = 0 we determine s = sj . Similarly we can find β−p,j , assuming β0,j = 1. Note however that the series in (2.3.2) is not in general convergent, and the preceding expressions must be understood as asymptotic expansions of the actual solutions in suitable sectors of the complex plane. Namely, we recall the following basic result, see for example Wasow [194]. Proposition 2.3.1. Given any sector Λ = {x ∈ C \ {0} : ϕ1 < arg x < ϕ2 } with ϕ2 − ϕ1 < π2 , there exist uΛ,j (x), j = 1, . . . , m, x ∈ C, linearly independent solutions of P u = 0, such that for every n ≥ 0, x ∈ Λ, x → ∞: n x2 sj −p −n β−p,j x + o(x ) . (2.3.3) uΛ,j (x) = x exp irj + ηj x 2 p=0 In the case some rj coincide, we may replace (2.3.3) with the weaker information, valid for any linear combination of the corresponding independent solutions: x2 u ˜j (x), uΛ,j (x) = exp irj 2

2−

|˜ uj (x)| eν|x|

,

x ∈ Λ,

(2.3.4)

for some ν > 0, > 0. Observe that the solutions uΛ,j in (2.3.3), (2.3.4) depend on the sector Λ. Consider now two sectors Λ+ , Λ− satisfying the hypotheses of

84


Proposition 2.3.1 and containing R+ , R− , respectively. We may conclude the − existence of two systems of solutions, u+ j , uj , j = 1, . . . , m, satisfying (2.3.3) or (2.3.4) in Λ+ , Λ− . Any classical solution u of P u = 0 in R can be written u=

m

+ μ+ j uj =

j=1

m

− μ− j uj ,

− μ+ j , μj ∈ C.

j=1

Assume as in Theorem 2.2.2 that Im rj > 0 for j = 1, . . . , m+ , Im rj < 0 for j = m+ + 1, . . . , m, m = m+ + m− . Then from (2.3.3), (2.3.4) we have that − u belongs to S(R), or equivalently to S (R), if and only if μ+ j = μj = 0 for + m < j ≤ m. Summing up: +

If u ∈ S(R), then u =

m

+

+ μ+ j uj

=

j=1

m

− μ− j uj .

(2.3.5)

j=1

From (2.3.5), (2.3.3) and (2.3.4) we obtain the following results, which complete the information from Theorem 2.2.2. Theorem 2.3.2. Let u ∈ S (R), hence u ∈ S(R), be a solution of P u = 0. Then 2

|u(x)| e−δx ,

for x ∈ R

(2.3.6)

for some positive constant δ > 0. Moreover m+ − m− ≤ dim(Ker P ∩ S(R)) ≤ m+ .

(2.3.7)

In view of (2.3.3), (2.3.4) we may take in (2.3.6) any δ > 0 with δ
0, but ind P = m+ − m− ≤ 0, the existence of nontrivial solutions u ∈ S(R) of P u = 0 depends on the coefficients aαβ in (2.3.1). We shall now prove that in the case m+ = 1 all the possible solutions in S(R) are of the form x2 (2.3.8) u(x) = Q(x)eir1 2 +η1 x , for some polynomials Q(x). This reduces the computation of the eigenvalues to a purely algebraic matter, and generalizes what we proved for the one-dimensional harmonic oscillator (2.2.19). Theorem 2.3.3. Consider P = (Dx − r1 x) . . . (Dx − rm x) +

α+β<m

aαβ xβ Dα

(2.3.9)

2.3. Asymptotic Integration and Solutions of Exponential Type

85

with Im r1 > 0, Im rj < 0 for j = 2, . . . , m, aαβ ∈ C. If u ∈ S(R) is a solution of P u = 0, then u is of exponential type, i.e., u has the form (2.3.8) where η1 = −i

aαβ r1α j=2 (r1 − rj )

$m

α+β=m−1

and Q(x) is a polynomial. In the proof we shall use the following classical Phragmen-Lindelöf result; see e.g. Titchmarsh [188, page 177]. Lemma 2.3.4. Let U (x), x ∈ C, be analytic for R ≤ |x| < ∞,

ϕ1 ≤ arg x ≤ ϕ2 ,

(2.3.10)

where R, ϕ1 , ϕ2 are real costants, R > 0. Let |U (x)| ≤ C exp[ν |x|η ]

(2.3.11)

in the same region, for some positive constants C, ν and η such that ϕ2 −ϕ1 < π/η. If U is bounded as x → ∞ on the lines arg x = ϕ1 and arg x = ϕ2 , then U is bounded uniformly in the region (2.3.10). Proof of Theorem 2.3.3. Let Λ1+ , Λ2+ , Λ1− , Λ2− , Λ+i , Λ−i be 6 sectors in C, with vertex at the origin and a positive central angle not exceeding π2 , such that Λ1+ ∪ Λ2+ ∪ Λ1− ∪ Λ2− ∪ Λ+i ∪ Λ−i = C \ {0}.

(2.3.12)

Moreover: Λ1+ and Λ2+ contain R+ ; Λ1− and Λ2− contain R− ; Λ+i contains iR+ ; Λ−i contains iR− . To be definite, let us define # " π Λ1+ = x ∈ C \ {0} : − < arg x < − 2 , 2 # " π 2 Λ+ = x ∈ C \ {0} : − + 2 < arg x < , 2 " # π π Λ+i = x ∈ C \ {0} : − 3 < arg x < + 3 , 2 2 and symmetrically Λ1− , Λ2− , Λ−i . Fix in particular 0 < < π/12. Let u ∈ S(R) be a solution of P u = 0. Then we may apply (2.3.5) with m+ = 1: + − − u(x) = μ+ 1 u1 (x) = μ1 u1 (x). More precisely, in the four sectors Λ1+ , Λ2+ , Λ1− , Λ2− , we have for u, up to multiplicative constants, the asymptotic expansion given by (2.3.3): n x2 s1 −p −n + η1 x β−p,1 x + o(x ) , (2.3.13) x exp ir1 2 p=0


86

with η1 as in Theorem 2.3.3. Take now an integer N such that N ≥ |s1 | and consider, for x ∈ C, x2 U (x) = x−N exp − ir1 − η1 x u(x) , (2.3.14) 2 which is an analytic function for x = 0. It follows from (2.3.13) that U (x) is bounded in Λ1+ ∪ Λ2+ ∪ Λ1− ∪ Λ2− , for |x| ≥ 1 say. We may actually prove that U is bounded in the whole C for |x| ≥ 1, by applying Lemma 2.3.4 to the remaining sectors Λ+i , Λ−i . Let us fix attention on Λ+i , |x| ≥ 1. We know already that U (x) is bounded on the lines arg x = π2 − 3 , arg x = π2 + 3 , since they belong to Λ1+ , Λ1− , respectively. On the other hand, we may apply Proposition 2.3.1 and (2.3.4) to the sector Λ+i as well. Since u(x) is a linear combination of the corresponding functions in (2.3.3), (2.3.4) we have in Λ+i an estimate 2

|U (x)| ≤ C exp[ν |x| ], for suitable positive constants C and ν. Hence (2.3.11) is satisfied with η = 2. Since we assume 6 < π/2, we may apply Lemma 2.3.4 and conclude that U (x) is bounded in Λ+i , and similarly in Λ−i . In conclusion: U (x) in (2.3.14) is bounded in C for |x| ≥ 1, therefore the function x2 − η1 x u(x) = U (x)xN Q(x) = exp − ir1 2 is analytic in C with a pole at ∞; thus it is indeed a polynomial. For the sake of completeness, we report the classical proof. It is sufficient to show that (dn Q/dxn )(0) = 0 for n ≥ n0 , with n0 sufficiently large. Now, by the Cauchy formula, dn Q (−1)n n! (0) = Q(z)z −n−1 dz dxn 2πi γ where γ is a closed path around the origin. Take n0 such that Q(z)z −n0 −1 ≤ −2

for large |z|. Then for any fixed n ≥ n0 and every δ > 0, there exists C |z| R > 0 such that, by choosing γ = {x ∈ C : |x| = R}, we have |(dn Q/dxn )(0)| < δ. This ends the proof of Theorem 2.3.3. The preceding Theorem 2.3.3 can be extended to the case when in (2.2.13), (2.3.1) we have one root with positive imaginary part, appearing with multiplicity m+ ≥ 2. Let us write r1 = · · · = rm+ = r0 , Im r0 > 0. We need however some conditions on the lower order terms, which are expressed by imposing on the operator P the following particular form: ⎛ ⎞ + ⎝ P = cαβ xβ Dxα ⎠(Dx − r0 x)m −j 0≤j<m+

α+β=m− −j

+

α+β≤m− −m+

cαβ xβ Dxα ,

(2.3.15)

2.3. Asymptotic Integration and Solutions of Exponential Type where we understand 1 ≤ m+ ≤ m− and cαβ xβ ξ α = α+β=m−

(ξ − rj x)

87

(2.3.16)

m+ <j≤m

with Im rj < 0 for j = m+ + 1, . . . , m. In the case m+ = 1 we recapture the operator P in Theorem 2.3.3, with aαβ = 0 for α + β = m − 1, hence η1 = 0. Theorem 2.3.5. Consider P as in (2.3.15), (2.3.16) with 1 ≤ m+ ≤ m− , Im r0 > 0, Im rj < 0 for j = m+ + 1, . . . , m, cαβ ∈ C. If u ∈ S(R) is a solution of P u = 0, then x2 (2.3.17) u(x) = Q(x) exp ir0 2 for some polynomial Q(x). The rough estimate (2.3.4), valid in the case of multiple roots, is not sufficient for the proof of Theorem 2.3.5. Taking advantage of the particular form of P in (2.3.15), however, we have from the asymptotic integration the following more precise result. Lemma 2.3.6. Let P be as in Theorem 2.3.5. Given any sector Λ ⊂ C with vertex at the origin and a positive central angle not exceeding π/2, there exist uΛ,j (x), j = 1, . . . , m+ , linearly independent solutions of P u = 0, of the form x2 (2.3.18) u ˜j (x), j = 1, . . . , m+ uΛ,j (x) = exp ir0 2 where u ˜j (x) are entire functions, satisfying for some integer N : N

|˜ uj (x)| |x| ,

x ∈ Λ, |x| ≥ 1.

(2.3.19)

In fact, when integrating asymptotically P u = 0, we have x2 + − + − x2 exp − ir0 P exp ir0 xs = f (s)xs−m +m + O(xs−m +m −1 ), 2 2 where f (s) =

0≤j<m+

⎛ ⎝

⎞ cαβ r0α ⎠ s(s − 1) . . . (s − m+ + j + 1).

α+β=m− −j

This is a polynomial of degree m+ , corresponding to the multiplicity of the root r0 . That grants the lack of lower order exponential terms. Moreover, if the equation f (s) = 0 has m+ distinct roots s1 , . . . , sm+ , we have x2 uΛ,j (x) = xsj exp ir0 (1 + o(1)), j = 1, . . . , m+ . 2 When some sj coincide, logarithmic terms appear; the estimate (2.3.19) keeps valid anyhow for a sufficiently large N .

88


Proof of Theorem 2.3.5. We cover C\{0} by six sectors as in (2.3.12). Let u ∈ S(R) be a solution of P u = 0. Then we may apply (2.3.5): +

u=

m

+

+ μ+ j uj

=

m

j=1

− μ− j uj .

j=1

More precisely, using Lemma 2.3.6 in the four sectors Λ1+ , Λ2+ , Λ1− , Λ2− , we have x2 u(x) = exp ir0 u ˜(x) 2 where the entire function u ˜(x) satisfies N

|˜ u(x)| |x| ,

x ∈ Λ1+ ∪ Λ2+ ∪ Λ1− ∪ Λ2− , |x| ≥ 1.

We may then argue exactly as in the proof of Theorem 2.3.3 and obtain (2.3.17).

2.4 H-Polynomials We pass now to consider Γρ -hypoelliptic partial differential operators, 0 < ρ ≤ 1, cf. Definition 2.1.15. In view of Proposition 2.2.1, we may again limit our attention to operators with polynomial coefficients: P = cαβ xβ Dxα . (2.4.1) |α|+|β|≤m

Setting z = (x, ξ), we write their symbols in the standard quantization: cγ z γ . p(z) =

(2.4.2)

|γ|≤m

In the following we shall assume m ≥ 1 and cγ = 0 for some γ with |γ| = m. The Γρ -hypoellipticity is expressed by |∂zγ p(z)| |p(z)| z−ρ|γ| ,

|z| ≥ R

(2.4.3)

for some R > 0. In fact, condition (2.1.29) implies for a polynomial p(z) condition (2.1.28), since for some γ with |γ| = m we have ∂ γ p(z) =constant= 0, hence 0 < c = |∂ γ p(z)| |p(z)| z−ρm ,

|z| ≥ R,

and therefore |z|

ρm

|p(z)| ,

|z| ≥ R.

(2.4.4)

2.4. H-Polynomials

89

So, under the assumption (2.4.3) we may apply Theorem 2.1.16 and deduce existence of parametrices for P in (2.4.1). In particular, P is globally regular and the following a priori estimate is valid for u ∈ S(Rd ):

u HΓρm ≤ C ( P u L2 + u L2 ) .

(2.4.5)

In the sequel of this section we want to clarify the algebraic meaning of (2.4.3). We shall argue on polynomials in arbitrary dimension n though in our applications n = 2d is even. Consider the subset of Cn , V = {ζ ∈ Cn : p(ζ) = 0},

(2.4.6)

d(z) = distanceCn (z, V ) = inf |z − ζ| .

(2.4.7)

and define, for z ∈ Rn , ζ∈V

Proposition 2.4.1. The following properties are equivalent for a polynomial p(z): (i) ζ ∈ V , ζ → ∞ implies |Im ζ| → +∞. (ii) z ∈ Rn , z → ∞ implies d(z) → +∞. (iii) |∂zγ p(z)/p(z)| → 0 when z → ∞ in Rn , if γ = 0. If (i), (ii), (iii) are satisfied, we say that p(z) is H-type. Let ρ be fixed, ρ > 0; we say that p(z) is ρ-H-type if the following equivalent properties are satisfied: (i)ρ We have:

|ζ|ρ 1 + |Im ζ|

for ζ ∈ V.

(ii)ρ There is a constant R > 0 such that ρ

|z| d(z)

for |z| ≥ R.

(iii)ρ There is a constant R > 0 such that |∂ γ p(z)| |p(z)| z−ρ|γ|

for |z| ≥ R,

i.e., p(z) is Γρ -hypoelliptic. Obviously, if p(z) is ρ-H-type for some ρ, 0 < ρ ≤ 1, then it is H-type. In the opposite direction, if p(z) is H-type, then it is ρ-H-type for some ρ > 0; moreover, the numbers ρ for which (i)ρ , (ii)ρ , (iii)ρ are valid form an interval ]0, ρ0 ], with ρ0 a rational number ≤ 1. To prove Proposition 2.4.1 we need two auxiliary results.

90


Lemma 2.4.2. Let m ≥ 1 be an integer. There is a constant C > 0, depending only on m and the dimension n, such that for all polynomials p(z) of degree m we have 1/|γ| |∂ γ p(z)/p(z)| ≤ C, (2.4.8) C −1 ≤ d(z) |γ|=0

for z ∈ Rn , p(z) = 0. Proof. Setting, for p(z) = 0, A(z) =

1/|γ|

|∂ γ p(z)/p(z)|

,

|γ|=0

we have

|∂ γ p(z)| ≤ A(z)|γ| |p(z)| .

From the Taylor expansion we obtain, for ζ ∈ Cn , (A(z) |ζ|)|γ| /γ!. |p(z + ζ) − p(z)| ≤ |p(z)| 1≤|γ|≤m

Choose c > 0 such that

c|γ| /γ! ≤ 1.

1≤|γ|≤m

If A(z) |ζ| < c we must have p(z + ζ) = 0 and therefore A(z)d(z) ≥ c. We obtain thus the left-hand side inequality in (2.4.8) for any C ≥ 1/c. To establish the right-hand side inequality, choose ζ ∈ Cn such that |ζ| ≤ d(z)/2 and regard p(z + tζ) as a polynomial Q(t) in t. The roots tj of Q(t) satisfy the inequality |tj ζ| ≥ d(z), and therefore |tj | ≥ 2. It follows that m (tj − 1)/tj ≤ (3/2)m . |p(z + ζ)/p(z)| = |Q(1)/Q(0)| =

(2.4.9)

j=1

Regarding p(z + ζ) as a holomorphic function of ζ ∈ Cn and applying Cauchy’s inequality on a polycylinder in the ball |ζ| ≤ d(z)/2, we obtain from (2.4.9) |∂ γ p(z)| ≤ Cd(z)−|γ| |p(z)| , for a suitable C depending only on m and n, which implies the right-hand side inequality of (2.4.8). The second auxiliary result is the following version of the classical SeidenbergTarski Theorem. For the proof see for example Hörmander [119, Vol. II]. Theorem 2.4.3. If A, subset of Rn+m = Rn ⊕ Rm , is semi-algebraic (i.e., finite union of finite intersections of sets defined by a polynomial equation or inequality), then the projection of A in Rm is also semi-algebraic.

2.4. H-Polynomials

91

Proof of Proposition 2.4.1. The equivalence between (i) and (ii) is easily established, while the equivalence between (ii) and (iii) is a consequence of Lemma 2.4.2. Let us detail the proof of (i)ρ ⇔ (ii)ρ . Observe first that (ii)ρ cannot be valid for ρ > 1. Indeed, the function d(z) is Lipschitz, therefore d(z) ≤ C(1 + |z|), for some C > 0. This forces ρ ≤ 1 in (ii)ρ . It is now clear that (ii)ρ implies (i)ρ ; in fact we have for ζ ∈ V : ρ

|Re ζ| ≤ C(1 + d(Re ζ)) ≤ C(1 + |Im ζ|). On the other hand, if (i)ρ holds we can take ζ ∈ V with |ζ − z| ≤ 2d(z), say, and write |z| ≤ |ζ| + |z − ζ| ≤ C 1/ρ (1 + |Im ζ|)1/ρ + 2d(z); since |Im ζ| ≤ 2d(z), we obtain (ii)ρ . The equivalence between (ii)ρ and (iii)ρ follows easily from Lemma 2.4.2. Finally, let us prove that (ii) implies (ii)ρ for some ρ > 0. Let us begin by considering the set A of points (z, η, θ, τ, δ) ∈ Rn × Rn × n R × R × R defined by the following equations and inequalities: p(η + iθ) = 0,

τ > 0,

|z − η|2 + |θ|2 ≤ τ −2 ,

δ > 0,

|z| δ = 1.

By Theorem 2.4.3 the image B of A by the projection (z, η, θ, τ, δ) → (z, τ, δ) is also semi-algebraic. It is easily seen that B is defined by the inequalities τ > 0,

d(z) ≤ τ −1 ,

δ > 0,

|z| δ = 1.

If (ii) holds, there is δ0 > 0 such that if 0 < δ < δ0 and |z| = δ −1 we have d(z) > 0. For all such δ the function τ (δ) = sup d(z)−1 |z|δ=1

is well defined and continuous. The image E of B by the projection (z, τ, δ) → (τ, δ) is a semi-algebraic set. Moreover, we can show that (τ (δ), δ) varies on the boundary of E when 0 < δ < δ0 . Shrinking this interval, if necessary, it follows that Q(τ (δ), δ) = 0 for a suitable polynomial Q in two variables. Then, τ (δ) admits a Puiseux expansion in some neighbourhood of the origin in the complex δ-plane: τ (δ) = ak (δ 1/q )k + ak+1 (δ 1/q )k+1 + . . . , where q > 0 and k are integers. We can assume ak = 0 and, by choosing for δ 1/q the branch which is real and positive for δ > 0, also ak > 0. In view of (ii), we


92

have τ (δ) → 0 as δ → 0, hence k > 0. It follows easily that, for some c > 0 and sufficiently large |z|, inf d(z) = τ (δ)−1 ≥ c |z|k/q . |z|δ=1

This implies (ii)ρ with ρ = k/q and also the fact that the set of numbers for which (ii)ρ holds is a closed interval with a rational number as upper limit. The proof of Proposition 2.4.1 is therefore complete. As we show now, to prove that p(z) is H-type, it is sufficient to check (iii) in Proposition 2.4.1 for first order derivatives. This simplifies a lot the reasoning in the applications. Proposition 2.4.4. The following properties of the polynomial p(z) are equivalent (and provide Γρ -hypoellipticity of p(z) for some ρ, 0 < ρ ≤ 1): (a) ∂zγ p(z)/p(z) → 0 when z → ∞ in Rn , if γ = 0. (b) For all θ ∈ Rn , p(z + θ)/p(z) → 1 when z → ∞ in Rn . (c) ∂zj p(z)/p(z) → 0 when z → ∞ in Rn , for j = 1, . . . , n. We need the following lemma, which allows us to write each γ-derivative of a polynomial p(z) as a linear combination of translations of p(z). Lemma 2.4.5. Let p(z) = |γ|≤m cγ z γ , z ∈ Rn , be a polynomial of degree m ≥ 0, m+n and N = n . Then for all (θ1 , θ2 , . . . , θN ) in an open dense subset of (Rn )N and all γ ∈ Nn there exist real numbers t1 , t2 , . . . , tN , such that ∂ γ p(z) = and satisfying further

N

k=1 tk

N

tk p(z + θk ),

(2.4.10)

k=1

= 0 if γ = 0.

Proof. For all k = 1, . . . , N , let θk ∈ Rn . Consider Taylor’s formula for p(z + θk ): p(z + θk ) = p(z) +

∂ β p(z) β θk , β!

β=0

and multiply the left-hand and right-hand sides by a real variable tk . Summing up on k from 1 to N we obtain N k=1

tk p(z + θk ) = p(z)

N k=1

tk +

N ∂ β p(z) tk θkβ . β!

β=0

(2.4.11)

k=1

In order to get (2.4.10) when γ = 0, it suffices that the following inhomogeneous linear system is solvable: ⎧N ⎪ ⎨k=1 tk = 0, N γ (2.4.12) k=1 tk θk = γ!, ⎪ ⎩N β k=1 tk θk = 0 for β = γ, β = 0.

2.4. H-Polynomials

93

This is an N × N system, cf. (0.3.15), whose coefficients matrix is A = (θkβ ), k = 1, . . . , N , |β| ≤ m. Here the column index is k, whereas β plays the role of row index and it is understood that the set of these multi-indices is ordered in some way. Now, the determinant of A is a polynomial in the components of each because, up to θk , k = 1, . . . , N . It also clear that it does not vanish identically σN the sign, it is given by (σ1 ,...,σN ) sign(σ1 , . . . , σN )θ1σ1 · · · θN , where (σ1 , . . . , σN ) is a permutation of {β ∈ Nn : |β| ≤ m}, and there are no similar terms in this sum. Hence the above system is solvable for every choice of θ1 , . . . , θN for which such a determinant does not vanish, which gives an open dense subset of (Rn )N . N Similarly, for γ = 0 one considers, in place of (2.4.12), the system k=1 tk = N 1 and k=1 tk θkβ = 0 for all β = 0, which concludes the proof. Proof of Proposition 2.4.4. Of course (a) ⇒ (c). We just observe that by Proposition 2.4.1 and (2.4.4) it follows from (a) that p(z) → ∞ as z → ∞. (c) ⇒ (b) Let n ≥ 3. Since p(z) = 0 on the simply connected open set |z| > R, for R large enough, we can consider a branch of log p(z), for |z| > R. For any fixed θ = (b1 , b2 , . . . , bn ) ∈ Rn we can write log p(z + θ) − log p(z) =

1

0

d log p(z + tθ) dt, dt

for |z| > R + |θ|. We have n ∂zj p(z + tθ) d n ∂zj p(z + tθ) log p(z + tθ) = |bj |, bj ≤ dt j=1 p(z + tθ) j=1 p(z + tθ) and it follows from (c) that p(z + θ) →1 p(z)

as z → ∞.

In the cases n = 1, 2 one can repeat the above argument separately on two simply connected open subsets which cover R and R2 , respectively. (b) ⇒ (a) From Lemma 2.4.5 it follows that for each γ ∈ Nn , γ = 0, there exist real numbers t1 , . . . , tN and vectors θ1 , . . . , θN in Rn , N = m+n , such that n ∂ γ p(z) =

N

tk p(z + θk ),

with

k=1

N

tk = 0.

k=1

For large |z| and γ = 0 we obtain ∂ γ p(z) p(z + θk ) tk tk = 0, = → p(z) p(z) N

N

k=1

k=1

as z → ∞.

94


2.5 Quasi-Elliptic Polynomials We may now discuss some relevant subclasses of the H-polynomials, namely the quasi-elliptic ones and, as a generalization in the next section, the multi-quasielliptic polynomials. Such polynomials, regarded as symbols of linear partial differential operators, are obviously Γρ -hypoelliptic for some ρ > 0, cf. (2.4.3). Their importance, however, resides in the fact that we may easily restore for them a notion of ellipticity, cf. Definition 1.3.1, by introducing adapted weight functions. In the present and next sections we fix attention on the algebraic aspects, and address to the next Section 2.7 for the related pseudo-differential calculus. We first define the quasi-elliptic polynomials. Let M = (M1 , . . . , Mn ) be an n-tuple of rational numbers Mj ≥ 1, j = 1, . . . , n. Assume also minj Mj = 1. Every polynomial p(z) in Rn can be written cγ z γ , (2.5.1) p(z) = γ,M ≤m

for a sufficiently large integer m, the M -order of p(z). In the sum we mean γ, M = γ1 M1 + · · · + γn Mn . The quasi-principal part of p(z), with respect to M , will be then defined as pM,m (z) = cγ z γ . (2.5.2) γ,M =m

Definition 2.5.1. We shall say that the polynomial p(z) in (2.5.1) is quasi-elliptic with respect to M if pM,m (z) = 0 for all z = 0. (2.5.3) If M = (1, . . . , 1), then pM,m (z) in (2.5.2) is pm (z), the standard principal part of p(z), and (2.5.3) means that p(z) is Γ-elliptic, cf. (2.2.5), (2.2.6). Let us recall that a function f (z) in Rn \ {0} is called quasi-homogeneous of degree k ∈ R, with respect to M , if for all t > 0, z = 0.

f (tM1 z1 , . . . , tMn zn ) = tk f (z1 , . . . , zn )

Observe that if f is quasi-homogeneous of degree k, then ∂ γ f is quasi-homogeneous of degree k −γ, M . According to the previous definition, the quasi-principal part pM,m (z) in (2.5.2) is quasi-homogeneous of degree m with respect to M . Let us define, for z ∈ Rn , |z|M =

n

|zj |

1/Mj

.

(2.5.4)

j=1

The function |z|M is quasi-homogeneous of degree 1. Any quasi-homogeneous function f (z) of degree k is identified by its values on the compact manifold {|z|M = 1}; namely we may write k z) (2.5.5) f (z) = |z|M f (˜

2.5. Quasi-Elliptic Polynomials

95

where

M

M

z˜ = (z1 / |z|M1 , . . . , zn / |z|Mn ), hence |˜ z |M = 1. If f (z) is continuous in Rn \ {0}, it follows that k

|f (z)| ≤ C |z|M

(2.5.6)

with C = max|z|M =1 |f (z)|. If in addition f (z) = 0, then |z|M ≤ −1 |f (z)| k

(2.5.7)

with = min|z|M =1 |f (z)|. Since p(z) in (2.5.1) can be regarded as a sum of quasi-homogeneous terms of degree ≤ m, we deduce from (2.5.6) that, for every R > 0, m

|z| ≥ R.

|p(z)| |z|M ,

(2.5.8)

On the other hand, using (2.5.7) and arguing as in the proof of Proposition 2.1.5 we have: Proposition 2.5.2. The polynomial p(z) in (2.5.1) is quasi-elliptic with respect to M if and only if there exists R > 0 such that |z|m M |p(z)| ,

|z| ≥ R,

(2.5.9)

where m is the M -order of p(z). Summing up, we have that quasi-elliptic polynomials p(z), of order m with m respect to M , are characterized by the asymptotic equivalence |p(z)| |z|M . We can now prove that quasi-elliptic polynomials are H-type. Proposition 2.5.3. Let p(z) in (2.5.1) be quasi-elliptic with respect to M . Then −|γ|

|∂zγ p(z)| |p(z)| |z|M

|z| ≥ R,

,

(2.5.10)

for some R > 0, and p(z) is ρ-H-type for ρ = min 1/Mj .

(2.5.11)

j

Proof. Since p(z) in (2.5.1) can be written as a sum of quasi-homogeneous terms of degree ≤ m, then ∂ γ p(z) is a sum of quasi-homogeneous terms of degree ≤ m − γ, M . It follows from the estimates (2.5.6) that m−γ,M

|∂zγ p(z)| |z|M

|z| ≥ R.

,

Hence by applying (2.5.9), −γ,M

|∂zγ p(z)| |p(z)| |z|M

,

|z| ≥ R,

(2.5.12)

96


which implies (2.5.10) since γ, M ≥ |γ|. On the other hand, we have zρ |z|M for ρ as in (2.5.11) and large |z|; therefore |∂zγ p(z)| |p(z)| z−ρ|γ| ,

|z| ≥ R,

i.e., p(z) is ρ-H-type. Example 2.5.4. Let us return to split z = (x, ξ) and consider, for x ∈ R, ξ ∈ R, p(x, ξ) = ξ h + rxk ,

(2.5.13)

where h, k are positive integers and r ∈ C. The associated ordinary differential operator is P = Dxh + rxk . (2.5.14) If k ≥ h, we fix the weight 1 for the x variable and the weight k/h for the ξ variable, i.e., M = (1, k/h). If h ≥ k, then M = (h/k, 1). The M -order of p(x, ξ) in (2.5.13) is max{h, k} and, in view of (2.5.3), quasi-ellipticity amounts to assuming ξ h + rxk = 0 for (x, ξ) = (0, 0). In particular, if h or k is odd, p(x, ξ) is quasi-elliptic if and only if Im r = 0. Example 2.5.5. For z = (x, ξ) ∈ R2d consider p(x, ξ) = |ξ|2 + V (x) where V (x) =

a β xβ ,

(2.5.15) (2.5.16)

|β|≤2k

for some integer k ≥ 1. Assume V2k (x) =

aβ xβ = 0 for x = 0.

(2.5.17)

|β|=2k

The associated operator is P = − + V (x).

(2.5.18)

We give now the weight 1 to the x variables and the weight k to the ξ variables, i.e., M = (1, . . . , 1, k, . . . , k). Then p(x, ξ) in (2.5.15), (2.5.16), (2.5.17) is quasi2 elliptic if and only if |ξ| + V2k (x) = 0 for (x, ξ) = (0, 0), that is V2k (x) does not take values in R− ∪ {0} for x = 0. Remark 2.5.6. In the estimates (2.5.8), (2.5.9), (2.5.10), the function |z|M can be replaced by any asymptotically equivalent function. Observe in particular that ⎞1/2 ⎛ n 2m/M m j⎠ ⎝1 + zj |z|M |p(z)| , |z| ≥ R. (2.5.19) j=1

Note that m/Mj , j = 1, . . . , n, are integers; in fact, the quasi-ellipticity assumption m/Mj , j = 1, . . . , n, in the expression (2.5.3) forces the presence of the monomials zj of p(z).

2.6. Multi-Quasi-Elliptic Polynomials

97

2.6 Multi-Quasi-Elliptic Polynomials We now extend the arguments of the preceding section. Namely, we try to reproduce the estimates (2.5.8), (2.5.9), (2.5.10) by replacing the function |z|m M |p(z)| with weight functions suited to more general H-polynomials. In particular, looking m to the weight in the left-hand side of (2.5.19), asymptotically 2γ 1/2 equivalent to |z|M , one is led in a natural way to consider the function ( γ z ) , where γ runs over indices appearing in the expression of p(z). To be precise, we begin by defining the Newton polyhedron of a polynomial p(z), and then recalling terminology and notation from the general theory of the convex polyhedra. Definition 2.6.1. The Newton polyhedron P of a polynomial p(z) = cγ z γ , z ∈ Rn , |γ|≤m

is the convex hull of A ∪ {0} with A = {γ ∈ Nn : cγ = 0}.

(2.6.1)

We recall that, in general, a convex polyhedron P ⊂ Rn is defined as the convex hull of a finite set of points in Rn . One can show that P can be obtained as the convex hull of a finite subset V (P) ⊂ Rn of convex-linearly independent points, called the vertices of P and univocally determinated by P. Moreover there exists a finite set of normal vectors N (P) = N0 (P) ∪ N1 (P) ⊂ Rn such that |ν| = 1,

for all ν ∈ N0 (P),

P = {z ∈ Rn : ν · z ≥ 0, ∀ν ∈ N0 (P)} ∩ {z ∈ Rn : ν · z ≤ 1, ∀ν ∈ N1 (P)}. N0 (P) and N1 (P) are univocally determined by P, if P has non-empty interior. The boundary of P is made of faces Fν which are the convex hull of the vertices of P lying on the hyperplane Hν orthogonal to ν ∈ N (P) and of equation ν · z = 0, ν · z = 1,

if ν ∈ N0 (P), if ν ∈ N1 (P).

If P ⊂ (R+ ∪ {0})n and V (P) ⊂ Nn , as we have for the Newton polyhedron of a polynomial, we define ΛP (z) =

1/2 z

2γ

,

z ∈ Rn .

(2.6.2)

γ∈V (P)

Definition 2.6.2. A complete polyhedron is a convex polyhedron P ⊂ (R+ ∪ {0})n such that (1) V (P) ⊂ Nn ;


98

(2) (0, . . . , 0) ∈ V (P); (3) V (P) = {(0, . . . , 0)}; (4) N0 (P) = {e1 , . . . , en }, with ej = (0, . . . , 0,

1

, 0, . . . , 0) ∈ Rn for j =

j−th entry

1, . . . , n;

(5) N1 (P) ⊂ Rn+ , i.e., every ν ∈ N1 (P) has strictly positive components νj , j = 1, . . . , n. One easily proves: zμ0 ΛP (z) zμ1

for all z ∈ Rn ,

with μ0 =

min

γ∈V (P)\{0}

|γ|,

μ1 = max |γ| = max |γ| . γ∈V (P)

γ∈P

(2.6.3)

μ0 and μ1 are called the minimum and the maximum order of P. We also introduce the formal order of P: μ = max

# "1 : j = 1, . . . , n, ν ∈ N1 (P) . νj

(2.6.4)

We have 0 < μ0 ≤ μ1 ≤ μ. Theorem 2.6.3. The Newton polyhedron of an H-type polynomial is complete. Proof. Let p(z) satisfy the equivalent properties (i), (ii), (iii) in Proposition 2.4.1. Let us prove by induction on n ≥ 2 that its Newton polyhedron P is complete. If n = 2 let l s bj (z1 )z2j p(z) = j=1

with s1 < s2 < . . . < sl . By (iii) in Proposition 2.4.1 we have that ∂z1 p(z) → 0, p(z)

as z → ∞.

In particular we have that ∂z1 p(z) ∂z1 bl (z1 ) = lim = 0. z2 →∞ p(z) bl (z1 ) It follows that sl > 0 and that bl (z1 ) does not depend on z1 . Therefore P contains the point (0, sl ) and all other points (q1 , q2 ) ∈ P are such that q2 < sl . In a similar way we have that there exists (r, 0) ∈ P, with r > 0, such that q1 < r for all other points (q1 , q2 ) ∈ P. The completeness of P follows now easily from the convexity of P.


99

Let us now assume that the theorem is true in dimension n ≥ 2 and let us prove it in dimension n + 1. Let us first show that there are no faces passing through the origin and not lying on the coordinate hyperplanes. In fact if Fν is such a face, ν should have at least two components, say, νq and νn+1 different from zero and at least one, say νq , negative. The reason is that Fν intersects (R+ )n+1 . The intersection of Fν with the coordinate hyperplane −1 zn+1 = 0 is the face Fσ with σ = |ν | ν and ν = (ν , νn+1 ), of the Newton polyhedron of p(z , 0) with z = (z , zn+1 ). It is obvious that p(z , 0) is H–type, so by induction σ should not have negative components, in contrast to the fact that −1 σq = |ν | νq < 0. This shows that there are no faces through the origin that do not lie on the coordinate hyperplanes. Now we prove that ν ∈ Rn+1 for all ν ∈ N1 (P) such that Fν intersects a + coordinate hyperplane, say zn+1 = 0. Set z = (z , zn+1 ) and ν = (ν , νn+1 ) and observe that p(z , 0) is H-type and the intersection of Fν with the hyperplane zn+1 = 0 is the face Fν of the Newton polyhedron of p(z , 0) with normal ν . By induction we have that ν ∈ Rn+ . So we have to prove that νn+1 > 0. Let p(z) =

m

j bj (z )zn+1

r

j=1

with r1 < r2 < . . . < rm . It is clear that the exponents of the monomials z α in b1 (z ) are such that α · ν ≤ 1 and that if α belongs to Fν , then α · ν = 1. Hence r1 = 0. Moreover, if γ is in Fν ∩ Nn+1 with γn+1 > 0, then there exists rj = γn+1 rj and γ is the exponent of a monomial in bj (z ). Now rj !bj (z ) = ∂zn+1 p(z , 0), thus by (iii) in Proposition 2.4.1 we have that bj (z )/b1 (z ) → 0 as z → ∞. In particular, if we set z = (a1 tν1 , . . . , an tνn ) and let t → ∞ for different choices of a1 , . . . , an , we obtain that γ · ν < 1. But then 1 = γ · ν = γ · ν + γn+1 νn+1 implies that νn+1 > 0. Finally we prove that ν ∈ Rn+1 for all ν ∈ N1 (P). In fact if this were + not the case, we would have that there exists ν ∈ N1 (P) with a non-positive component, say, νn+1 ≤ 0. Let w = (w , wn+1 ) be an internal point to Fν . Let Fσ , σ = (σ , σn+1 ), be any face intersecting the coordinate hyperplane zn+1 = 0. We must have w · σ ≤ 1 because w ∈ P and, because σn+1 > 0 by what we have just proved, we have w · σ < 1. So (w , 0) is internal to the intersection of P with the hyperplane zn+1 = 0 and hence, in particular, we must have w · ν < 1, but this is impossible because w · ν = 1 − wn+1 νn+1 ≥ 1, since νn+1 ≤ 0. Remark 2.6.4. The Newton polyhedron P may be complete without p(z) being Htype. For example p(x, ξ) = (x−ξ)2 , z = (x, ξ) ∈ R2 , is not H-type but its Newton polyhedron is the triangle with vertices (2, 0), (0, 2), (0, 0), which is complete. Example 2.6.5. As in the preceding Section 2.5 consider cγ z γ . p(z) = γ,M ≤m

100

Chapter 2. Γ-Pseudo-Differential Operators and H-Polynomials m/M

j If we assume quasi-ellipticity, the terms zj appear in the expression, with non-zero coefficients, cf. Remark 2.5.6. Hence the Newton polyhedron is given by

n # " Mj zj /m ≤ 1 , P = z ∈ Rn : z ≥ 0, j=1

which is complete, according to the previous Theorem 2.6.3, with vertices V (P) = {0, me1 /M1 , . . . , men /Mn }. We have μ0 = min{m/Mj }, μ1 = μ = m. The function ΛP (z) is given by 1+

n

2m/Mj

1/2

zj

m

|z|M

j=1

for large |z|, cf. Remark 2.5.6. The Γ-elliptic case, considered in Section 2.2, corresponds to taking M1 = . . . = Mn = 1 in the preceding formulas. Theorem 2.6.6. Let p(z) be a polynomial and let P be its Newton polyhedron. Define as in (2.6.2) 1/2 2γ ΛP (z) = z . γ∈V (P)

Then we have |p(z)| ΛP (z),

z ∈ Rn .

(2.6.5)

The proof is not so obvious as for (2.5.8), because of the lack of the quasihomogeneous structure. We need the following lemma. Lemma 2.6.7. Given z ∈ (R+ ∪ {0})n , a finite subset A ⊂ (R+ ∪ {0})n and a convex linear combination β = α∈A cα α, we have zβ ≤

cα z α .

(2.6.6)

α∈A

Proof. We argue by induction on the number N of the elements of A. If N = 1, the estimate (2.6.6) is obvious. If N > 1, choosing any γ ∈ A, let A = A \ {γ}. If cγ = 1, then cα = 0 for all α ∈ A and there is nothing to prove. If cγ < 1, then c = α∈A cα > 0. If we set γ = c1 α∈A cα α, we have β = c γ + (1 − c )γ. Let us consider the function f (t) = z tγ

+(1−t)γ

t = z γ (z γ )1−t .

We have that f ≥ 0, so that f is convex and

z β ≤ c z γ + (1 − c )z γ = c z γ + cγ z γ .


101

On the other hand, by the inductive hypothesis, we have zγ ≤ cα z α α∈A

with cα = cα /c . Therefore z β ≤ c

cα z α + cγ z γ =

α∈A

cα z α .

α∈A

Proof of Theorem 2.6.6. Observe first that |z γ | . ΛP (z)

(2.6.7)

γ∈V (P)

To prove (2.6.5), it will be sufficient to estimate every term of the type z β with Newton polyhedron P. Such β can be regarded as a convex β ∈ Nn belonging to the linear combination β = α∈A cα α with A = V (P). Hence applying Lemma 2.6.7 we obtain β z |z γ | . γ∈V (P)

In view of (2.6.7), this concludes the proof of Theorem 2.6.6

Having as model (2.5.9) in Proposition 2.5.2, we now define the multi-quasielliptic polynomials. Definition 2.6.8. We say that the polynomial p(z) is multi-quasi-elliptic if the corresponding Newton polyhedron is complete and for some R > 0: ΛP (z) |p(z)| ,

|z| ≥ R.

(2.6.8)

Theorem 2.6.9. Let p(z) be multi-quasi-elliptic. Then |∂zγ p(z)| |p(z)| ΛP (z)−|γ|/μ ,

|z| ≥ R,

(2.6.9)

for some R > 0, and p(z) is ρ-H–type for ρ = μ0 /μ

(2.6.10)

where μ0 and μ are the minimum and the formal order of P, cf. (2.6.3), (2.6.4). Proof. Let us show that, for each β ∈ Nn , β ∂ p(z) ΛP (z)1−|β|/μ ,

for all z ∈ Rn .

(2.6.11)

Let γ ∈ A with A given by (2.6.1), and consider β ∈ Nn . We have to prove that β γ ∂ z ΛP (z)1−|β|/μ .

102


If we do not have β ≤ γ, then ∂ β z γ = 0 and there is nothing to prove. Since μ ≥ μ1 ≥ |γ|, cf. (2.6.3), the conclusion is obvious also for β = γ. Assume that β < γ. Then for each ν ∈ N1 (P) we have 0 < (γ − β) · ν ≤ 1 − β · ν ≤ 1 − Hence

μ (γ − β) ∈ P μ − |β|

and, by Lemma 2.6.7, which gives

|β| . μ

μ γ−β μ−|β| z ΛP (z),

|∂ β z γ | ΛP (z)1−|β|/μ .

This completes the proof of (2.6.11). Using the multi-quasi-ellipticity assumption, from (2.6.11) we obtain (2.6.9). Finally, since zμ0 ΛP (z), we have from (2.6.9), |∂ γ p(z)| |p(z)| z−|γ|μ0 /μ ,

(2.6.12)

i.e., p(z) is ρ-H-type with ρ as in (2.6.10).

Example 2.6.10. Quasi-elliptic polynomials are multi-quasi-elliptic, since the estimate (2.5.9) is equivalent to (2.6.8), cf. (2.5.19) and Example 2.6.5. A simple example of multi-quasi-elliptic polynomial, which is not quasi-elliptic, is given for z = (x, ξ) ∈ R2 by ξ h1 + xk2 ξ h2 + xk1 , (2.6.13) where hj , kj , j = 1, 2, are positive integers, h1 > h2 , k1 > k2 , and The associated ordinary differential operator is P = Dxh1 + xk2 Dxh2 + xk1 .

k2 k1

+

h2 h1

> 1.

(2.6.14)

The Newton polyhedron is given by P = {(x, ξ) ∈ R2 : x ≥ 0, ξ ≥ 0,

x (k1 − k2 ) (h1 − h2 ) ξ + ξ ≤ 1, x+ ≤ 1}, k1 h2 k1 k2 h 1 h1 (2.6.15)

which is complete, with vertices V (P) = {(0, 0), (k1 , 0), (0, h1 ), (h2 , k2 )}. We have μ0 = min{h1 , k1 }, μ1 = h2 + k2 , h 2 k1 k2 h1 μ = max . , k1 − k2 h1 − h2

(2.6.16)

(2.6.17)


103

If we assume that hj , kj , j = 1, 2, are even, we have ΛP (x, ξ) ξ h1 + xk2 ξ h2 + xk1 ,

|x| + |ξ| ≥ R,

(2.6.18)

for every R > 0, and the polynomial in (2.6.13) is obviously multi-quasi-elliptic. An alternative definition of multi-quasi-ellipticity is given by the following proposition. Proposition 2.6.11. The polynomial p(z) is multi-quasi-elliptic if and only if the corresponding Newton polyhedron P is complete and there exists R > 0 such that ΛP (z) |pP,μ (z)| ,

|z| ≥ R,

(2.6.19)

where pP,μ (z), for short pμ (z) in the following, is the P-principal part of p(z): pμ (z) =

cγ z γ

(2.6.20)

Fν (P).

(2.6.21)

γ∈F (P)

with

F (P) =

ν∈N1 (P)

We recall that μ denotes the formal order of P and for ν ∈ N1 (P) we have Fν (P) = {z ∈ P : ν · z = 1}. We need the following lemma, whose proof is an obvious application of Lemma 2.6.7. Lemma 2.6.12. For every γ ∈ (R+ ∪ {0})n , in particular for all γ ∈ Nn , we have |z γ | ΛP (z)k(P,γ) where

(2.6.22)

k(P, γ) = inf{t > 0, t−1 γ ∈ P} = max ν · γ. ν∈N1 (P)

Proof of Proposition 2.6.11. We split p(z) = pμ (z) + p˜(z) with

p˜(z) =

cγ z γ .

γ∈P\F (P)

From Lemma 2.6.12 we have for some C > 0 |˜ p(z)| ≤ CΛP (z)δ ,

z ∈ Rn ,

with δ=

max

γ∈P\F (P)

k(P, γ) < 1,

(2.6.23)

104


in view of (2.6.20), (2.6.21) and (2.6.22), (2.6.23). For any fixed > 0, we then obtain in the region |z| ≥ R: |p(z) − pμ (z)| = |˜ p(z)| ≤ ΛP (z) provided R is so large that ΛP (z)δ−1 ≤ C −1 in the same region. Taking sufficiently small , we get the equivalence of (2.6.19) and (2.6.8). Hence Proposition 2.6.11 is proved. We may now explain the name “multi-quasi-elliptic polynomials” attributed to polynomials satisfying (2.6.19), i.e., (2.6.8). Proposition 2.6.13. Let p(z) be a polynomial of the variable z = (x, ξ) ∈ R2 . Assume p(z) is multi-quasi-elliptic, with complete Newton polyhedron P. Then we can always find a product of quasi-elliptic polynomials of type (2.5.13), whose P-principal part coincides with that of p(x, ξ); precisely:

p(x, ξ) = c0 (ξ h1 + r1 xk1 ) · · · (ξ hM + rM xkM ) +

cαβ xβ ξ α ,

(2.6.24)

(β,α)∈P\F (P)

for suitable positive integers hj , kj prime to each other and complex constants c0 , rj , cαβ , with c0 = 0, Im rj = 0 for j = 1, . . . , M . Correspondently, P can be decomposed into an algebraic sum of triangles: P = Ph1 ,k1 + . . . + PhM ,kM ,

(2.6.25)

where Phj ,kj has vertices {(0, 0), (0, hj ), (kj , 0), j = 1, . . . , M }. Proof. To prove (2.6.24), (2.6.25) we consider the principal part pμ of p, by ordering (β, α) ∈ F (P): M ai xβi ξ αi pμ (x, ξ) = i=0

so that αM > αM −1 > . . . > α0 = 0, 0 = βM < βM −1 < . . . < β0 . Note that a0 = 0, aM = 0. We begin to determine the integers h, k, with no common factor, and t ≥ 1 by means of the conditions k = |βM − βM −1 | = . . . = |βM −t+1 − βM −t | , h = αM − αM −1 = . . . = αM −t+1 − αM −t , k = |βM −t − βM −t−1 | or else h = αM −t − αM −t−1 . We can then write pμ (x, ξ) = q(x, ξ)(ξ h + rxk ) +

(β,α)∈P\F (P)

c˜β xβ ξ α ,

2.6. Multi-Quasi-Elliptic Polynomials with q(x, ξ) =

M −1

105

bi xβi ξ αi −h ,

i=0

where r, bi ∈ C are obtained by solving the system ⎧ aM = rbM −1 ⎪ ⎪ ⎪ ⎪ ⎪ a M −1 = bM −1 + rbM −2 ⎪ ⎪ ⎪ ⎪ ⎪ ··· ⎨ aM −t+1 = bM −t+1 + rbM −t ⎪ ⎪ ⎪ aM −t = bM −t ⎪ ⎪ ⎪ ⎪ ⎪ ··· ⎪ ⎪ ⎩ a0 = b0 and c˜αβ are suitable complex constants. Writing respectively Q and Ph,k for the Newton polyhedron of q(x, ξ) and ξ h + rxk , we have P = Q + Ph,k . It is easy to see that the multi-quasi-ellipticity of p implies the multi-quasi ellipticity of q and ξ h + rxk , which in turn forces Im r = 0, cf. Example 2.5.4. Iterating the procedure, we get (2.6.24), (2.6.25). Example 2.6.14. In Rn , n > 2, there exist multi-quasi-elliptic polynomials which cannot be factorized in terms of quasi-elliptic polynomials. Consider for example in R4 , with z = (x1 , x2 , ξ1 , ξ2 ), p(x, ξ) = ξ12 + ξ22 + V (x1 , x2 ),

(2.6.26)

corresponding to the operator in R2 , − + V (x1 , x2 ).

(2.6.27)

Let us assume that the potential V is of the form (2.6.13) with respect to the variables x1 , x2 , say to be definite V (x1 , x2 ) = x61 + x41 x42 + x62 .

(2.6.28)

Then p(x, ξ) in (2.6.26) is multi-quasi-elliptic, but it cannot be factorized as before. Finally, we observe that multi-quasi-elliptic polynomials do not exhaust the class of the H-polynomials, as the following example shows. Example 2.6.15. Let, for (x, ξ) ∈ R2 , p(x, ξ) = ξ 2 + ixξ − x3 ,

(2.6.29)


106

corresponding in the standard quantization to the ordinary differential operator P = Dx2 + ixDx − x3 .

(2.6.30)

We have that p is not multi-quasi-elliptic. In fact, let P be the Newton polyhedron of p. Then P is complete, with vertices (0, 0), (3, 0), (0, 2). Assuming p multi-quasi-elliptic, we would have |p(x, ξ)|2 = (ξ 2 − x3 )2 + x2 ξ 2 1 + ξ 4 + x6 for x2 + ξ 2 ≥ R > 0. But this is contradicted on the curve ξ 2 = x3 . Let us show that p(x, ξ) is H-type. Using Proposition 2.4.4 (c), we may limit ourselves to checking that ∂x p(x, ξ)/p(x, ξ) → 0

and

∂ξ p(x, ξ)/p(x, ξ) → 0

(2.6.31)

for (x, ξ) → ∞. In turn, (2.6.31) reduces readily to prove that 2

ξ 2 / |p(x, ξ)| → 0,

2

x4 / |p(x, ξ)| → 0

as (x, ξ) → ∞.

(2.6.32)

If x < 0 the proof of (2.6.32) is easy. Assume x ≥ 0 and, without loss of generality, also ξ ≥ 0. We discuss the validity of (2.6.32) separately in three regions: " "1 " # 1 # 1 # D1 = ξ 2 ≤ x3 , D2 = x3 ≤ ξ 2 ≤ 2x3 , D3 = x3 ≤ ξ 2 . 2 2 2 In D1 we have In D2 we estimate In D3 we have

2

|p(x, ξ)| ≥ (ξ 2 − x3 )2 x6 . 2

|p(x, ξ)| ≥ x2 ξ 2 x5 ξ 10/3 . 2

|p(x, ξ)| ≥ (ξ 2 − x3 )2 ξ 4 .

Taking into account that ξ 2 x3 in D1 and x4 ξ 8/3 in D3 , we obtain (2.6.32).

2.7 ΓP -Pseudo-Differential Operators Quasi-elliptic and multi-quasi-elliptic polynomials in R2d are Γρ -hypoelliptic symd bols in Γm ρ (R ) for some ρ with 0 < ρ ≤ 1, see Definition 2.1.15, Proposition 2.5.3 and Theorem 2.6.9. We may then apply Theorem 2.1.16 to the corresponding operators, obtaining in particular global regularity. However, the algebraic estimates (2.5.10) and (2.6.9) suggest that we may organize a more precise symbolic calculus, provided |z|M and ΛP (z) are admissible weight functions. Namely, we have

2.7. ΓP -Pseudo-Differential Operators

107

to check that Φ(z) = Ψ(z) = |z|M , or Φ(z) = Ψ(z) = ΛP (z)ρ , for some range of exponents ρ, are sub-linear and temperate weights, cf. (1.1.1), (1.1.2), satisfying the strong uncertainty principle (1.1.10). To this end, re-starting from scratch, we consider a generic complete polyhedron P ⊂ (R+ ∪ {0})n , cf. Definition 2.6.2, and we define as in (2.6.2): 1/2 2γ z , z ∈ Rn , (2.7.1) ΛP (z) = γ∈V (P)

where V (P) is the set of the vertices of P. As we observed after Definition 2.6.2 we have zμ0 ΛP (z) zμ1 , z ∈ Rn , (2.7.2) where μ0 and μ1 are the minimum and the maximum order of P, 0 < μ0 ≤ μ1 . Another obvious remark is that, for every t ∈ R, ΛP (tz) ≤ CΛP (z),

z ∈ Rn ,

(2.7.3)

where the constant C depends on t. The next lemma summarizes the main properties of the function ΛP (z). We use below the notation of Section 2.6, in particular the formal order μ of P is defined by "1 # μ = max : j = 1, . . . , n, ν ∈ N1 (P) . (2.7.4) νj Lemma 2.7.1. Let P be a complete polyhedron in Rn and let ΛP (z) be defined as in (2.7.1). Then for every α ∈ Nn , β ∈ Nn the following estimate holds: α α+β ξ ∂ ΛP (z) ΛP (z)1−|β|/μ , z ∈ Rn , (2.7.5) where μ is the formal order of P. Proof. We argue by induction on k = |α + β|. For k = 0 the estimate (2.7.5) is trivially satisfied with C0,0 = 1. For a fixed k ∈ N, let us assume that (2.7.5) holds for any α, β ∈ Nn with |α + β| ≤ k. Consider now α, β ∈ Nn such that |α + β| = k + 1. From (2.7.1) we obtain:

2χ α+β 2 z 2χ−α−β . (ΛP (z) ) = (α + β)! (2.7.6) ∂ α+β χ∈V (P) 2χ≥α+β

So, by Leibniz’ formula, we get ∂

α+β

ΛP (z) =

1 2ΛP (z)

2χ z 2χ−α−β (α + β)! α + β χ∈V (P)

2χ≥α+β

−

δ≤β,η≤α (η,δ)=(0,0), (η,δ)=(α,β)

) α β η+δ α−η+β−δ ∂ ΛP (z)∂ ΛP (z) , δ η

(2.7.7)


108 whence

β α+β z ∂ ΛP (z) ≤

1 2ΛP (z)

2χ−α 2χ z (α + β)! α + β χ∈V (P)

2χ≥α+β

+

δ≤β,η≤α (η,δ)=(0,0), (η,δ)=(α,β)

) β−δ α−η+β−δ α β δ η+δ z ∂ ΛP (z) z ∂ ΛP (z) . (2.7.8) δ η

From inductive assumption, we have δ η+δ |η| z ∂ ΛP (z) ΛP (z)1− μ , and

z ∈ Rn

β−δ α−η+β−δ |α−η| z ∂ ΛP (z) ΛP (z)1− μ ,

z ∈ Rn .

(2.7.9) (2.7.10)

We can prove now that 2χ−α |α| z ΛP (z)2− μ ,

z ∈ Rn .

(2.7.11)

The argument follows closely the proof of (2.6.11). In fact, if 2χ = α (β = 0), z 2χ−α ≡ 1 and |α| = 2 |χ| ≤ 2μ1 ≤ 2μ (μ1 := maxχ∈V (P) |χ|), so that |α|

ΛP (z)2− μ ≥ 1 and the inequality (2.7.11) is trivially verified. Let us consider now the case when 2χ > α. As χ ∈ V (P) ⊂ P, we have χ · ν ≤ 1 and, from the definition of μ, α · ν ≥ μ1 |α|, when ν ∈ N1 (P). Since 2μ − |α| > 0, μ the previous inequalities yield 2μ−|α| (2χ − α) · ν ≤ 1, as ν ∈ N1 (P), and then μ μ (2χ − α) ∈ P. So z 2χ−α 2μ−|α| ≤ ΛP (z), whence the estimate (2.7.11) 2μ−|α| follows. So estimates (2.7.9), (2.7.10) and (2.7.11), jointly with (2.7.8), give (2.7.5) for |α + β| = k + 1 and conclude the proof. Let us detail some consequences of Lemma 2.7.1. Through the following we will set for brevity K := {γ ∈ Nn : γj ∈ {0, 1}, j = 1, . . . , n}. Proposition 2.7.2. Let ΛP (z) be defined as in (2.7.1). Then for any m ∈ R, α ∈ Nn , γ ∈ K we have γ α+γ z ∂ ΛP (z)m ΛP (z)m−|α|/μ , z ∈ Rn . (2.7.12) Proof. By induction one can easily prove the estimates m |z γ ∂ α+γ ΛP (z) | |α+γ|

≤

k=1

Cm,k ΛP (z)m−k

Cα,γ,k z γ1 ∂ α1 +γ1 ΛP (z) · · · z γk ∂ αk +γk ΛP (z) ,


109

for any α ∈ Nn and γ ∈ K with |α + γ| > 0. For any 1 ≤ k ≤ |α + γ|, the multi-indices in the second sum of the right-hand side span over all the 2k-tuples (γ1 , . . . , γk , α1 , . . . αk ) ∈ Kk ×(Nn )k such that γ1 +. . .+γk = γ and α1 +. . .+αk = α, cf. the proof of Lemma 1.3.5. In order to complete the proof, it suffices now to apply the estimates (2.7.5) to each term |z γJ ∂ αJ +γJ ΛP (z)|, J = 1, . . . , k. Proposition 2.7.3. For ΛP (z) defined as before, there exist positive constants C and such that for all z, ζ ∈ Rn : 1 1 (i) ΛP (z) μ − ΛP (ζ) μ ≤ C |z − ζ|; (ii) C −1 ≤

ΛP (z) ΛP (ζ)

1

≤ C if |z − ζ| ≤ ΛP (z) μ .

Proof. By means of Taylor’s expansion we have 1

1

ΛP (ζ) μ − ΛP (z) μ =

1 1 (ζ − z)α ∂ α ΛP (θ) μ , α!

(2.7.13)

|α|=1

where θ = tζ + (1 − t)z, for some 0 < t < 1. Then: 1 1 1 1 |ζ − z| ∂ α ΛP (θ) μ , ΛP (ζ) μ − ΛP (z) μ ≤ α!

(2.7.14)

|α|=1

1 proving assertion (i), since for |α| = 1, ∂ α ΛP (θ) μ ≤ cα in view of (2.7.12). 1

Let us assume now that |ζ − z| ≤ ΛP (z) μ . From (2.7.14) again, we have 1 1 1 ΛP (ζ) μ − ΛP (z) μ ≤ C1 ΛP (z) μ that is (1 − C1 )μ ΛP (z) ≤ ΛP (ζ) ≤ (1 + C1 )μ ΛP (z), which for a suitable > 0 shows (ii). Corollary 2.7.4. Let 0 < ρ ≤ μ1 . Then Φ(z) = Ψ(z) = ΛP (z)ρ , z = (x, ξ) ∈ R2d , are sub-linear and temperate weights, cf. (1.1.1), (1.1.2), satisfying the strong uncertainty principle (1.1.10). Proof. Since in (2.7.1) 0 ∈ V (P) (cf. (2) in Definition 2.6.2), we have ΛP (z) ≥ 1. More precisely, ΛP (z) zμ0 with μ0 > 0, cf. (2.7.2). Sub-linear growth in (1.1.1) follows from (i) in Proposition 2.7.3, by fixing there ζ = 0. It is clear that it suffices to verify the temperance estimate (1.1.2) for a fixed power of ΛP (z). Now, by (i) in Proposition 2.7.3, 1

1

ΛP (ζ) μ ΛP (z) μ + |z − ζ| ,

z ∈ R2d , ζ ∈ R2d , 1

(2.7.15) 1

which gives the temperance estimate (1.1.2) for ΛP (z) μ when |z − ζ| ΛP (z) μ . 1 On the other hand, when ΛP (z) μ |z − ζ|, (2.7.15) yields 1

1

ΛP (ζ) μ |z − ζ| ≤ ΛP (z) μ (1 + |z − ζ|),


110 1

1

since ΛP (z) μ ≥ 1. Hence (1.1.2) is proved for ΛP (z) μ . Finally, since ρ > 0 the strong uncertainty principle (1.1.10) is satisfied as well, by the lower bound in (2.7.2). Remark 2.7.5. Note that the slow variation conditions (1.3.8) are also satisfied by Φ(z) = Ψ(z) = ΛP (z)ρ , and M (z) = ΛP (z)m , m ∈ R. Indeed they reduce to prove that there exists > 0 such that ΛP (z) ΛP (ζ) ΛP (z)

1

|ζ − z| ≤ ΛP (z) μ ,

for

(2.7.16)

which coincides with (ii) in the previous Proposition 2.7.3. Example 2.7.6. As in Section 2.5, let M = (M1 , . . . , Mn ) be an n-tuple of rational numbers Mj ≥ 1 with minj Mj = 1, and define |z|M =

n

|zj |

1/Mj

.

(2.7.17)

j=1

We may choose m ∈ N in such a way that all the m/Mj , Mj = 1, . . . , n, are integers. Then we have from Remark 2.5.6 and Example 2.6.5: n 1/2 2m/Mj |z|m Λ (ζ) = 1 + z , P j M

|z| ≥ R > 0,

(2.7.18)

j=1

with

n # " Mj zj /m ≤ 1 . P = z ∈ Rn : z ≥ 0,

(2.7.19)

j=1

For such P the formal order μ is given by m. 1 In view of Corollary 2.7.4, taking Φ(z) = Ψ(z) = ΛP (z) m |z|M we obtain a couple of sub-linear and temperate weights. Note that slow variation and temperance can be deduced for |z|M directly from (2.7.17); however, in view of 1 Proposition 2.7.2, the equivalent function ΛP (z) m satisfies the estimates (2.7.12), providing more precise information. In the sequel, ΛP (z) is defined as in (2.7.1), corresponding to a complete polyhedron P in Rn , with n = 2d. We understand z = (x, ξ), x ∈ Rd , ξ ∈ Rd . As before μ denotes the formal order of P. d Definition 2.7.7. Let m ∈ R and 0 < ρ ≤ μ1 . We denote by Γm ρ,P (R ), or for short m ∞ 2d 2d Γρ,P , the class of functions a ∈ C (R ) such that, for all α ∈ N ,

|∂ α a(z)| ΛP (z)m−ρ|α| , m We also write Γm P for the class Γ 1 ,P . μ

z ∈ R2d .

(2.7.20)


111

In view of Corollary 2.7.4, we may develop a symbolic calculus for the corresponding pseudo-differential operators, by applying the results of Chapter 1. Note that we assume ρ > 0, and the strong uncertainty principle (1.1.10) is satisfied, cf. (2.7.2). We may also define Sobolev spaces corresponding to ΛP (z) and obtain Fredholmness on them for the ΛP -elliptic operators. The results can be obviously particularized to the case ΛP (z) |z|m M . We leave all this to the reader. m In the following we prefer to limit attention to M Γm ρ,P , subclass of Γρ,P , whose definition below is suggested by the sharp estimates (2.7.12). On the one hand, parametrices of operators with multi-quasi-elliptic polynomials as symbols belong to the corresponding pseudo-differential subclass OPM Γm ρ,P , so nothing is lost in the application. On the other hand, operators in the subclass OPM Γ0ρ,P turn out to be bounded on Lp (Rd ), 1 < p < ∞, as we shall prove in Section 2.8. d Definition 2.7.8. For m ∈ R and 0 < ρ ≤ μ1 , we denote by M Γm ρ,P (R ), for short ∞ 2d M Γm ρ,P , the class of the functions a(z) ∈ C (R ) such that

z γ ∂ γ a(z) ∈ Γm ρ,P

(2.7.21)

for any γ ∈ K = {γ ∈ N2d : γj ∈ {0, 1}, j = 1, . . . , 2d}. We also write M Γm P for . the class M Γm 1 ,P μ

Example 2.7.9. Given a complete polyhedron P in R2d , consider the polynomial cβ z β , cβ ∈ C, z ∈ R2d . (2.7.22) p(z) = β∈P

We have p ∈ M Γ1P . In fact, denoting by Q the Newton polyhedron of p(z), we have Q ⊂ P, hence ΛQ (z) ≤ ΛP (z). From Theorem 2.6.6 we then obtain |p(z)| ΛQ (z) ≤ ΛP (z),

z ∈ R2d .

From the first part of the proof of Theorem 2.6.9, cf. in particular (2.6.11), we have the more precise information |∂ α p(z)| ΛP (z)1−|α|/μ ,

z ∈ R2d .

This shows that p ∈ Γ1P . On the other hand, for any γ ∈ N2d we have z γ ∂ γ p(z) = cγβ z β β∈P

for new coefficients cγβ ∈ C. The preceding arguments give then z γ ∂ γ p(z) ∈ Γ1P . Hence we obtain p ∈ M Γ1P . Note that this conclusion does not require the multiquasi-ellipticity of p(z) in (2.7.22). We list some propositions; they are obvious variants of the standard symbolic calculus and proofs are omitted.


112

Proposition 2.7.10. The following statements are equivalent: (1) a(z) ∈ M Γm ρ,P ; m−ρ|α|

(2) z γ ∂ α+γ a(z) ∈ Γρ,P

for all α ∈ N2d and γ ∈ K;

(3) for every α ∈ N2d and γ ∈ K, |z γ ∂ α+γ a(z)| ΛP (z)m−ρ|α| , Proposition 2.7.11. For m, m ∈ R, 0 < ρ ≤ ρ ≤

1 μ

z ∈ R2d .

(2.7.23)

the following properties hold:

m (1) if m ≤ m , then M Γm ρ,P ⊂ M Γρ ,P ;

m+m m ; (2) if a(z) ∈ M Γm ρ,P and b(z) ∈ M Γρ,P , then (ab)(z) ∈ M Γρ,P m−ρ|α|

α (3) if a(z) ∈ M Γm ρ,P , then ∂ a(z) ∈ M Γρ,P

Proposition 2.7.12. Let m ∈ R and 0 < ρ ≤

1 μ.

, for any α ∈ N2d .

Then the following inclusions hold:

m 0 Γm−N ⊂ M Γm ρ,P ⊂ Γρ,P , ρ,P

where N0 := 2d

1 μ0

(2.7.24)

− ρ , cf. (2.7.2). m

j , Definition 2.7.13. We say that a sequence {aj }j∈N of symbols aj (z) ∈ M Γρ,P such that mj > mj+1 and limj→∞ mj = −∞, is an asymptotic expansion for 0 a(z) ∈ M Γm ρ,P and write a(z) ∼ aj (z),

if, for any integer N ≥ 1, a(z) −

j

j 0, (2.7.25) |a(z)| ΛP (z)m , |z| ≥ R. We will write EΓm ρ,P for the class of the P-elliptic symbols of order m and m m m m m m EM Γm ρ,P = EΓρ,P ∩ M Γρ,P . We also set EΓP = EΓ1/μ,P , EM ΓP = EM Γ1/μ,P . Example 2.7.16. We know from Example 2.7.9 that the polynomial cβ z β , cβ ∈ C, z ∈ R2d , p(z) = β∈P


113

can be regarded as an element of M Γ1P . The P-ellipticity assumption in (2.7.25), |p(z)| ΛP (z),

|z| ≥ R,

is then equivalent to the multi-quasi-ellipticity of p(z) in the sense of Definition 2.6.8. Concerning the P-elliptic symbols we have the following proposition; since it has a key role in the following, we shall give the details of the proof. ∞ 2d Proposition 2.7.17. Consider a(z) ∈ EΓm ρ,P and let ψ(z) be a function in C (R ) which is identically zero for |z| ≤ R and identically 1 for |z| ≥ R , with 0 < ψ(z) −m m R < R sufficiently large. Then ψ(z) a(z) ∈ EΓρ,P . If a(z) ∈ EM Γρ,P , then a(z) ∈ EM Γ−m ρ,P .

Proof. Since a(z) fulfills estimate (2.7.25) with a suitable positive R, taking any ∈ ψ(z) satisfying the prescribed assumptions with R > R > R, we have ψ(z) a(z) ∞ 2d C (R ). The first statement then follows from Lemma 1.3.5. ∈ Γ−m For the second one, it suffices to prove that z γ ∂ γ ψ(z) ρ,P for any non-zero a(z) γ ∈ K. By use of Leibniz’ rule and by induction we see that for γ = 0, γ ∈ K, zγ ∂ γ

|ν| z γ ∂ γ ψ(z) ψ(z) z γ−ν ∂ γ−ν ψ(z) Ck,ν,ν 1 ,...,ν k = + a(z) a(z) a(z)k+1 0=ν≤γ k=1

1

1

k

k

× z ν ∂ ν a(z) . . . z ν ∂ ν a(z),

(2.7.26)

where, for any ν ≤ γ and 1 ≤ k ≤ |ν|, the last sum in the right-hand side is taken over all the k-tuples (ν 1 , . . . , ν k ) ∈ K × . . . × K such that ν 1 + . . . + ν k = ν, while Ck,ν,ν 1 ,...,ν k are suitable constants depending on ν ≤ γ, ν 1 , . . . , ν k and k. γ−ν

−m(k+1)

γ−ν

z ∂ ψ(z) ∈ Γρ,P for every ν ≤ γ, k ≥ 0 Since a(z) ∈ EΓm ρ,P , then a(z)k+1 −∞ 2d (it belongs in particular to Γρ,P = S(R ) when ν < γ); on the other hand, J

J

z ν ∂ ν a(z) ∈ Γm ρ,P for J = 1, . . . , k. This just shows that z γ ∂ γ ψ(z) ∈ Γ−m ρ,P and completes the proof. a(z)

Remark 2.7.18. As an immediate consequence of Propositions 2.7.2 and 2.7.10, we see that any real power of a weight function ΛP (z)m is a P-elliptic symbol of order m; more precisely ΛP (z)m ∈ EM Γm for any m ∈ R. 1 ,P μ

We pass then to consider the related classes of pseudo-differential operators. From now on we split z = (x, ξ), x ∈ Rd , ξ ∈ Rd . Firstly, we list some notions and related results which will be useful in the sequel. We shall associate to each symbol a(x, ξ) ∈ Γm ρ,P the pseudo-differential operator of τ -type, cf. (1.2.1): ¯ dξ, ¯ u ∈ S(Rd ), (2.7.27) Au(x) = ei(x−y)ξ a((1 − τ )x + τ y, ξ)u(y) dy

114


where τ ∈ R is fixed. According to the results of Section 1.2, different choices of τ define the same class of operators, which we shall denote by OPΓm ρ,P . We recall and a (x, ξ) and a (x, ξ) are the τ and the τ -symbol of A that if A ∈ OPΓm τ τ 1 2 1 2 ρ,P respectively, then aτ2 (x, ξ) ∼

1 (τ1 − τ2 )|α| ∂ξα Dxα aτ1 (x, ξ), α! α

(2.7.28)

see Remark 1.2.6. According to the notations of Section 1.2, a0 (x, ξ) corresponding to the standard quantization τ = 0 is called the left symbol of A, a1 (x, ξ) corresponding to τ = 1 the right symbol of A and a 12 (x, ξ) corresponding to τ = 12 the Weyl symbol of A. We refer to Section 1.2 for details concerning the pseudodifferential calculus in classes OPΓm ρ,P . Now we are interested in the operators in OPΓm ρ,P with τ -symbol a(x, ξ) ∈ M Γm . ρ,P Proposition 2.7.19. Let A ∈ OPΓm ρ,P and τ1 ∈ R. If the τ1 -symbol aτ1 (x, ξ) of A , then the τ -symbol aτ2 (x, ξ) also belongs to M Γm belongs to M Γm 2 ρ,P ρ,P , for every τ2 ∈ R. Proof. For any τ2 ∈ R, the τ2 -symbol of A can be expressed in terms of its τ1 symbol by means of the asymptotic expansion (2.7.28). By Proposition 2.7.11, ∂ξα Dxα aτ1 (x, ξ) ∈ M Γm−2ρN for |α| = N and then aτ2 (x, ξ) ∈ M Γm ρ,P by Proposiρ,P tion 2.7.12. m We set OPM Γm ρ,P for the subclass of OPΓρ,P consisting of the operators with τ m symbol in M Γρ,P . Thanks to the above proposition, OPM Γm ρ,P is independent of τ.

m Proposition 2.7.20. If A ∈ OPM Γm ρ,P and B ∈ OPM Γρ,P , then

; (1) AB ∈ OPM Γm+m ρ,P (2) tA ∈ OPM Γm ρ,P ; (3) A∗ ∈ OPM Γm ρ,P .

m Proof. Let a(x, ξ) ∈ M Γm ρ,P and b(x, ξ) ∈ M Γρ,P be the Weyl symbols of A and B

of C = AB we have respectively. Then for the Weyl symbol c(x, ξ) ∈ Γm+m ρ,P c(x, ξ) ∼

(−1)|β| α,β

α!β!

2−|α+β| ∂ξα Dxβ a(x, ξ)∂ξβ Dxα b(x, ξ),

cf. Theorem 1.2.17. In view of Proposition 2.7.11,

−2ρN ∂ξα Dxβ a(x, ξ)∂ξβ Dxα b(x, ξ) ∈ M Γm+m , ρ,P

(2.7.29)


115

for |α + β| = N , and then c(x, ξ) ∈ M Γm+m by Proposition 2.7.12. This proves ρ,P the first statement. For statements (2) and (3), it suffices to observe that the Weyl symbols t a(x, ξ) and a∗ (x, ξ) of t A and A∗ respectively are related to the Weyl symbol a(x, ξ) of A by the formulas t

a(x, ξ) = a(x, −ξ),

(2.7.30)

a∗ (x, ξ) = a(x, ξ),

(2.7.31)

∗

cf. (1.2.14), (1.2.16). Thus a(x, ξ), a (x, ξ) ∈ t

M Γm ρ,P .

Given a(x, ξ) ∈ Γm ρ,P , we may also define the operator Aa with Anti-Wick symbol a(x, ξ), cf. Section 1.7. As an easy consequence of the formula (1.7.17) for the asymptotic expansion of the Weyl symbol of A, we obtain the following result. m Proposition 2.7.21. If a(x, ξ) ∈ M Γm ρ,P , then Aa ∈ OPM Γρ,P .

We also recall from Section 1.3.1 that the notion of ellipticity does not depend on the quantization, cf. Proposition 1.3.4. Proposition 2.7.22. Let τ1 , τ2 ∈ R be given. If the τ1 -symbol aτ1 (x, ξ) of A ∈ m OPΓm ρ,P belongs to EΓρ,P , the same holds for the τ2 -symbol aτ2 (x, ξ). The parametrix of an operator with symbol a(x, ξ) ∈ EΓm ρ,P can be constructed as in Section 1.3.1. For the parametrix of an operator with symbol in EM Γm ρ,P the following holds: Proposition 2.7.23. Let P ∈ OPM Γm ρ,P be given with P-elliptic symbol. Then there exists an operator Q with symbol in EM Γ−m ρ,P such that P Q = I + R,

QP = I + S,

(2.7.32)

where R, S are regularizing and I is the identity operator. Proof. We know already from Section 1.3.1 that there exists Q ∈ OPΓ−m ρ,P with P-elliptic symbol satisfying the identities (2.7.32). ∞ Moreover, if p(x, ξ) ∈ EM Γm ρ,P is the Weyl symbol of P and ψ(x, ξ) is a C function as in Proposition 2.7.17 (with z = (x, ξ)) we may write Q = RB1 , where B1 is the operator with Weyl symbol b1 (x, ξ) = ψ(x,ξ) p(x,ξ) while R is an operator in OPΓ0ρ,P whose Weyl symbol r(x, ξ) has the following asymptotic expansion: r(x, ξ) ∼

(−1)j rj (x, ξ),

(2.7.33)

j≥0

where r1 (x, ξ) is the Weyl symbol of the operator R1 := B1 P − I and, for any j ≥ 0, rj (x, ξ) is the Weyl symbol of the j-th power R1j of R1 (see the proof of

116


Theorem 1.3.6). By Proposition 2.7.17, we know that b1 (x, ξ) ∈ M Γ−m ρ,P . Since moreover r1 (x, ξ) has the asymptotic expansion r1 (x, ξ) ∼

|α+β|>0

(−1)|β| −|α+β| α β ψ(x, ξ) β α 2 ∂ D p(x, ξ) ∂ξ Dx α!β! p(x, ξ) ξ x −2ρ|α+β|

β α and, for all α, β, ∂ξα Dxβ ψ(x,ξ) p(x,ξ) ∂ξ Dx p(x, ξ) ∈ M Γρ,P

(2.7.34)

, in view of Propositions

2.7.11 and 2.7.12, it follows that r1 (x, ξ) ∈ M Γ−2ρ ρ,P . From Proposition 2.7.20, we −2ρj deduce rj (x, ξ) ∈ M Γρ,P and, by virtue of (2.7.33), r(x, ξ) ∈ M Γ0ρ,P . So Q = RB1 ∈ OPM Γ−m ρ,P is the required parametrix.

Proposition 2.7.23 gives in particular the existence of a parametrix with sym−1 bol in EM Γ−1 P = EM Γ1/μ,P and the global regularity for the operators with polynomial coefficients P = cαβ xβ D α , (β,α)∈P

obtained by standard quantization from a multi-quasi-elliptic polynomial cγ z γ , z = (x, ξ), p(z) = γ∈P

cf. Examples 2.7.9 and 2.7.16 (actually, in view of Proposition 2.7.22, the multiquasi-ellipticity of the symbol classes does not depend on the quantization). On the other hand, global regularity for such operators follows from Theorem 2.1.16 and Theorem 2.6.9. The advantage of providing a parametrix of P with symbol in EM Γ−1 P will be evident in the next section.

2.8 Lp -Estimates As in the preceding section, P is a complete polyhedron in R2d with formal order μ and ΛP (z) is the corresponding weight function. For 0 < ρ ≤ 1/μ, m ∈ R, the m classes of symbols M Γm ρ,P and the classes of operators OPM Γρ,P are defined as 2d before. Namely, a(z) ∈ M Γm and γ ∈ K2d = {γ ∈ ρ,P means that for every α ∈ N 2d N : γj ∈ {0, 1}, j = 1, . . . , 2d}, |z γ ∂ α+γ a(z)| ΛP (z)m−ρ|α| ,

z = (x, ξ) ∈ R2d .

(2.8.1)

Theorem 2.8.1. Any operator A ∈ OPM Γ0ρ,P extends to a bounded operator from Lp (Rd ) to itself, for all 1 < p < ∞. Actually, to obtain Lp -boundedness, we shall use much weaker assumptions than a ∈ M Γ0ρ,P on the symbol of A. Namely it will be sufficient to assume for a(x, ξ) ∈ C ∞ (R2d ): |ξ γ ∂xλ ∂ξν+γ a(x, ξ)| ξ−|ν| ,

(x, ξ) ∈ R2d ,

(2.8.2)

2.8. Lp -Estimates

117

for some > 0, all λ ∈ Nd , ν ∈ Nd , γ ∈ Kd = {γ ∈ Nd : γj ∈ {0, 1}, j = 1, . . . , d}. Note that if (2.8.1) is satisfied with m = 0, then (2.8.2) is valid with = ρμ0 , where μ0 is the minimum order of P, cf. (2.7.2). The proof of Theorem 2.8.1 will use the classical result of Lizorkin and Marcinkiewicz concerning Fourier multipliers, that we begin to recall here below. Theorem 2.8.2. Let the function m(ξ) be continuous together with its derivatives ∂ξγ m(ξ), for any γ ∈ Kd . If there is a constant B > 0 such that |ξ γ ∂ξγ m(ξ)| ≤ B,

ξ ∈ Rd , γ ∈ Kd ,

(2.8.3)

then for every 1 < p < ∞ we can find a constant Ap > 0, depending only on p, B and the dimension d, such that

m(D)u Lp ≤ Ap u Lp for all u ∈ S(Rd ). Proof of Theorem 2.8.1. We use for A the left-symbol representation dξ, ¯ ϕ ∈ S(Rd ), Aϕ(x) = a(x, D)ϕ(x) = eixξ a(x, ξ)ϕ(ξ)

(2.8.4)

with a(x, ξ) ∈ M Γ0ρ,P . For m = (m1 , . . . , md ) ∈ Zd let us consider + * 1 Qm = x ∈ Rd : |xj − mj | ≤ , j = 1, 2, . . . , d , 2 + * 2 ∗ d Qm = x ∈ R : |xj − mj | ≤ , j = 1, 2, . . . , d , 3 * + d = x ∈ R : |x − m | ≤ 1, j = 1, 2, . . . , d , Q∗∗ j j m and cut-off functions ψm (x) ∈ C0∞ (Rd ) such that 0 ≤ ψm (x) ≤ 1, supp ψm ⊂ Q∗∗ m and ψm = 1 in Q∗m . Setting ϕ1,m = ψm ϕ and ϕ2,m = (1 − ψm )ϕ, we have |a(x, D)ϕ(x)|p dx ≤ 2p−1 {I1,m + I2,m }, (2.8.5) m∈Zd

where

p

|a(x, D)ϕi,m (x)| dx,

Ii,m =

i = 1, 2.

(2.8.6)

Qm

In fact, since Rd = ∪m∈Zd Qm and the measure of Qm ∩ Qn vanishes when n = m, we can write for ϕ(x) ∈ S(Rd ): p

a(x, D)ϕ pp = |a(x, D)ϕ(x)| dx. (2.8.7) m∈Zd

Qm


118

Moreover, for any m ∈ Zd , p |a(x, D)ϕ(x)| dx = Qm

p

|a(x, D)(ϕ1,m + ϕ2,m )(x)| dx

Qm

≤ 2p−1 {I1,m + I2,m }.

(2.8.8)

Therefore (2.8.5) is proved. To continue the proof, we shall use the following lemma. Lemma 2.8.3. Let a(x, ξ) ∈ M Γ0ρ,P , χ(x) ∈ C0∞ (Rd ) and set aχ (x, ξ) = χ(x)a(x, ξ); we can then consider the Fourier transform of aχ (x, ξ) with respect to the x variable ¯ (2.8.9) aχ (η, ξ) = e−ixη aχ (x, ξ) dx. For any N > 0 there is a positive constant CN,χ such that γ γ aχ (η, ξ) ≤ CN,χ (1 + |η|)−N , ξ ∂ξ

(2.8.10)

for all η, ξ ∈ Rd and γ ∈ Kd . Proof. Since aχ (x, ξ) has compact support with respect to the x variable, differentiation and integration by parts in (2.8.9) give β γ |β| e−ixη ∂xβ ∂ξγ aχ (x, ξ) dx, aχ (η, ξ) = (−i) ¯ (2.8.11) η ∂ξ where β is an arbitrary multi-index. β β γ γ aχ (η, ξ) ≤ η ξ ∂ξ ν

Then by Leibniz’ formula we get β−ν ∂x χ(x) ξ γ ∂xν ∂ γ a(x, ξ) dx. ¯ ξ

(2.8.12)

ν≤β

Since ξ γ ∂xν ∂ξγ a(x, ξ) ≤ Cγ,ν ΛP (x, ξ)−ρ|ν| ≤ Cγ,ν (cf. (2.8.1) or (2.8.2)), we obtain β γ γ aχ (η, ξ) ≤ Cβ,χ , η ξ ∂ξ

(2.8.13)

for every η, ξ ∈ Rd and γ ∈ Kd , where the positive constant on the right-hand side is given by β−ν β ∂x χ(x) dx. Cγ,ν ¯ (2.8.14) Cβ,χ = max γ∈Kd ν ν≤β

Thus by summing on |β| ≤ N we obtain the desired conclusion.

Continuing the proof of Theorem 2.8.1. In order to estimate I1,m in (2.8.5), (2.8.6), let us consider a cut-off function θ(x) ∈ C0∞ (Rd ) such that θ(x) = 1 for x ∈ Q0

2.8. Lp -Estimates

119

and define θm (x) = θ(x − m) for any m ∈ Zd . By setting now am (x, ξ) := θm (x)a(x, ξ) ∈ M Γ0ρ,P we get p p |am (x, D)ϕ1,m | dx ≤ |am (x, D)ϕ1,m | dx. (2.8.15) I1,m = Rd

Qm

If we set now χ(x) = θm (x) in Lemma 2.8.3, we can show that for every N > 0 there exists a positive constant CN such that γ γ am (η, ξ) ≤ CN (1 + |η|)−N η, ξ ∈ Rd , γ ∈ Kd . (2.8.16) ξ ∂ξ Notice that, from the definition of the functions θm (x) and (2.8.14), the constant CN in (2.8.16) depends on θ(x) but not on m ∈ Zd . Then by Theorem 2.8.2 for every N > 0 there exists MN > 0, which does not depend on m, such that

am (η, D)u Lp ≤ MN (1 + |η|)−N u Lp ,

(2.8.17)

for every u ∈ S(Rd ), η ∈ Rd and m ∈ Zd . On the other hand, from Fubini’s theorem we have ixξ u(ξ) dξ ¯ = eixη am (η, D)u(x) dη. ¯ (2.8.18) am (x, D)u(x) = e am (x, ξ) So by using (2.8.17) and Minkowski’s inequality in integral form, we have p p I1,m ≤ eixη am (η, D)ϕ1,m (x) dη ¯ dx ≤

am (η, D)ϕ1,m Lp dη ¯ p p p p ≤ MN

ϕ1,m pLp ¯ = MN CN ϕ1,m pLp , (2.8.19) (1 + |η|)−N dη where, for N > d, CN := (1 + |η|)−N dη ¯ < ∞. To estimate the second term I2,m we use the kernel representation, cf. (1.2.2), (1.2.3), (1.2.4): a(x, D)ϕ(x) = K0 (x, y)ϕ(y) dy (2.8.20) Rd

with K0 (x, y) = K(x, x − y), where, in the distribution sense, K(x, t) = (2π)−d eitξ a(x, ξ) dξ.

(2.8.21)

Rd

We need the following lemma. Lemma 2.8.4. If a ∈ M Γ0ρ,P , then for every sufficiently large positive integer N , there is a positive constant CN such that −N

|K(x, t)| ≤ CN |t|

,

t = 0.

(2.8.22)

120


Proof. Let ν ∈ Nd be an arbitrary multi-index. We obtain from (2.8.21) in the distribution sense ν −d eitξ ∂ξν a(x, ξ) dξ. (−it) K(x, t) = (2π) Rd

Since a ∈ M Γ0ρ,P , then (2.8.2) is valid. Hence for sufficiently large ν we have |tν | |K(x, t)| ≤ Cν ,

x ∈ Rd , t ∈ Rd ,

for a positive constant Cν . The estimates (2.8.22) follow immediately.

End of the proof of Theorem 2.8.1. By (2.8.20), (2.8.21) and Lemma 2.8.4, there is a constant C2N such that, for all x ∈ Qm , K(x, x − y)ϕ2,m (y)dy |a(x, D)ϕ2,m (x)| = d R = K(x, x − y)ϕ2,m (y)dy Rd \Q∗m −2N ≤ C2N |x − y| |ϕ2,m (y)| dy. Rd \Q∗ m

/ Q∗m : We may further estimate under the integral, for x ∈ Qm and y ∈ |x − y|−2N ≤ CN (λ + |x − y|)−N (λ + |m − y|)−N , for any fixed λ > 0 and for √a suitable positive constant CN (depending on λ). d Take in particular λ = 1 + 2 . By Minkowski’s inequality in integral form we obtain

1/p 1/p p |a(x, D)ϕ2,m (x)| dx I2,m = Qm

p )1/p (λ + |x − y|)−N |ϕ2,m (y)| ≤ dy dx (λ + |m − y|)N Qm Rd \Q∗ m 1/p p (λ + |x − y|)−N p |ϕ2,m (y)| ≤ C2N CN dx dy (λ + |m − y|)N p Rd \Q∗ Qm m 1/p |ϕ2,m (y)| −N p = C2N CN (λ + |x − y|) dx dy, (λ + |m − y|)N Rd \Q∗ Qm m C2N CN

so that for N sufficiently large and new constants CÑ > 0, 1/p ˜ I2,m ≤ CN (λ + |m − y|)−N |ϕ2,m (y)| dy. Rd \Q∗ m

2.8. Lp -Estimates

121

Applying Hölder’s inequality we easily deduce the following estimate: p |ϕ2,m (y)| I2,m ≤ HN,p N p dy, 2 Rd \Q∗ m (λ + |m − y|)

(2.8.23)

where HN,p is a positive constant depending only on N , p and the dimension d. d From the definition of ϕ1,m and Q∗∗ m , m ∈ Z , we obtain p

ϕ1,m pLp = |ϕ1,m (x)| dx ≤ Cd ϕ pLp , (2.8.24) m∈Zd

m∈Zd

Q∗∗ m

where the constant Cd > 0 depends only on the dimension d. Moreover p p |ϕ2,m (y)| |ϕ2,m (y)| N p dy ≤ N p dy 2 Rd \Q∗ Ql (λ + |m − y|) 2 m (λ + |m − y|) m∈Zd m∈Zd l=m 1 p ≤ |ϕ(y)| dy Np (1 + |m − l|) 2 Ql m∈Zd l∈Zd 1 = ϕ pLp (2.8.25) Np (1 + |m|) 2 m∈Zd 1 and m∈Zd N p < ∞ for suitably large N . By (2.8.6) and the estimates (1+|m|)

2

(2.8.5), (2.8.19), (2.8.23), (2.8.24), (2.8.25) we then get:

ϕ pLp ,

a(x, D)ϕ pLp ≤ Cp,d

(2.8.26)

which ends the proof.

We may now define L -Sobolev spaces related to a complete polyhedron P. p

Definition 2.8.5. For s ∈ R and 1 < p < ∞, HPs,p is the space of all the temperate distributions u ∈ S (Rd ) such that ΛsP (x, D)u ∈ Lp (Rd ). Here ΛsP (x, D) is the pseudo-differential operator with left symbol ΛP (x, ξ)s ∈ M Γs1 ,P , that is μ ΛsP (x, D)u(x) = eixξ ΛP (x, ξ)s u (ξ) dξ, ¯ u ∈ S(Rd ). Set HPp = HP1,p . Since ΛP (x, ξ)s is P-elliptic of order s (see Remark 2.7.18), ¯ −s (x, D) ∈ OPM Γ−s such in view of Proposition 2.7.23, there exists an operator Λ ρ,P that ¯ −s (x, D)Λs (x, D) = I + Rs , Λ (2.8.27) P 2

2

where Rs is regularizing. When ΛP (x, ξ) = (1 + |x| + |ξ| )1/2 and p = 2 the spaces HPs,p coincide with the spaces HΓs (Rd ) of Section 2.1. We impose a Banach topology on HPs,p by setting, for any u ∈ HPs,p ,

u s,p,P = ΛsP (x, D)u Lp + Rs u Lp , where Rs is defined by (2.8.27).

(2.8.28)

122


Proposition 2.8.6. HPs,p is a Banach space with respect to the norm · s,p,P defined by (2.8.28). Proof. In the following we write for short Λs = ΛsP (x, D) and Λ−s = Λ−s (x, D). Of course, · s,p,P is a semi-norm in HPs,p . Let us suppose u s,p,P = 0. By (2.8.28) it follows that Λs u = 0 and Rs u = 0; then u = Λ−s (Λs u) − Rs u = 0, in view of (2.8.27). To prove the completeness, let {uν }ν∈N be a Cauchy sequence in HPs,p with respect to · s,p,P . Then it follows that {Λs uν }ν∈N and {Rs uν }ν∈N are Cauchy sequences in Lp (Rd ); let v and w be their limits in Lp (Rd ), respectively. We prove now that {uν }ν∈N converges to u := Λ−s v − w in HPs,p . Setting vν := Λs uν and wν := Rs uν , we get Λs uν = Λs Λ−s vν − Λs wν . Since vν → v in Lp (Rd ) and Λs Λ−s ∈ OPM Γ01 ,P , Λs Λ−s vν → Λs Λ−s v in Lp (Rd ) in view of μ

Theorem 2.8.1. On the other hand, since Rs is a regularizing operator, uν → Λ−s v − w in S (Rd ) yields wν = Rs uν → Rs (Λ−s v − w) in S(Rd ). Thus, for the uniqueness of the limit, wν → w in S(Rd ) and then Λs wν → Λs w in S(Rd ). This shows that Λs u ∈ Lp (Rd ) and that Λs uν = Λs Λ−s vν − Λs wν → Λs Λ−s v − Λs w = Λs u in Lp (Rd ). The proof is therefore concluded. In fact the previous Sobolev spaces can be better described by means of any pseudo-differential operator with positive Anti-Wick elliptic symbol. Firstly we recall the following version of Proposition 1.7.12, cf. Theorem 2.1.7. Lemma 2.8.7. Let a(x, ξ) ∈ Γm ρ,P be P-elliptic of order m and assume that a(x, ξ) > 0, for any x, ξ ∈ Rd . Then the operator Aa with Anti-Wick symbol a is an isomor−m phism of S(Rd ) extending to an isomorphism of S (Rd ). Moreover A−1 a ∈ OPΓρ,P is P-elliptic of order −m. Proposition 2.8.8. Let a(x, ξ) ∈ M Γsρ,P be an arbitrary P-elliptic symbol of order s such that a(x, ξ) > 0, for any x, ξ ∈ Rd . Then for any 1 < p < ∞, p d d p d HPs,p = A−1 a (L (R )) = {u ∈ S (R ) : Aa u ∈ L (R )},

where Aa is the operator with Anti-Wick symbol a. Moreover a norm on HPs,p equivalent to (2.8.28) is given by Aa u Lp . Proof. Firstly suppose that Aa u ∈ Lp (Rd ) and write Λs u = Λs A−1 a (Aa u), with 0 Λs = ΛsP (x, D). Then Λs u ∈ Lp (Rd ) since Λs A−1 a ∈ OPM Γρ,P . Conversely let us take u ∈ S (Rd ) such that Λs u ∈ Lp (Rd ). By (2.8.27) we may write Aa u = Aa Λ−s (Λs u) − Aa Rs Λ−s (Λs u) + Aa Rs (Rs u). (2.8.29) So using Theorem 2.8.1, we conclude that Aa u ∈ Lp (Rd ) and there is a positive constant C independent of u such that

Aa u Lp ≤ C( Λs u Lp + Rs u Lp ),

(2.8.30)

2.8. Lp -Estimates

123

since Aa Λ−s ∈ OPM Γ0ρ,P and Aa Rs Λ−s , Aa Rs are regularizing. On the other hand, since Aa is an isomorphism of S (Rd ), Aa u Lp is a norm on HPs,p and it is easy to show that HPs,p is complete with respect to it; thus the equivalence with the norm (2.8.28) follows from the Open Mapping Theorem. Remark 2.8.9. We may as well reset Definition 2.8.5 by replacing ΛsP (x, D) with any elliptic pseudo-differential operator of order s. Namely, for any T with Pelliptic symbol in M Γsρ,P , HPs,p = {u ∈ S (Rd ) : T u ∈ Lp (Rd )}.

(2.8.31)

If moreover Q is a parametrix of T and QT = I + R, then T u Lp + Ru Lp is a norm in HPs,p equivalent to (2.8.28). Remark 2.8.10. Notice also that for any t < s and 1 < p < ∞ the following inclusions hold: S(Rd ) ⊂ HPs,p ⊂ HPt,p ⊂ S (Rd ), with compact embeddings. Using the above description of the spaces HPs,p we can plainly deduce the action of pseudo-differential operators in classes OPM Γm ρ,P . Namely the following holds. Proposition 2.8.11. If A ∈ OPM Γm ρ,P , then A : HPs+m,p → HPs,p , continuously, for all s ∈ R and 1 < p < ∞. Proof. In the following we set Gt for the operator with Anti-Wick symbol ΛP (x, ξ)t ∈ M Γt1 ,P ; since ΛP (x, ξ)t is positive and P-elliptic of order t, in view of Proposition μ

p d 2.8.8 we have HPt,p = G−1 t (L (R )) for any 1 < p < ∞. −1 Since Q := Gs AGs+m ∈ OPM Γ0ρ,P and, for any u ∈ HPs+m,p , Gs+m u ∈ Lp (Rd ), then Gs Au = Q(Gs+m u) ∈ Lp (Rd ); moreover there is a positive constant C such that

Gs Au Lp = Q(Gs+m u) Lp ≤ C Gs+m u Lp .

This shows the continuity of A and concludes the proof.

Using the results of the preceding section, we may now obtain precise results of Lp -regularity for the solutions of the P-elliptic equations. Theorem 2.8.12. Let a(x, ξ) ∈ M Γm ρ,P be P-elliptic, cf. Definition 2.7.15. Let A be the pseudo-differential operator with symbol a, in the standard left quantization. Let 1 < p < ∞ and s ∈ R. If u ∈ S (Rd ) and Au ∈ HPs,p , then u ∈ HPs+m,p . Moreover for any t < s + m there is a positive constant C, independent of u, for which (2.8.32)

u s+m,p,P ≤ C( Au s,p,P + u t,p,P ).


124

Proof. From Proposition 2.7.23 there exists an operator B ∈ OPM Γ−m ρ,P such that BA = I + R with R regularizing. For any u ∈ S (Rd ) we have then u = BAu − Ru. If Au ∈ HPs,p we get BAu ∈ HPs+m,p in view of Proposition 2.8.11, whereas Ru ∈ S(Rd ), hence u ∈ HPs+m,p . The estimate (2.8.32) follows, because of the continuous embedding S(Rd ) ⊂ HPt,p . From the theory of the Fredholm operators in the Banach spaces, cf. Remark 1.6.8, we have also readily the following theorem. To be precise, we shall denote here by As , s ∈ R, the restriction of A : S (Rd ) → S (Rd ) to HPs,p , or equivalently the extension of A : S(Rd ) → S(Rd ) to HPs,p . Theorem 2.8.13. Consider A ∈ OPM Γm ρ,P with P-elliptic symbol. Then, for 1 < p < ∞, s ∈ R: (i) As ∈Fred (HPs,p , HPs−m,p ); (ii) ind As is independent of s ∈ R and 1 < p < ∞;

(iii) If T ∈ OPM Γm with m < m, then As + Ts ∈ Fred (HPs,p , HPs−m,p ) and ind (As + Ts ) = ind As . Similarly to Theorem 2.1.12, we have a more explicit definition of HPs,p when s ∈ N; let us fix attention for simplicity on the case s = 1. Theorem 2.8.14. Writing HPp for the space HP1,p , 1 < p < ∞, we have: * + HPp = u ∈ S (Rd ) : xβ D α u ∈ Lp (Rd ), (β, α) ∈ V (P) with equivalent norm

xβ Dα u Lp ,

(2.8.33) (2.8.34)

(β,α)∈V (P)

where V (P) is the set of vertices of P. Proof. Let us assume that u belongs to HP1,p , 1 < p < ∞; since xβ Dxα ∈ OPM Γ11 ,P μ

as long as (β, α) ∈ V (P), we have that

xβ Dα u Lp ≤ Cα,β u 1,p,P

for a positive constant Cα,β independent of u ( · 1,p,P is the norm (2.8.28) in HP1,p ). Conversely, we argue similarly to the proof of Theorem 2.1.12. Namely, we consider the operator T = W−1 P where W−1 is any P-elliptic operator in OPM Γ−1 ρ,P , 0 < ρ ≤ μ1 (for instance the pseudo-differential operator with left symbol ΛP (x, ξ)−1 ) β α β α and P is the differential operator P = (β,α)∈V (P) x Dx (x Dx ). Since P ∈ 2 OPM Γ 1 ,P and is P-elliptic, for some regularizing R we have that μ

T u Lp + Ru Lp ≤ C

(β,α)∈V (P)

W−1 x D (x D u) Lp + Ru Lp β

α

β

α

2.8. Lp -Estimates

125 ≤C

x D u Lp + u Lp , β

α

(β,α)∈V (P)

as W−1 xβ Dxα are operators in OPM Γ0ρ,P whenever (β, α) ∈ V (P). This ends the proof.

Remark 2.8.15. The proof of Theorem 2.8.14 shows that we may take, as an equivalent norm in HPp ,

xβ D α u Lp ,

u HPp = (β,α)∈P

where the sum is now extended to all multi-indices (β, α) ∈ P. In conclusion, we want to apply the preceding results to the operator with polynomial coefficients cαβ xβ Dα (2.8.35) P = (β,α)∈P

obtained by standard left quantization from a multi-quasi-elliptic polynomial, cf. Examples 2.7.9 and 2.7.16. Let us refer to the space HPp , 1 < p < ∞, defined as in (2.8.33). We have that (2.8.36) P : HPp −→ Lp is Fredholm, and the following a priori estimates are valid for a suitable C > 0: (2.8.37)

xβ Dα u Lp ≤ C P u Lp + u Lp . (β,α)∈P

These Lp -estimates are in particular valid for the Γ-elliptic differential operators 2 of Section 2.2, so for example for the harmonic oscillator H = − + |x| in Rd we have d 2

xj u Lp + Dx2 j u Lp ≤ C Hu Lp + u Lp . (2.8.38) j=1

Another relevant example is given by the more general Schrödinger operator P = − + V (x),

(2.8.39)

where V (x) is a multi-quasi-elliptic polynomial V (x) = cβ xβ ,

(2.8.40)

β∈Q

with Q a complete polyhedron in Rd . The symbol of P in (2.8.39) is given by the polynomial in R2d , 2 |ξ| + cβ xβ . (2.8.41) β∈Q


126

The corresponding Newton polyhedron P is easily computed, and turns out to be complete in R2d . Assuming further V (x) > 0

(2.8.42)

we deduce multi-quasi-ellipticity of the polynomial (2.8.41) with respect to P. The preceding results then apply and give in particular the estimates d j=1

Dx2j u Lp +

xβ u Lp ≤ C (− + V (x))u Lp + u Lp .

(2.8.43)

β∈Q

Examples of potentials V (x) satisfying the preceding assumptions are quasi-elliptic polynomials d 2N xj j , (2.8.44) V (x) = j=1

for positive integers Nj , and polynomials in R2 of the form V (x1 , x2 ) = xh1 1 + xh1 2 xk22 + xk21 ,

(2.8.45)

where the integers hj , kj , j = 1, 2, are even and satisfy the conditions in Example 2.6.10.

Notes d The classes Γm (Rd ) and Γm ρ (R ), in Section 2.1, were studied by Shubin, see [183] and the references there to previous works. We also cite Helffer [109]. Such classes play an important role in semi-classical analysis, besides [183] see Robert [170] and references therein. Passing to consider the Γ-elliptic differential operators in Sections 2.2 and 2.3, we observe that they appeared earlier in Grushin [104], [105] as a tool for study of the local properties of partial differential equations with multiple characteristics. In this regard we mention also Helffer and Rodino [110], [111], Parenti and Rodino [158] and Mascarello and Rodino [142] for applications to the local regularity problem and to Parenti and Parmeggiani [157] and Mughetti and Nicola [149] for applications to lower bound estimates. In particular, the proofs of Theorems 2.3.3 and 2.3.5 were first given in [111], whereas we cite [142, Chapter 7] for details on the results of asymptotic integration used in Section 2.3; see also Gramchev and Popivanov [96]. The H-type and ρ-H-type polynomials of Section 2.4 were introduced by Hörmander [113], [114] in the context of the local regularity; see also Treves [189], Rodino [172]. We cite De Donno [70] for an alternative proof of Proposition 2.4.4. A main reference for Section 2.6, concerning multi-quasi-elliptic polynomials and generalizing Section 2.5, on the quasi-elliptic case, is the monograph of Boggiatto, Buzano and Rodino [19]; relevant previous works on the subject are Cattabriga

Notes

127

[46], Friberg [83], Gindikin and Volevich [91], Zanghirati [200]. For recent contributions on multi-quasi-elliptic polynomials see Gindikin and Volevich [92], Calvo [35], Bouzar and Chaili [25]. The source for Example 2.6.15 is Pini [163]; more general examples of H-polynomials, which are not multi-quasi-elliptic, can be found in De Donno and Oliaro [71]. The ΓP -pseudo-differential operators were first introduced in Boggiatto [17], and then studied in detail in [19]. The presentation in Sections 2.7, 2.8 follows Morando [148]; in the local context, the same Lp -estimates were proved by Garello and Morando [85], inspired by the original idea of Cattabriga [47]. Concerning the theorem of Lizorkin and Marcinkiewicz, we refer for the proof to Lizorkin [134] and Stein [184]. In our proof of the Lp -boundedness in Section 2.8, we also used some arguments from Wong [197]. For boundedness on Lp and on other function spaces arising in Fourier Analysis we also refer to Stein [185], the recent contribution by Cordero and Nicola [54] and the references therein. Finally, we would like to cite Garetto [86], where the Γ-pseudo-differential calculus is recast in the context of Colombeau generalized functions, based on the space S(Rd ), see also Garetto, Gramchev and Oberguggenberger [87], Garetto and Hörmann [88], Hörmann, Oberguggenberger and Pilipovic [120].

Chapter 3

G-Pseudo-Differential Operators Summary As in Chapter 2, the basic example here is a partial differential operator with polynomial coefficients in Rd , that is P =

cαβ xβ Dα ,

wherein the sum (α, β) ∈ Nd × Nd runs over a finite subset of indices. The symbol in the standard quantization is p(x, ξ) =

cαβ xβ ξ α =

cγ z γ

with z = (x, ξ) ∈ R2d , γ = (β, α) ∈ N2d . As we have seen in Chapter 2, Hörmander’s condition, i.e., Γ-hypoellipticity |∂zγ p(z)| |p(z)|z−ρ|γ|

(3.0.1)

for z ∈ R2d , |z| ≥ R > 0, 0 < ρ ≤ 1, is sufficient to obtain the global regularity of P . However, the condition is not necessary. In fact, the aim of the present chapter is to discuss classes of symbols p which do not satisfy (3.0.1) and nevertheless give rise to globally regular operators P . As an elementary example in this connection, consider a polynomial q(ξ) depending only on the ξ-variables and satisfying for some integer m ≥ 1: ξm |q(ξ)| ξm

for ξ ∈ Rd .

(3.0.2)

It follows from (3.0.2) that q(ξ) is elliptic of order m with respect to the ξ-variables, but the estimates (3.0.1) are not satisfied in the whole space R2d . On the other

Chapter 3. G-Pseudo-Differential Operators

130

hand, the global regularity of the partial differential operator with constant coefficients q(D) is easily proved directly. To be definite, fix attention on the onedimensional case: q(ξ) =

m

cj ξ j ,

ξ ∈ R, cj ∈ C, cm = 0.

j=0

Then the lower bound (3.0.2) is satisfied if and only if the algebraic equation q(ξ) = 0 has no real roots, which amounts to requiring that all the solutions of the ordinary differential equation q(D)u =

m

cj D j u = 0

j=0

have exponential decay/growth; it follows easily that q(D)u ∈ S(R), u ∈ S (R) imply u ∈ S(R), i.e., q(D) is globally regular. The lower bound in (3.0.2), corresponding to the absence of real roots, is essential to get the conclusion; for example, if q(ξ) = ξ and q(D) = D in R, then the equation Du = 0 admits u(x) = const. as solution, with u ∈ S (R), u ∈ S(R). A relevant example of operator q(D) in Rd , d > 1, with symbol satisfying (3.0.2), is the free particle Schrödinger operator −Δ − λ, λ ∈ R+ ∪ {0}. In Section 3.1 we re-consider the previous examples in the framework of the socalled G-calculus. Namely, we define the class Gm,n (Rd ), m ∈ R, n ∈ R, of all the symbols a(x, ξ) satisfying for α ∈ Nd , β ∈ Nd , |∂ξα ∂xβ a(x, ξ)| ξm−|α| xn−|β| , x ∈ Rd , ξ ∈ Rd .

(3.0.3)

The definition of G-ellipticity, generalizing (3.0.2), is given by xn ξm |a(x, ξ)| for |x|2 + |ξ|2 ≥ R2 > 0.

(3.0.4)

We refer to Section 3.1 for the corresponding pseudo-differential calculus. With the notations in Chapter 1, we have Φ(x, ξ) = x, Ψ(x, ξ) = ξ, that gives a couple of sub-linear temperate weights, cf. (1.1.1) and (1.1.2). d m,n (Rd ), In Section 3.2 we study classical G-symbols a ∈ Gm,n cl(ξ,x) (R ) ⊂ G m possessing three principal parts: σψ (a), homogeneous with respect to ξ; σen (a), m,n homogeneous with respect to x; σψ,e (a), homogeneous with respect to x and ξ separately. The definition of G-ellipticity can be reset in terms of principal parts. As an example, consider p(x, ξ) = cαβ xβ ξ α , (3.0.5) |α|≤m, |β|≤n

Summary

131

d which we may regard as an element of Gm,n cl(ξ,x) (R ), and assume m,n σψ,e (p) =

cαβ xβ ξ α = 0

for x = 0, ξ = 0,

(3.0.6)

cαβ xβ ξ α = 0

for all x ∈ Rd and ξ = 0,

(3.0.7)

cαβ xβ ξ α = 0 for x = 0 and all ξ ∈ Rd .

(3.0.8)

|α|=m |β|=n

σψm (p) =

|α|=m |β|≤n

σen (p) =

|α|≤m |β|=n

Namely, (3.0.6), (3.0.7), (3.0.8) hold simultaneously if and only if the G-ellipticity condition (3.0.4) is satisfied. When n = 0, i.e., the case of an operator with constant coefficients, we recapture (3.0.2). Besides construction of parametrix and global regularity, the previous assumptions of G-ellipticity provide Fredholm m,n (Rd ), defined for nonproperty and a priori estimates in the Sobolev spaces HG negative integers m, n by

u HGm,n =

xβ D α u L2 < ∞. (3.0.9) |α|≤m |β|≤n

Section 3.3 is devoted to a detailed analysis of G-elliptic ordinary differential operators. Similarly to Section 2.3, we apply here the theory of Asymptotic Integration, and we deduce some exponential decay properties of the solutions, which will be extended later in the book to semi-linear equations, cf. Sections 6.3, 6.4, 6.5. Section 3.4 concerns other applications of the general calculus of Chapter 1 to the problem of global regularity. Namely we take here, for some ρ with 0 < ρ < 1, the couple of weights Φ(x, ξ) = 1, Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )ρ/2

(3.0.10)

Φ(x, ξ) = (1 + |x|2 + |ξ|2 )ρ/2 , Ψ(x, ξ) = 1.

(3.0.11)

or alternatively

In the framework of the corresponding calculus, we prove global regularity of the operator P = −Δ + V (x), (3.0.12) where the potential V (x) is a polynomial satisfying x2ρ V (x),

|∂ α V (x)| V (x).

(3.0.13)

As an example, consider in R2 , V (x1 , x2 ) = (1 + x21 )(1 + x22 ).

(3.0.14)


132

Similarly, we have a global regularity for the operator P = V (D) + |x|2 ,

(3.0.15)

where the polynomial V (ξ) satisfies, in the variable ξ, the same estimates (3.0.13). Notice that the assumption (3.0.13) for V is weaker than Hörmander’s property in Rd , cf. Example (3.0.14). Hence the operator P is not locally regular in the Schwartz’ sense, in general.

3.1 G-Pseudo-Differential Calculus Other basic examples of the general classes of symbols considered in Chapter 1 are the G-classes defined below. With respect to the Γ-classes studied in Chapter 2, now the asymptotic behaviour in the (x, ξ)-space of the two weights Φ, Ψ is not optimal separately, nevertheless the strong uncertainty principle (1.1.10) is satisfied thanks to a favourable combination of decay properties. Definition 3.1.1. We define Gm,n (Rd ), m ∈ R, n ∈ R, as the set of all functions a(x, ξ) ∈ C ∞ (R2d ) satisfying, for all α, β ∈ Nd , the estimates |∂ξα ∂xβ a(x, ξ)| ξm−|α| xn−|β| ,

(x, ξ) ∈ R2d .

(3.1.1)

The weights Φ(x, ξ) = x, Ψ(x, ξ) = ξ are sub-linear (1.1.1) and temperate (1.1.2), and the strong uncertainty principle (1.1.10) is obviously satisfied, since xξ (1 + |x|2 + |ξ|2 )1/2 . Hence all the results of Chapter 1 apply for the corresponding pseudo-differential operators u(ξ) dξ, ¯ (3.1.2) Au(x) = a(x, D)u(x) = eixξ a(x, ξ) with action a(x, D) : S(Rd ) → S(Rd ), S (Rd ) → S (Rd ). Namely, we shall denote by OPGm,n (Rd ) the class of the operators in (3.1.2) with symbol a ∈ Gm,n (Rd ). The definition of OPGm,n (Rd ) does not depend on the quantization. In the standard quantization, if a ∈ Gm,n (Rd ), b ∈ Gm ,n (Rd ), then the operator c(x, D) = a(x, D)b(x, D) belongs to OPGm+m ,n+n (Rd ), with symbol c(x, ξ) ∼ (α!)−1 ∂ξα a(x, ξ)Dxα b(x, ξ), (3.1.3) α

where it is worth observing that the orders of the term ∂ξα a(x, ξ)Dxα b(x, ξ) are given by m + m − |α|, n + n − |α|. Similarly we may argue for transposed operators and formal adjoints. The corresponding definition of ellipticity is the following. Definition 3.1.2. Let a ∈ Gm,n (Rd ), m ∈ R, n ∈ R. We say that a is G-elliptic if there exists R > 0 such that xm ξn |a(x, ξ)| for |x|2 + |ξ|2 ≥ R2 .

(3.1.4)

3.1. G-Pseudo-Differential Calculus

133

We then repeat from Chapter 1 the results on the existence of parametrices, cf. Theorem 1.3.6, Corollary 1.3.9, particularized to G-elliptic symbols. Theorem 3.1.3. Let a ∈ Gm,n (Rd ), m ∈ R, n ∈ R, be G-elliptic. Then there exists b ∈ G−m,−n (Rd ) such that b(x, D) is a parametrix of a(x, D), i.e., a(x, D)b(x, D) = I + S1 ,

b(x, D)a(x, D) = I + S2

where S1 and S2 are regularizing operators. Hence a(x, D) is globally regular, cf. Definition 1.3.8. Passing to consider the corresponding Sobolev spaces, we may define them by means of the pseudo-differential operators as,t (x, D) = xt Ds , s ∈ R, t ∈ R, that is the standard quantization of the symbol as,t (x, ξ) = xt ξs ∈ Gs,t (Rd ). Note that as,t (x, ξ) is G-elliptic, and as,t (x, D) : S(Rd ) → S(Rd ) is an isomorphism, extending to an isomorphism on S (Rd ), with inverse b−s,−t (x, D) = D−s x−t ; the asymptotic expansion of b−s,−t (x, ξ) ∈ G−s,−t (Rd ) can be computed in terms of (3.1.3). Definition 3.1.4. For s ∈ R, t ∈ R we define the Hilbert space s,t HG (Rd ) = D−s x−t L2 (Rd ) = {u ∈ S (Rd ) : xt Ds u ∈ L2 (Rd )} (3.1.5) with the scalar product (u, v)H s,t = xt Ds u, xt Ds v L2 G

(3.1.6)

and corresponding norm

u H s,t = xt Ds u L2 . G

(3.1.7)

According to Proposition 1.5.3, for every T ∈ OPGm,n (Rd ) with G-elliptic symbol we have s,t HG (Rd ) = {u ∈ S (Rd ) : T u ∈ L2 (Rd )} (3.1.8) with scalar product equivalent to (3.1.6) (u, v)T = (T u, T v)L2 + (Ru, Rv)L2

(3.1.9)

where R is a regularizing operator associated to a parametrix of T . From Proposition 1.5.5, we then obtain the following boundedness result. Theorem 3.1.5. Every A ∈ OPGm,n (Rd ), m ∈ R, n ∈ R, defines for all s ∈ R, t ∈ R a continuous operator s,t s−m,t−n (Rd ) → HG (Rd ). A : HG

(3.1.10)


134

s,t s ,t We have compact immersion HG (Rd ) → HG (Rd ) for s > s , t > t , and s,t s ,t compactness of the map A : HG (Rd ) → HG (Rd ) for A ∈ OPGm,n (Rd ) whenever s − s > m, t − t > n. Moreover s,t s,t HG (Rd ) = S(Rd ), HG (Rd ) = S (Rd ). (3.1.11) s∈R, t∈R

s∈R, t∈R

For a couple m, n of non-negative integers, an equivalent definition of the space s,t (Rd ) is given by HG m,n HG (Rd ) = {u ∈ S (Rd ) : xβ Dα u ∈ L2 (Rd ) for |α| ≤ m, |β| ≤ n},

with equivalent norm

u HGm,n =

xβ Dα u L2 .

(3.1.12)

(3.1.13)

|α|≤m |β|≤n

s,0 (Rd ) gives the standard Sobolev space Observe also that for t = 0 the space HG s d H (R ). We may now improve Theorem 3.1.3 adding precise Sobolev estimates.

Theorem 3.1.6. Consider A ∈ OPGm,n (Rd ), m ∈ R, n ∈ R, with G-elliptic symbol s,t s+m,t+n and assume u ∈ S (Rd ), Au ∈ HG (Rd ), s ∈ R, t ∈ R. Then u ∈ HG (Rd ) and for every s < s + m, t < t + n, (3.1.14)

u H s+m,t+n ≤ C Au H s,t + u H s ,t , G

G

G

for a positive constant C depending on s and t. Moreover, the operator A, cons+m,t+n s,t sidered as a map HG (Rd ) → HG (Rd ) is Fredholm, cf. Theorem 1.6.9. If m, n is a couple of positive integers, we may refer to the equivalent norm (3.1.13) and re-write (3.1.14) for s = s = 0, t = t = 0 as

xβ D α u L2 ≤ C ( Au L2 + u L2 ) . (3.1.15) |α|≤m |β|≤n

We may as well consider G-hypoelliptic symbols and related operators. Theorem 3.1.7. Assume a ∈ Gm,n (Rd ), m ∈ R, n ∈ R, 0 < ρ ≤ 1, i.e., for all ρ 2d (α, β) ∈ N , |∂ξα ∂xβ a(x, ξ)| ξm−ρ|α| xn−ρ|β| , (x, ξ) ∈ R2d .

(3.1.16)

Assume that a is hypoelliptic, i.e., there exist m , n with m ≤ m, n ≤ n and R > 0 such that

ξm xn |a(x, ξ)|

for |x|2 + |ξ|2 ≥ R2

(3.1.17)

3.1. G-Pseudo-Differential Calculus

135

and for every (α, β) ∈ N2d , |∂ξα ∂xβ a(x, ξ)| |a(x, ξ)|ξ−ρ|α| x−ρ|β| .

(3.1.18)

Then, denoted by A = a(x, D) the pseudo-differential operator with symbol a, there exists B = b(x, D) with symbol b ∈ G−m ,−n (Rd ) such that AB = I + S1 ,

BA = I + S2 ,

with S1 and S2 regularizing operators. The estimate (3.1.14) remains valid with m, n replaced by m , n . Let us now compare G-classes with Γ-classes. We may say that G-classes are more general, in the sense that ΓM (Rd ), M ∈ R, is included in Gm,n (Rd ) for a suitable couple m, n, take for example m = n = M , if M > 0, and m = n = 0 for M < 0, and use the obvious estimates x ≤ (1 + |x|2 + |ξ|2 )1/2 , ξ ≤ (1 + |x|2 + |ξ|2 )1/2 , (1 + |x|2 + |ξ|2 )1/2 ≤ xξ. Also, the generalized classes m m Γm ρ , Γρ,P , M Γρ,P in Chapter 2 are included in the classes defined by (3.1.16) and Γ-hypoellipticity implies G-hypoellipticity, cf. (3.1.18). Summing up, all the results of global regularity in Chapter 2 are recaptured by the above Theorem 3.1.7. However, Γ-elliptic symbols and P-elliptic symbols are not G-elliptic, and we cannot identify in the G-frame the peculiar properties of the corresponding operators, in particular the sharp Sobolev estimates in Sections 2.1, 2.7 and 2.8. In the opposite direction, the G-symbols do not satisfy in general the estimates required for Γ-symbols. Relevant examples are given by Fourier multipliers, i.e., operators with symbols depending only on the ξ-variables. Namely, consider a(ξ) ∈ C ∞ (Rd ) satisfying for all α ∈ Nd and some m ∈ R, |∂ α a(ξ)| ξm−|α| , ξ ∈ Rd .

(3.1.19)

We have a ∈ Gm,0 (Rd ) and a is G-elliptic when ξm |a(ξ)|,

ξ ∈ Rd .

(3.1.20)

It is worth emphasizing that estimate (3.1.20) is assumed to be valid in the whole Rd , which implies a(ξ) = 0 for all ξ ∈ Rd . In fact, the vanishing of a(ξ) at a point ξ0 would contradict (3.1.4) for any choice of the constant R > 0. Note also that a symbol a(ξ) satisfying (3.1.19) does not belong to any of the Γ-classes, but when a(ξ) is a polynomial, and in this case Γ-ellipticity fails anyhow for m ≥ 1. If (3.1.20) is satisfied, we may apply Theorems 3.1.3 and 3.1.6. The parame1 trix of a(D) is actually an inverse, with symbol b(ξ) = a(ξ) ∈ G−m,0 (Rd ). Then a(D) : S(Rd ) → S(Rd ) is an isomorphism extending to an isomorphism a(D) : S (Rd ) → S (Rd ) and s,t s−m,t (Rd ) → HG (Rd ) a(D) : HG

(3.1.21)


136 where s ∈ R, t ∈ R, with inverse

s−m,t s,t b(D) : HG (Rd ) → HG (Rd ).

(3.1.22)

Let us discuss in detail some examples of Fourier multipliers, considering first differential operators with constant coefficients. Example 3.1.8. Partial differential operators. For m a positive integer, consider p(ξ) = cα ξ α , cα ∈ C, (3.1.23) |α|≤m

giving the operator P = p(D) =

cα Dα .

(3.1.24)

|α|≤m

Define pm (ξ) =

cα ξ α .

(3.1.25)

|α|=m

Then the G-ellipticity (3.1.20) is equivalent to assume simultaneously: (i) pm (ξ) = 0 for ξ = 0; (ii) p(ξ) = 0 for all ξ ∈ Rd . In fact, from the proof of Proposition 2.1.5 we know that (i) is equivalent to the ellipticity of p(ξ) as a polynomial in ξ-variables: |ξ|m |p(ξ)| for |ξ| ≥ R,

(3.1.26)

with R > 0 sufficiently large. On the other hand (3.1.26) and (ii), jointly, are equivalent to (3.1.27) ξm |p(ξ)|, ξ ∈ Rd . Under the G-ellipticity assumption (3.1.27), the previous conclusions hold, in particular P in (3.1.24) is globally regular. Note that if p(ξ0 ) = 0 for some ξ0 ∈ Rd , then global regularity fails, because u(x) = eiξ0 x satisfies u ∈ S (Rd ), u ∈ S(Rd ) and P (eiξ0 x ) = p(ξ0 )eiξ0 x = 0. Example 3.1.9. Ordinary differential operators. In the one-dimensional case, consider the symbol m−1 p(ξ) = ξ m + cj ξ j , cj ∈ C, ξ ∈ R, j=0

with corresponding ordinary differential operator P = p(D) = Dxm +

m−1 j=0

cj Dxj .

3.2. Polyhomogeneous G-Operators

137

The G-ellipticity reduces to assume p(ξ) = 0 for all ξ ∈ R, that is all roots rj ∈ C, j = 1, ..., m of the algebraic equation p(ξ) = 0 satisfy Im rj = 0. Consequently a basis for the classical solutions of P u = 0 is given by functions of the form u(x) = Q(x)eirj x for some polynomial Q(x), with Re irj = 0, giving exponential growth in R+ or in R− . Hence the only solution u ∈ S(R) of P u = 0 is the trivial solution, as expected from the previous arguments. We get also that P u = f ∈ S(R) admits a unique s+m,t s,t (R) → HG (R) is an isomorphism solution u ∈ S(R). More precisely, P : HG for all s ∈ R, t ∈ R. Example 3.1.10. The free particle. As a particular case of the above examples, consider in Rd the operator of the free particle P = −Δ − λ with symbol

p(ξ) = |ξ|2 − λ.

The symbol is G-elliptic and the operator globally regular, if and only if λ ∈ R+ ∪ {0}. Example 3.1.11. The intermediate-long-wave operator. The following non-polynomial symbol appears in the study of solitary waves, see later in this book, Section 6.4: p(ξ) = ξCtgh ξ + γ, ξ ∈ R, with Ctgh ξ = Ch ξ/Sh ξ = (eξ + e−ξ )/(eξ − e−ξ ) and γ ∈ R. We have p ∈ G1,0 (R) and the G-ellipticity condition is satisfied if γ > −1.

3.2 Polyhomogeneous G-Operators It is natural to introduce the subclass of Gm,n (Rd ) of the polyhomogeneous (or classical) symbols. The definition is somewhat more complicated than in the Γcase, since homogeneity has to be considered separately in the x and ξ variables. Building blocks of the calculus are the following different types of homogeneous symbols. Type 1 symbols: orders m, n; bi-homogeneity with respect to x, ξ. We assume that a(x, ξ) ∈ C ∞ ((Rd \ {0}) × (Rd \ {0})) is positively homogeneous of degree m ∈ R, n ∈ R with respect to x and ξ separately, i.e., a(sx, tξ) = tm sn a(x, ξ) for all t > 0, s > 0. m,n the class of such functions. We shall also denote by Hξ,x


138

If we take χ ∈ C ∞ (Rd ), χ(x) = 1 for |x| ≥ 2, χ(x) = 0 for |x| ≤ 1, then χ(x)χ(ξ)a(x, ξ) ∈ Gm,n (Rd ). Type 2 symbols: (i) orders m, n; homogeneity m with respect to ξ. We assume that a(x, ξ) ∈ C ∞ (Rd × (Rd \ {0})) is positively homogeneous of degree m with respect to ξ, i.e., a(x, tξ) = tm a(x, ξ), for all s > 0. We assume moreover for all α, β ∈ Nd , |∂ξα ∂xβ a(x, ξ)| xn−|β|

for x ∈ Rd , ξ ∈ Sd−1 ,

where Sd−1 = {ξ ∈ Rd : |ξ| = 1}. Taking χ as before, we have χ(ξ)a(x, ξ) ∈ Gm,n (Rd ). (ii) orders m, n; homogeneity n with respect to x. Same definition as in (i), interchanging the role of x and ξ. Type 3 symbols: (i) orders m, n; homogeneity m with respect to ξ, polyhomogeneity with respect to x. Let a(x, ξ) satisfy the same conditions as in Type 2, (i), and assume m,n−j , cf. Type 1, j = 0, 1, ..., such that the existence of bn−j (x, ξ) ∈ Hξ,x ∞ a(x, ξ) ∼ j=0 bn−j (x, ξ), in the sense that, for all α, β ∈ Nd , α β χ(x)bn−j (x, ξ) xn−N −|β| , x ∈ Rd , ξ ∈ Sd−1 . ∂ξ ∂x a(x, ξ) − 0≤j 0

for x = 0, ξ = 0.

Applying then (3.2.22), (3.2.23), (3.2.24), we obtain similar estimates for a ˜: |˜ a(x, ξ)| ≥ C > 0,

for |x| ≥ T, |ξ| ≥ T,

with suitable new constants C and T > 0. To handle a ˜(x, ξ) in the strip |x| ≤ T , we write a)(x, ξ) + rψ (x, ξ). a ˜(x, ξ) = χ(ξ)σψ0 (˜ From (3.2.27), in view of the homogeneity with respect to ξ we have a)(x, ξ)| ≥ > 0, |χ(ξ)σψ0 (˜

for |ξ| ≥ 2, |x| ≤ T,

d for a suitable depending on T . Since rψ ∈ G−1,0 cl(ξ,x) (R ), the same estimates are valid for a ˜(x, ξ), if we shrink > 0 and impose |ξ| ≥ R for a sufficiently large R. The same argument applies in the strip |ξ| ≤ T , by writing

a ˜(x, ξ) = χ(x)σe0 (˜ a)(x, ξ) + re (x, ξ) and using the assumption (3.2.28). This gives (3.2.26) and concludes the proof of Theorem 3.2.9.


144

d Theorem 3.2.10. Let a ∈ Gm,n cl(ξ,x) (R ), m ∈ R, n ∈ R, satisfy (3.2.16), (3.2.17),

d (3.2.18). Then there exists b ∈ G−m,−n cl(ξ,x) (R ) such that b(x, D) is a parametrix of a(x, D). Hence A = a(x, D) is globally regular, satisfies the estimates (3.1.14) and s+m,t+n s,t is Fredholm, considered as a map HG (Rd ) → HG (Rd ).

Proof. In view of Theorem 3.2.9, the symbol a is G-elliptic, and we may apply Theorem 3.1.3 and Theorem 3.1.6. It remains to prove that the symbol b(x, ξ) of the parametrix is polyhomogeneous. To this aim, set m −1 , x ∈ Rd , ξ = 0, b−m ψ (x, ξ) = σψ (a)(x, ξ) n b−n e (x, ξ) = (σe (a)(x, ξ))

−1

,

x = 0, ξ ∈ Rd ,

which are symbols of Type 3, in view of (3.2.16) and (3.2.17), and −1 m,n (x, ξ) = σ (a)(x, ξ) , x = 0, ξ = 0, b−m,−n ψ,e ψ,e −m,−n d in view of (3.2.18). We then construct ˜b ∈ G−m,−n well defined in Hξ,x cl(ξ,x) (R ) as ˜b(x, ξ) = χ(ξ)b−m (x, ξ) + χ(x)b−n (x, ξ) − χ(x)χ(ξ)b−m,−n (x, ξ). e ψ,e ψ

In view of (3.2.13), (3.2.14), (3.2.15), we obtain ˜b(x, D)a(x, D) = I + S1 ,

a(x, D)˜b(x, D) = I + S2 ,

−m,−n d d where S1 , S2 ∈ OPG−1,−1 cl(ξ,x) (R ). The expression of the symbol b(x, ξ) ∈ Gcl(ξ,x) (R ) of the parametrix follows then from iterative arguments, as in the proof of Theorem 1.3.6.

Example 3.2.11. Reconsidering Example 3.2.7, we have from Theorem 3.2.9 that the symbol cαβ xβ ξ α p(x, ξ) = |α|≤m |β|≤n

of the operator P in (3.2.8) is G-elliptic if cαβ xβ ξ α = 0 for x ∈ Rd , ξ = 0,

(3.2.30)

|α|=m |β|≤n

that is, we have local ellipticity in the classical sense, and additionally cαβ xβ ξ α = 0 for x = 0, ξ ∈ Rd ,

(3.2.31)

|α|≤m |β|=n

cαβ xβ ξ α = 0 for x = 0, ξ = 0.

|α|=m |β|=n

In the case n = 0 we recapture the conclusions in Example 3.1.8.

(3.2.32)

3.3. G-Elliptic Ordinary Differential Operators

145

Note that (3.2.32) is necessary to get G-ellipticity, and to conclude global regularity through Theorem 3.2.10. Consider in fact the operator in OPG1,1 cl(ξ,x) (R) P = Dx − x,

x ∈ R,

with symbol ξ−x. Conditions (3.2.30) and (3.2.31) are obviously satisfied, however 1,1 = 0, hence we do not have G-ellipticity. A solution u ∈ S (R) of P u = 0 is σψ,e x2

given by u(x) = ei 2 ∈ S(R), and global regularity fails. We refer to Section 3.3 for a detailed discussion of G-ellipticity for ordinary differential operators.

3.3 G-Elliptic Ordinary Differential Operators Let us first fix attention on ordinary differential operators with polynomial coefficients. We may re-write in this case the operator P ∈ OPGm,n cl(ξ,x) (R), m ∈ N, n ∈ N, in the form m P = Qj (x)Dxj , x ∈ R, (3.3.1) j=0

where Qj , j = 0, . . . , m, are polynomials of degree ≤ n: Qj (x) =

n

cjk xk ,

cjk ∈ C, x ∈ R.

(3.3.2)

k=0

We may read the G-ellipticity of the symbol of P in terms of (3.2.30), (3.2.31), (3.2.32) in Example 3.2.11. Namely, (3.2.30) corresponds to assuming that Qm (x) = 0 for all x ∈ R.

(3.3.3)

Note that this implies analyticity in R of all the classical solutions of P u = 0. As for condition (3.2.31), this is equivalent to assuming that m

cjn ξ j = 0

for all ξ ∈ R,

(3.3.4)

j=0

granting exponential asymptotic behaviour of the classical solutions, see below. Finally, (3.2.32) amounts to assuming that cmn = 0.

(3.3.5)

Theorem 3.3.1. Under the conditions (3.3.3), (3.3.4), (3.3.5), we may regard the operator in (3.3.1), (3.3.2) as a Fredholm map m,n (R) → L2 (R), P : HG

with ind P = 0.

(3.3.6)

146


Proof. The Fredholm property of P follows from Theorem 3.2.10. It remains to prove the assertion on the index. Assume then cmn = 1, without loss of generality, and consider the operator P0 = Qm (x)P00 with P00 = Dxm +

m−1

cjn Dxj .

j=0

The symbols of P and P0 have the same principal parts, hence we have P − P0 ∈ OPGm−1,n−1 (R). Therefore, P −P0 is a compact operator, so that ind P = ind P0 , cl(ξ,x) cf. Theorems 3.1.5 and 1.6.6. On the other hand P00 belongs to OPGm,0 cl(ξ,x) (R) with G-elliptic symbol, in view of (3.3.4). As observed in Example 3.1.9, P00 : m,n 0,n (R) → HG (R) is then an isomorphism. The same conclusion is valid for the HG multiplication operator 0,n Qm : H G (R) → L2 (R), because of (3.3.3), and in view of Corollary 1.6.5 we conclude ind P = ind Qm P00 = ind Qm + ind P00 = 0. We may go further in the analysis of the solutions of the equation P u = 0 by applying the theory of asymptotic integration. Let us first observe that the solutions u(x) extend to the complex domain, with possibly a finite number of singularities at the points x ∈ C where Qm (x) = 0. Consider then the roots r1 , . . . , rm of the equation m cjn ξ j = 0. (3.3.7) j=0

In view of (3.3.4), we have Im rj = 0 for all j = 1, . . . , m. Assume for simplicity that all the rj , j = 1, . . . , m, are distinct. We recall the following basic result on the asymptotic expansions of the solutions of P u = 0 in sectors of the complex plane, see for example Wasow [194]. Proposition 3.3.2. Given any sector Λ = {x ∈ C \ {0} : ϕ1 < arg x < ϕ2 } with ϕ2 − ϕ1 < π, there exist uΛ,j (x), j = 1, . . . , m, linearly independent solutions of P u = 0, such that for x ∈ Λ, x → ∞, uΛ,j (x) ∼ xsj exp[irj x]

∞

β−p,j x−p .

(3.3.8)

p=0

We can determine sj , β−p,j , j = 1, . . . , m, p = 0, 1, . . ., by letting the righthand side of (3.3.8) be a formal solution of P u = 0, cf. Section 2.3. The asymptotic meaning of (3.3.8) is then the same as in Proposition 2.3.1. Here we understand x ∈ C, |x| > R, with R sufficiently large, so that Qm (x) = 0 and the solutions uΛ,j are holomorphic in Λ ∩ {|x| ≥ R}.

3.3. G-Elliptic Ordinary Differential Operators

147

Taking now two sectors Λ+ , Λ− containing R+ , R− respectively, we may − conclude the existence of u+ j , uj , j = 1, . . . , m, satisfying (3.3.8) in R+ , R− respectively. Every classical solution u(x) can be written as u=

m

+ μ+ j uj =

j=1

m

− μ− j uj ,

− μ+ j , μj ∈ C.

j=1

Assume Im rj > 0 for j = 1, . . . , m+ ,

Im rj < 0 for j = m+ + 1, . . . , m,

m+ + m− = m. (3.3.9) = 0 for m+ < Then u belongs to S(R), or equivalently to S (R), if and only if μ+ j + j ≤ m and μ− j = 0 for 1 ≤ j ≤ m , that is + − u= μ+ μ− j uj = j uj . 1≤j≤m+

m+ <j≤m

Summing up, we have obtained the following result. Theorem 3.3.3. Let u ∈ S (R) be a solution of P u = 0, with P as in (3.3.1), (3.3.2) satisfying (3.3.3), (3.3.4), (3.3.5). Then u ∈ S(R) and |u(x)| e−δ|x| ,

x ∈ R,

(3.3.10)

for some constant δ > 0. Moreover u(x) has holomorphic extension for x ∈ C in the strip {|Im x| < T } for some T > 0. Letting m+ , m− be defined as in (3.3.9), we have (3.3.11) dim (Ker P ∩ S(R)) ≤ min{m+ , m− }. In particular, if m+ = 0 or m− = 0, then the equation P u = 0, u ∈ S (Rd ), admits only the trivial solution. By a slight improvement of Proposition 3.3.2, cf. (2.3.4), we may see that Theorem 3.3.3 is valid regardless of the multiplicity of the roots rj in (3.3.9). Let us pass now to consider differential operators in OPGm,n cl(ξ,x) (R), m ∈ N, n ∈ N, with C ∞ -coefficients. Actually, for the sake of brevity, we shall limit the analysis to the first-order operator L = Dx − r(x),

x ∈ R.

(3.3.12)

∞ We have L ∈ OPG1,0 cl(ξ,x) (R) if r ∈ C (R) admits an asymptotic expansion

r(x) ∼

∞

r−j (x),

(3.3.13)

j=0

where r−j (x) is positively homogeneous of degree −j. In particular r0 (x) = r0+ if x > 0,

r0 (x) = r0− if x < 0,

(3.3.14)


148

for constants r0+ , r0− ∈ C. To check (3.2.16), (3.2.17), (3.2.18), we compute σψ1 = 1,0 0 σψ,e = ξ and σe0 = ξ − r0+ if x > 0, = ξ − r− if x < 0. Hence G-ellipticity for the symbol of (3.3.12) amounts to requiring Im r0+ = 0,

Im r0− = 0.

In fact, the classical solutions of Lu = 0 are the functions x r(t) dt , C ∈ C. u(x) = C exp i

(3.3.15)

(3.3.16)

0

x Since 0 r(t) dt = r0+ x + O(log x) for x → +∞, = r0− x + O(log |x|) for x → −∞, we have that the solutions in (3.3.16) with C = 0 belong to S(R), or equivalently to S (R), if and only if Im r0+ > 0, Im r0− < 0. x u = 0, which is an equation of the type So for example u(x) = e−x solves u + x + − Lu = 0 with r0 = i, r0 = −i. Applying the results of the previous sections, we may then summarize as follows:

Theorem 3.3.4. Under the condition (3.3.15), we may regard the operator L in (3.3.12), (3.3.13), (3.3.14) as a Fredholm map 1,0 L : HG (R) → L2 (R)

(3.3.17)

with

1 sign Im r0+ − sign Im r0− . (3.3.18) 2 Here we used the fact that ind L = dim Ker L − dim Ker L∗ , and L∗ = Dx − r(x), cf. Theorem 1.6.9. 1,0 Note that HG (R) in (3.3.17) coincides with the standard Sobolev space 1 H (R), and that the exponential bound (3.3.10) remains valid for the solutions u ∈ S (R) of Lu = 0. ind L =

3.4 Other Classes of Globally Regular Operators A couple Φ, Ψ of sub-linear and temperate weights can satisfy the strong uncertainty principle (1.1.10) even in the case when Φ(x, ξ) = 1, provided good estimates in the whole (x, ξ) variables are valid for the other weight Ψ(x, ξ). A basic example is given by (3.4.1) Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )ρ/2 , Φ(x, ξ) = 1, with 0 < ρ ≤ 1. We may as well interchange the role of Φ and Ψ in the above argument. Rather than detailing the calculus for the corresponding classes S(M ; Φ, Ψ), we immediately propose some examples of operators with polynomial coefficients, to which this calculus applies.

3.4. Other Classes of Globally Regular Operators

149

Theorem 3.4.1. Consider the operator P = −Δ + V (x)

(3.4.2)

where V (x) is a positive polynomial in Rd satisfying, for some ρ > 0, N > 0 and every β ∈ N, x2ρ V (x) xN , |∂ V (x)| V (x), β

x ∈ Rd ,

x∈R . d

(3.4.3) (3.4.4)

The operator P is globally regular. Example 3.4.2. As an example of potential V (x) satisfying (3.4.3), (3.4.4) consider in R2 , (3.4.5) V (x1 , x2 ) = (1 + x21 )(1 + x22 ). The estimates (3.4.3) are valid for ρ = 1, N = 4. Note that also (3.4.4) is satisfied, however V (x) has not the Hörmander property with respect to x ∈ Rd . Proof of Theorem 3.4.1. We refer directly to the general calculus of Chapter 1, by taking Ψ, Φ as in (3.4.1) with ρ as in (3.4.3), where we may assume ρ ≤ 1. Let M (x, ξ) be defined as the symbol of P : M (x, ξ) = |ξ|2 + V (x),

(3.4.6)

which is a temperate weight. We now prove that M ∈ S(M ; Φ, Ψ), according to Definition 1.1.1. In fact |∂ξj M (x, ξ)| = |2ξj | M (x, ξ)Ψ(x, ξ)−1 , |∂ξ2j M (x, ξ)| = 2 M (x, ξ)Ψ(x, ξ)−2 , in view of (3.4.3) and (3.4.1). Moreover |∂xβ M (x, ξ)| = |∂xβ V (x)| V (x) ≤ M (x, ξ) in view of (3.4.4). Other derivatives of the symbol vanish, hence (1.1.4) is satisfied. On the other hand, M (x, ξ) is obviously elliptic in S(M ; Φ, Ψ) and, applying Corollary 1.3.9, we deduce the global regularity of P . Theorem 3.4.3. Consider the operator Q = V (D) + |x|2

(3.4.7)

where V (D) is a partial differential operator with constant coefficients and symbol V (ξ) satisfying the estimates (3.4.3), (3.4.4) with x replaced by ξ. The operator Q in (3.4.7) is globally regular.


150

Example 3.4.4. Rephrasing the example in Theorem 3.4.1, we have that Q = (1 + Dx21 )(1 + Dx22 ) + |x|2

(3.4.8)

is globally regular in R2 . As for the proof of Theorem 3.4.3, we may repeat the arguments in the proof of Theorem 3.4.1, by taking Ψ(x, ξ) = 1,

Φ(x, ξ) = (x, ξ)ρ = (1 + |x|2 + |ξ|2 )ρ/2 , M (x, ξ) = V (ξ) + |x|2 .

(3.4.9) (3.4.10)

As alternative proof, we may observe that Q in (3.4.7) is obtained from P in (3.4.2) by Fourier conjugation, Q = P = F−1 P F. Since global regularity is invariant under Fourier conjugation, Theorem 3.4.3 follows from Theorem 3.4.1.

Notes G-pseudo-differential operators were introduced by Parenti [156] and then studied in detail by Cordes [59]. An exhaustive treatment of polyhomogeneous G-operators is in Schulze [178, Section 1.4], which we followed closely in the previous Section 3.2. Concerning the results on asymptotic integration used in Section 3.3, besides Wasow [194], see also Mascarello and Rodino [142], Gramchev and Popivanov [96]. Finally, the examples in Section 3.4 are included in Buzano [27], Buzano and Ziggioto [33]. In the following we list some other important aspects of G-theory, which are not treated in the present book. First, the G–calculus can be extended to a suitable class of non-compact manifolds, see Schrohe [175]. In this setting, G-operators appear in the study of boundary value problems, see for example Kapanadze and Schulze [125], Harutyunyan and Schulze [108], as well as in the analysis of equations on manifolds with edges that have conical exits at infinity, see for example Calvo and Schulze [37]. For a polyhomogeneous G-calculus on manifolds, we refer also to Melrose [144], [145]. Other important topics are given by the G-hyperbolic equations, i.e., hyperbolic equations with coefficients globally defined in space variables, having G-type asymptotic behaviour. Their study was initiated in [59] and carried on by Coriasco [61], [62], Coriasco and Panarese [64], presenting the theory of the G-Fourier integral operators; see also Coriasco and Maniccia [63] for the related notion of G-wave front set, Coriasco and Rodino [65] for G-hyperbolic equations with multiple characteristics. Finally, we refer to Dasgupta and Wong [68] for the Lp -theory of the G-pseudo-differential operators. Let us return, in conclusion, to the problem of the characterization of the globally regular operators with polynomial coefficients: P = cαβ xβ Dα .

Notes

151

Namely, one would like to give necessary and sufficient conditions on the symbol p(x, ξ) = cαβ xβ ξ α for the global regularity of P . The results in Chapters 1,2,3, give sufficient conditions, but do not provide a complete answer to the problem. We refer to Camperi [38] for generalized G-operators included in the calculus of Hörmander [119, Vol. III], leading to further classes of globally regular operators with polynomial coefficients. As striking examples, giving evidence of the difficulty of the problem in general, we quote the operators in R2 , R = Dx2 1 + x21 − λ, i.e., the harmonic oscillator with respect to x1 , which is globally regular in the (x1 , x2 )-variables when λ = 2n + 1, n = 0, 1, . . ., cf. Gramchev, Pilipovic and Rodino [95], and the twisted Laplacian 1 Q = −Δ + (x21 + x22 ) + x1 Dx2 − x2 Dx1 , 4 also globally regular, cf. Wong [199] and also Dasgupta and Wong [69]. The symbols of R and Q fail to be elliptic or hypoelliptic in Γ and G classes, neither enter Hörmander’s calculus [119, Vol. III].

Chapter 4

Spectral Theory Summary In this chapter we study some problems of Spectral Theory for pseudo-differential operators with hypoelliptic symbols in the classes S(M ; Φ, Ψ) considered in Chapter 1; see in particular Sections 1.1 and 1.3.1. For a symbol a ∈ S(M ; Φ, Ψ) we denote by aw its Weyl quantization, as a continuous operator on S (Rd ), cf. Section 1.2, Proposition 1.2.13. Consider then the unbounded operator A in L2 (Rd ), with domain S(Rd ), defined as Au = aw u, for u ∈ S(Rd ). Clearly, the restriction of aw to the subspace {u ∈ L2 (Rd ) : aw u ∈ L2 (Rd )}

(4.0.1)

defines a closed extension of A. It is called the maximal realization of A. In particular, the operator A is closable, i.e., the closure of its graph in L2 (Rd ) × L2 (Rd ) is still a graph of a linear operator, which is called closure, or minimal realization, of A, and denoted by A. Moreover, the maximal realization of A extends A. In Section 4.1 we recall some basic facts about unbounded operators in Hilbert spaces. Then, assuming the strong uncertainty principle (1.1.10), in Section 4.2 we will prove that when a is hypoelliptic, cf. Definition 1.3.2, the above minimal and maximal realizations coincide, i.e., the domain of A is the space in (4.0.1). In Section 4.2 we also study the spectrum of operators with a hypoelliptic Weyl symbol a ∈ Hypo(M, M0 ; Φ, Ψ). The main result in this connection states that, if a is real-valued and M0 (x, ξ) → +∞ at infinity, then the spectrum of A is given by a sequence of real eigenvalues either diverging to +∞ or −∞. The eigenvalues are all of finite multiplicity and the eigenfunctions belong to S(Rd ). Moreover L2 (Rd ) has an orthonormal basis made of eigenfunctions of A. This generalizes the results obtained for the harmonic oscillator in Section 2.2. Basic

154

Chapter 4. Spectral Theory

examples of operators which this result applies to are the generalized harmonic oscillator A = −Δ + |x|2k , k ∈ N, k = 0, or A = xs (−Δ + I)k xs ,

s > 0, k ∈ N, k = 0.

Section 4.3 is devoted to complex powers of pseudo-differential operators with a hypoelliptic symbol a(x, ξ) taking values in a closed angle Re z ≥ −R Im z, when |x| + |ξ| ≥ R, for some R > 0. Here the basic assumption at the operator level is that the operator A is non-negative in the sense of Komatsu, i.e., (−∞, 0) is contained in the resolvent set of A and −1 < ∞. sup λ A + λI B(L2 (Rd ))

λ∈R+

z

This guarantees that the complex powers A , Re z > 0, are well defined, (in the sense of Balakhrishnan), as densely defined operators on L2 (Rd ). We will show z that A is in fact a pseudo-differential operator with hypoelliptic symbol given essentially by az plus a remainder satisfying suitable estimates. Notice that we allow the symbol to tend to zero and infinity in different directions and we also allow the spectrum of A to have zero as an accumulation point. We are also interested in the asymptotic behaviour of the eigenvalues of pseudo-differential operators A = aw , for a real elliptic symbol a diverging to +∞ at infinity at least algebraically. A classical strategy to attack this problem consists in studying the trace (that is the sum of the eigenvalues) of some parameter dependent function Ft (A) of the operator A. Then one deduces the asymptotic behaviour of the eigenvalues of A via a suitable Tauberian theorem. In Sections 4.5, 4.6 we will apply the above program with the function Ft (x) = e−tx , where t ≥ 0 is here regarded as a parameter (this is known as the heat method). Namely, denote by λj , j ∈ N, the sequence of eigenvalues (counted according to their multiplicity) of A. We have λj → +∞ as j → ∞. We will show that the heat semigroup e−tA is in fact a regularizing operator for t > 0, and its symbol u(t, x, ξ) is given by e−ta(x,ξ) , plus a remainder which will be conveniently estimated. Moreover the trace of e−tA can be expressed in terms of its symbol by the formula ∞

e−tλj =

u(t, x, ξ) dx ¯ dξ. ¯

j=0

An explicit computation of this integral allows us to deduce the asymptotic behaviour of the above trace as t → 0. Karamata’s Tauberian Theorem finally yields the asymptotic behaviour for the so-called counting function N (λ) = {j ∈ N : λj ≤ λ}, and therefore that of the λj ’s. In fact, for simplicity we will develop this program only for the classes m,n d d OPΓm ρ (R ), m > 0, 0 < ρ ≤ 1, studied in Chapter 2 and the classes OPGcl(ξ,x) (R ),

4.1. Unbounded Operators in Hilbert spaces

155

m > 0, n > 0 of Chapter 3. For operators in the former class or even in the second class when m = n we will obtain a formula of the type N (λ) ∼ Cλα as λ → +∞, d for certain explicit values of α, C > 0, whereas operators in OPGm,n cl(ξ,x) (R ), α m = n > 0, will fulfill N (λ) ∼ Cλ log(λ). More refined methods yield an estimate of the remainder term. We will discuss this briefly in the Notes at the end of the present chapter. Preliminary results about trace-class operators are also recalled in detail in Section 4.4.

4.1 Unbounded Operators in Hilbert spaces In this section we recall some basic results for unbounded operators in Hilbert spaces. Let H be a separable complex Hilbert space of infinite dimension. Let A be a linear operator, from a dense linear subspace Dom(A) ⊂ H, into H. We refer to Dom(A) as the domain of A. We do not require it to be closed, nor the operator A to be bounded. The graph of A is the linear subspace {(u, Au) ∈ H × H : u ∈ Dom(A)} of H × H. A is called closed if its graph is a closed subspace, namely, if un ∈ Dom(A) → u ∈ H and Aun → v ∈ H imply u ∈ Dom(A) and Au = v. A is called closable if the closure of its graph is still a graph of a (linear) operator, which is then denoted by A. Clearly, it is the smallest closed extension of A. It is easy to see that, in fact, it suffices that A admits a closed extension for A to be closable. We also recall that the adjoint operator A∗ has domain Dom(A∗ ) given by all v ∈ H such that the map Dom(A) ⊂ H → H, u → (Au, v)H , is bounded. Then A∗ v is defined as the unique w ∈ H such that (Au, v)H = (u, w)H ,

for all u ∈ Dom(A).

Thus we have (Au, v)H = (u, A∗ v)H ,

for all u ∈ Dom(A) and v ∈ Dom(A∗ ).

One can prove that A∗ is a closed operator. Moreover it is densely defined if A is closable. A is called self-adjoint if A = A∗ . A weaker definition than self-adjointness is that of symmetric operator: A is called symmetric if (Au, v)H = (u, Av)H ,

for u, v ∈ Dom(A),

this is, if A∗ is an extension of A. Proposition 4.1.1. A densely defined symmetric operator has symmetric closure.

156


Proof. Let un ∈ Dom(A) be a sequence such that un → 0 and Aun → v. Let w ∈ Dom(A). We have (v, w)H = lim (Aun , w)H = lim (un , Aw)H = 0. n→∞

n→∞

Thus v = 0, because Dom(A) is dense in H. This implies that if (u, v1 ), (u, v2 ) are two elements in the closure of the graph of A, then v1 = v2 , and therefore A is closable. Let u, v ∈ Dom(A). Then there exist two sequences un , vn ∈ Dom(A) such that un → u, vn → v, Aun → Au, Avn → Av. Hence (Au, v)H = lim (Aun , vn )H = lim (un , Avn )H = (u, Av)H . n→∞

n→∞

This proves that A is symmetric.

A densely defined symmetric operator A is called essentially self-adjoint if A is self-adjoint. It is equivalent to saying that A∗ = A, because for any closable ∗ operator A we have A = A∗ . Given a closed densely defined operator A on a complex Hilbert space H we define the resolvent set of A as the set ρ(A) of complex numbers λ such that A−λI is a bijection Dom(A) → H, with a bounded inverse RA (λ) = (A − λI)−1 (which has domain H). RA (λ) is called the resolvent (operator) of A. The spectrum of A is by definition the complementary set of ρ(A) in C: σ(A) = C \ ρ(A). Let us prove that self-adjoint operators have real spectrum. Proposition 4.1.2. If A is self-adjoint, then σ(A) ⊂ R. Proof. Let λ ∈ C \ R. Then λ = ξ + iη for real ξ, η, with η = 0. Because A is self-adjoint we have, for u ∈ Dom(A), A − (ξ ± iη)I u2 2 = (A − ξI)u 2 2 + η2 u 2 2 . L L L

(4.1.1)

This implies that A − λI is one-to-one and with closed range. In fact, if un is a sequence in Dom(A) such that (A − λ)un → v, then by (4.1.1) we have that un and Aun are convergent. But A is closed, hence un → u ∈ Dom(A) and (A − λI)un → (A − λI)u. This implies that v = (A − λI)u belongs to the range of A − λI, which is therefore closed. Since A is self-adjoint, the range of A − λI is then the orthogonal space of Ker(A − λI) = {0}, so that A − λI is onto. Hence A − λI : Dom(A) → H is

4.1. Unbounded Operators in Hilbert spaces

157

invertible. Moreover its inverse is bounded, because from (4.1.1) we obtain, for u = (A − λI)−1 v,

v L2 ≥ |η|(A − λI)−1 v L2 . This means that λ ∈ ρ(A).

We say that A has compact resolvent if there exists λ ∈ ρ(A) such that RA (λ) is compact. Proposition 4.1.3. If A has compact resolvent, then RA (λ) is compact for all λ ∈ ρ(A). Proof. Let λ0 ∈ ρ(A) be such that RA (λ0 ) is compact. Consider λ ∈ ρ(A) and let A − λI = A − λ0 I − (λ − λ0 )I.

(4.1.2)

Multiplication of (4.1.2) from the left by RA (λ0 ) and from the right by RA (λ) yields RA (λ) − RA (λ0 ) = (λ − λ0 )RA (λ)RA (λ0 ). This identity shows that RA (λ) is compact.

Self-adjoint operators with compact resolvent have very simple spectrum. In order to show this, we use the following well-known result about the spectrum of compact operators, combined with the subsequent lemma. Theorem 4.1.4. Let T be a compact operator on a complex Hilbert space H. Then σ(T ) is an at most countable set with no accumulation point different from 0. Each non-zero λ ∈ σ(T ) is an eigenvalue with finite multiplicity. If T is also selfadjoint, then all the eigenvalues are real and H has an orthonormal basis made of eigenvectors of T . Lemma 4.1.5. Let A be a closed densely defined operator on a Hilbert space H such that ρ(A) = ∅. Then for any λ0 ∈ ρ(A) we have * + ρ(A) = {λ0 } ∪ λ ∈ C : λ = λ0 and (λ − λ0 )−1 ∈ ρ RA (λ0 ) . Proof. Let B = A − λ0 I. Then we have RA (λ0 ) = B −1 and λ ∈ ρ(A) ⇐⇒ λ − λ0 ∈ ρ(B). We have to prove that λ ∈ ρ(B) ⇐⇒ λ = 0, or λ = 0 and λ−1 ∈ ρ(B −1 ). Let λ ∈ ρ(B), λ = 0. Then S := BRB (λ) = I + λRB (λ)

158


is bounded, and

B −1 S = RB (λ) = λ−1 (S − I).

This means that −1

(B −1 − λ−1 I)S = −λ−1 I.

−1

−1

(4.1.3)

−1

Hence B − λ I is onto. But B − λ I is also one-to-one, because from (B −1 − λ−1 I)u = 0 we obtain λu = Bu, which implies u = 0 because λ ∈ ρ(B). Then B −1 − λ−1 I is invertible. Moreover (4.1.3) shows that −1 −1 B − λ−1 I = −λS, which is bounded. Let now λ = 0, λ−1 ∈ ρ(B −1 ), and set S = B −1 RB−1 (λ−1 ) = I + λ−1 RB−1 (λ−1 ). Then S is bounded and so

BS = λ(S − I),

(4.1.4)

(B − λI)S = −λI.

This implies that B − λI is onto. Moreover it is one-to-one, because from (B − λI)u = 0 we obtain B −1 u = λ−1 u, which implies u = 0 because λ−1 ∈ ρ(B −1 ). Then we have that B − λI is invertible. Moreover (4.1.4) shows that (B − λI)−1 = −λ−1 S , which is bounded. Now we can describe the spectrum of a self-adjoint operator with compact resolvent. Theorem 4.1.6. Let A be a densely defined self-adjoint operator on a complex Hilbert space H (of infinite dimension). If A has compact resolvent, then σ(A) is a sequence of real isolated eigenvalues, diverging to ∞. Each eigenvalue has finite multiplicity and H has an orthonormal basis made of eigenfunctions of A. Proof. Fix any λ0 ∈ ρ(A), then RA (λ0 ) is compact. By Theorem 4.1.4 and Lemma 4.1.5, σ(A) must be at most countable. This meansthat there exists μ ∈ ρ(A) ∩ R. Then RA (μ) is compact self-adjoint. Moreover Ker RA (μ) = 0 because RA (μ) is one-to-one. The result now follows from Theorem 4.1.4 and Lemma 4.1.5 (applied now with μ in place of λ0 ).

4.2 Pseudo-Differential Operators in L2 : Realization and Spectrum We deal now with pseudo-differential operators having symbols in the classes S(M ) = S(M ; Φ, Ψ) and Hypo(M, M0 ) = Hypo(M, M0 ; Φ, Ψ); see Definitions 1.1.1 and 1.3.2. Moreover, we use here the Weyl quantization, cf. (1.0.10).

4.2. Pseudo-Differential Operators in L2 : Realization and Spectrum

159

Given two symbols a ∈ S(M1 ) and b ∈ S(M2 ), the Weyl symbol of the composition aw bw will be denoted by a#b. Hence we recall from Theorem 1.2.17 that a#b ∈ S(M1 M2 ) and the remainder term can be rewritten as RN (a, b) = a#b −

N {a, b}j j

j=0

(2i) j!

∈ S(M1 M2 hN +1 ),

(4.2.1)

for all N ∈ N, where {a, b}0 := ab, ⎡ ⎤

j n ∂ ∂ ∂ ∂ {a, b}j : = ⎣ − a(x, η)b(y, ξ)⎦ ∂η ∂y ∂x i i i ∂ξi y=x i=1

= j!

η=ξ |β|

(−1)

(α!β!)−1 ∂ξα ∂xβ a(x, ξ)∂ξβ ∂xα b(x, ξ),

(4.2.2)

|α+β|=j

for j > 0, and h is the Planck function in (1.1.8). More precisely, for each N, k ∈ N there exists an integer lN,k such that

RN (a, b) k,S(M1 M2 hN +1 ) a lN,k ,S(M1 ) b lN,k ,S(M2 )

(4.2.3)

for all a ∈ S(M1 ) and all b ∈ S(M2 ). This is easily seen, e.g., from (4.2.1) as a consequence of the Closed Graph Theorem. In the sequel we will also use the following notation: for k ∈ N, a ∈ C ∞ (R2d ) we set |a|k (x, ξ) =

sup |α|+|β|≤k

|∂ξα ∂xβ a(x, ξ)|Ψ(x, ξ)−|α| Φ(x, ξ)−|β| .

(4.2.4)

Hence, a symbol a ∈ Hypo(M, M0 ) satisfies |a|k |a| for every k ∈ N, when |x| + |ξ| ≥ R, with R large enough. Now, given a ∈ S(M ), we regard its Weyl quantization as an unbounded operator in L2 (Rd ) with dense domain S(Rd ), namely, we consider the operator A with domain S(Rd ) defined as Au = aw u, for u ∈ S(Rd ). When not specified, aw is understood as a continuous operator on S (Rd ). As we saw in the summary of the present chapter, the restriction of aw to the linear subspace (4.0.1) is a closed extension of A, the maximal realization of A. Hence A is closable. As we observed there, the closure is also called minimal realization. We now study the relationship between the formal adjoint and adjoint in L2 (Rd ) for a pseudo-differential operator with symbol a ∈ S(M ). Given a ∈ S(M ), let A be as above, and let A∗ be its adjoint in L2 (Rd ), as defined in the previous section. Consider moreover the formal adjoint (aw )∗ ∗ (= aw ) defined in Section 1.2.2, namely the operator (aw ) : S(Rd ) → S(Rd ) such that ∗ (4.2.5) ((aw ) u, v)L2 = (u, aw v)L2 for u, v ∈ S(Rd ).

160


Observe that A is symmetric if and only if it is formally self-adjoint, which is equivalent to saying that a is real-valued. We now show that A∗ coincides with ∗ the maximal realization of the formal adjoint (aw ) , when regarded as an operator 2 d d in L (R ) with dense domain S(R ). Denote by A+ : S (Rd ) → S (Rd ) the continuous extension to S (Rd ) of the pseudo-differential operator (aw )∗ . Proposition 4.2.1. Let a ∈ S(M ; Φ, Ψ) and let A, A+ be as above. Then the adjoint A∗ in L2 (Rd ) has domain * + (4.2.6) Dom(A∗ ) = u ∈ L2 (Rd ) : A+ u ∈ L2 (Rd ) , and coincides with the restriction of A+ : S (Rd ) → S (Rd ) to such a linear subspace. Proof. Given u ∈ Dom(A∗ ), A∗ u is the unique element of L2 (Rd ) such that (A∗ u, v)L2 = (u, Av)L2 = (A+ u, v), This means that and that

for v ∈ S(Rd ).

A∗ = A+ |Dom(A∗ ) * + Dom(A∗ ) ⊂ u ∈ L2 (Rd ) : A+ u ∈ L2 (Rd ) .

Vice-versa, let u ∈ L2 (Rd ) be such that A+ u ∈ L2 (Rd ). Then we have (A+ u, v)L2 = (u, Av)L2 ,

for v ∈ S(Rd ).

This implies that v → (u, Av)L2 is continuous and therefore u ∈ Dom(A∗ ).

We next result shows that, for operators with hypoelliptic symbols, the minimal and maximal realizations coincide. We need the following easy preliminary results. Lemma 4.2.2. Given two smooth functions a, b ∈ C ∞ (R2d ), we have k |a|j+l |b|j+k−l hj , {a, b}j k

l=0

for all k, j ∈ N. Proof. This follows from the very definition (4.2.2) and Leibniz’ formula.

Lemma 4.2.3. Let φj be a bounded sequence in S(1; Φ, Ψ), converging in some w 2 d symbol class S(M ; Φ, Ψ) to φ ∈ S(1; Φ, Ψ). Then φw j u → φ u in L (R ) for all 2 d u ∈ L (R ). Proof. In view of the uniform estimate (1.4.1), it suffices to prove the desired result w d when u ∈ S(Rd ). In that case we have φw j u → φ u in S(R ) by Remark 1.2.8.


161

Theorem 4.2.4. Consider a pseudo-differential operator A with Weyl symbol a ∈ Hypo(M, M0 ; Φ, Ψ). Assume there exists N0 ∈ N such that hN0

inf {1, M0 } . M

(4.2.7)

Then A has domain {u ∈ L2 (Rd ) : aw u ∈ L2 (Rd )} and coincides with the restriction of aw to this subspace. Proof. We have to show that for any u ∈ L2 (Rd ) such that aw u ∈ L2 (Rd ), there exists a sequence uj in S(Rd ) such that uj → u

and

aw uj → aw u

in L2 (Rd ).

By arguing as in the proof of Theorem 1.3.6 one sees that there exists q ∈ Hypo(M0−1 , M −1 ) such that r := 1 − q#a ∈ S(hN0 +1 ). By Proposition 1.1.5 there exists a sequence of symbols χj ∈ S(R2d ) which is bounded in S(1) and converges to 1, say, in S(1 + |x| + |ξ|). Then we set uj = χw j u. We have that uj ∈ S(Rd ), because χj ∈ S(R2d ). From the definition of r we have w w w w w aw uj = aw χw j q a u + a χj r u

= (a#χj #q)w aw u + (a#χj #r)w u.

(4.2.8)

Thanks to (4.2.3) and Lemma 4.2.2, and using the fact that a and q are hypoelliptic symbols, it follows that for all k ∈ N there exist l, l ∈ N such that the following estimates hold true: |a#χj #q|k ≤

N0

1 + |RN (a#χj , q)| {a#χ , q} j 0 n k n n! k 2 n=0

M hN0 +1 M0 m≤l m≤l

M hN0 +1 N0 +1 sup |q|m + sup |a|m sup |χj |m + M h M0 m≤l m≤l m≤l N0 +1 Mh −1 |a| + M hN0 +1 |a| + M0 N0 +1 Mh 1+ 1, M0 sup |a#χj |m sup |q|m +

162


for all |x| + |ξ| ≥ R and j ∈ N, with R large enough. As a consequence the symbols a#χj #q belong to a bounded subset of S(1). Moreover from (4.2.7) and (4.2.3) we have also

a#χj #r k,S(1) ≤ a#χj #r k,S(M hN0 +1 ) a l,S(M ) χj l,S(1) r l,S(hN0 +1 ) , for some l = lk ∈ N and all j ∈ N. This means that also the sequence a#χj #r belongs to a bounded set of S(1). On the other hand, a#χj #q and a#χj #r converge to a#q and a#r respectively, in suitable symbol classes, by (4.2.1), (4.2.2) and (4.2.3). 2 d w From Lemma 4.2.3 we obtain that uj = χw j u → u in L (R ), whereas a uj in (4.2.8) converges to w

w

(a#q) aw u + (a#r) u = (a#(q#a + r))w u = aw u in L2 (Rd ), which concludes the proof.

We now study the spectrum of operators with hypoelliptic symbols. Proposition 4.2.5. Assume the strong uncertainty principle (1.1.10). Consider a pseudo-differential operator A with Weyl symbol in Hypo(M, M0 ; Φ, Ψ), with M0 (x, ξ) → +∞ at infinity. Then its closure A in L2 has either spectrum σ(A) = C or has compact resolvent. Proof. Let σ(A) = C. Then there exists λ0 ∈ ρ(A) such that A−λ0 I has a bounded inverse RA (λ0 ). By Theorem 1.3.6 there exists a parametrix B ∈ S(M0−1 ) of A. This means that S = BA − I is regularizing. In particular, by Theorem 1.4.2, B and S extend to compact operators on L2 (Rd ). This implies that RA (λ0 ) = B + (λ0 B − S)RA (λ0 ) is compact.

Remark 4.2.6. It may happen that σ(A) = C. For example the ordinary differential operator A = iD + x has Weyl symbol iξ + x, which is elliptic in the class S(M ; Φ, Ψ), with M (x, ξ) = Φ(x, ξ) = Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )1/2 , and has spectrum equal to C. Indeed, any λ ∈ C is an eigenvalue, with eigenfunction 2 u(x) = e−(x−λ) /2 . Hence, we get the following result. Corollary 4.2.7. Assume the strong uncertainty principle. Consider a pseudodifferential operator A having real-valued Weyl symbol in Hypo(M, M0 ; Φ, Ψ), with M0 (x, ξ) → +∞ at infinity. Then A is essentially self-adjoint, and its closure A in L2 has compact resolvent.


163

Proof. The fact that A is essentially self-adjoint follows from Theorem 4.2.4 and Proposition 4.2.1. The remaining part of the statement is a consequence of Propositions 4.1.2 and 4.2.5. Bounds on the spectrum of pseudo-differential operators will be obtained in the next theorem, by using the following result. Lemma 4.2.8. Assume the strong uncertainty principle. Consider a real-valued positive symbol a ∈ Hypo(M, M0 ; Φ, Ψ). Then aw is bounded from below, i.e., there exists a positive constant C such that (aw u, u)L2 (Rd ) ≥ −C u 2L2 (Rd )

for u ∈ S(Rd ).

Proof. For every N ∈ N, we will prove the existence of a symbol c ∈ S(M 1/2 ) such that ∗ w , rN ∈ S(hN M ), (4.2.9) cw (cw ) − aw = rN which by Theorem 1.4.1 implies the desired lower bound, if N is so large that hN M 1. 1/2 Now, it follows from Lemma 1.3.5 that b := a1/2 ∈ Hypo(M 1/2 , M0 ). As a consequence of (4.2.3) and Lemma 4.2.2 we obtain that I − bw a w b w = r w , where r ∈ S(h) is real; cf. the proof of Lemma 4.2.3. Denote now by TN (x) the first N terms in the power series expansion of (1 − x)1/2 at 0. Since 1 − x − TN (x)2 is a polynomial divisible by xN , it turns out that I − rw − TN (rw )2 has symbol in S(hN ), so that TN (rw )2 − bw aw bw has symbol in S(hN ; Φ, Ψ). Let B a parametrix of bw , with real-valued Weyl symbol. If we set cw = BTN (r w ), since B and TN (r w ) are formally self-adjoint we obtain (4.2.9). Notice that, in comparison with the Sharp Gårding Inequality in Theorem 1.7.15, the conclusion in Lemma 4.2.8 is much stronger, but we have here the additional hypothesis that a is hypoelliptic. Now we can prove the spectral theorem for operators with hypoelliptic symbols. Theorem 4.2.9. Assume the strong uncertainty principle (1.1.10). Consider a pseudo-differential operator A, with real-valued Weyl symbol in the class Hypo(M, M0 ; Φ, Ψ), and assume M0 (x, ξ) → +∞ at infinity. Its closure A in L2 (Rd ) has spectrum given by a sequence of real eigenvalues either diverging to +∞ or −∞. The eigenvalues have all finite multiplicity and the eigenfunctions belong to S(Rd ). Moreover L2 (Rd ) has an orthonormal basis made of eigenfunctions of A.

164


Proof. The closure A of A is self-adjoint and has a compact resolvent by Corollary 4.2.7. Then we can apply Theorem 4.1.6. Hence there are infinite eigenvalues diverging to ∞. The corresponding eigenfunctions must belong to S(Rd ) by Corollary 1.3.9. If we show that A is semi-bounded we have that its eigenvalues diverge either to + or −∞. Since |a(x, ξ)| M0 (x, ξ) > 0, for |x| + |ξ| ≥ R, and a(x, ξ) is real-valued, a(x, ξ) has always the same sign on the connected set |x| + |ξ| ≥ R, i.e., it is either bounded from below or from above. Lemma 4.2.8 then implies that A is semibounded. So are its eigenvalues, which must diverge to +∞ or −∞ according to the sign of a. The proof therefore is complete.

4.3 Complex Powers This section is devoted to the study of the complex powers of hypoelliptic pseudodifferential operators. We begin by recalling some results on complex powers of a non-negative operator in the sense of Komatsu. Definition 4.3.1. A closed operator A on a Banach space X is called non-negative if (a) (−∞, 0) is contained in the resolvent set of A; −1 < ∞. (b) sup λ (A + λI) B(X)

λ∈R+

Remark 4.3.2. If A is a densely defined self-adjoint operator in a Hilbert space H, then A is non-negative if and only if (Au, u)H ≥ 0 for all u ∈ Dom(H) (see, e.g., Martínez Carracedo and Sanz Alix [141, Proposition 1.3.6]). * + Set C+ = z ∈ C : Re z > 0 and γk (z) =

(k − 1)! sin πz Γ(k) = , Γ(z)Γ(k − z) (k − 1 − z) · · · (1 − z)π

(4.3.1)

for k ∈ N, k = 0 and z ∈ C \ Z. Proposition 4.3.3. Consider a non-negative operator A on a Banach space X. Given z ∈ C+ , and u ∈ Dom(A[Re z]+1 ) we have that the integral ∞ k −1 z IA,k u = γk (z) λz−1 A (A + λI) u dλ (4.3.2) 0

is absolutely convergent for all integers k > Re z, as an improper Riemann integral taking values in X. Moreover these integrals are independent of k: z z u = IA,k u, IA,k+1

k > Re z.

4.3. Complex Powers

165

Proof. See [141, Proposition 3.1.3].

Following Balakhrishnan, we now define the complex powers of a non-negative operator. Definition 4.3.4. Given a non-negative operator A on a Banach space X and a z complex number z ∈ C+ , define a new operator JA on X as z ) = Dom A[Re z]+1 , Dom (JA z z u = IA,k u, for any k > Re z. JA Theorem 4.3.5. Assume that A is a non-negative, densely defined operator on a Banach space X, then z , z ∈ C+ , Az = JA is the unique family of operators which enjoys the following set of properties: z ; (a) Az extends JA

(b) Az is closed; (c) A1 = A; (d) Az Aw = Az+w for all z, w ∈ C+ . In particular, Ak = AA · · · A5, with k ∈ N. Moreover we have: 2 34 k−times

(e) (Spectral Mapping Theorem) the spectrum of Az is given by1 σ(Az ) = {λz : λ ∈ σ(A)} ; (f) For all u ∈ Dom(An ), with n ∈ N, n = 0, the mapping z → Az u * + is analytic in the strip z ∈ C : 0 < Re z < n . Proof. See [141, Theorems 3.1.5 and 3.1.8, Corollary 5.1.12 and Section 6.2] .

Let now a be a symbol in some class S(M ; Φ, Ψ) and consider the unbounded operator A in L2 (Rd ) with domain S(Rd ), defined by Au = aw u, u ∈ S(Rd ). In z Theorem 4.3.6 we show that under suitable hypotheses A is pseudo-differential. Theorem 4.3.6. Assume the strong uncertainty principle (1.1.10) and consider a hypoelliptic symbol a ∈ Hypo(M, M0 ; Φ, Ψ) such that Re a(x, ξ) ≥ −R |Im a(x, ξ)| ,

(4.3.3)

1 The complex power λz is the principal branch λz = exp z (log |λ| + i arg λ) , with −π < arg λ ≤ π.

166


for |x| + |ξ| ≥ R, where R is a positive constant such that estimates (1.3.2) and (1.3.3) are satisfied. Let A be as above and suppose that A is non-negative. Let a0 = a + χ,

(4.3.4)

where χ ∈ C0∞ (R2d ), χ ≥ 0 everywhere, and χ(x, ξ) > 0 for |x| + |ξ| ≤ R. Then for all z ∈ C+ there exists a hypoelliptic symbol a#z ∈ Hypo M Re z , M0Re z ; Φ, Ψ such that (i) for all k ∈ N and z ∈ C+ we have #z a − az0 (x, ξ) |a0 (x, ξ)|Re z h(x, ξ), k

for (x, ξ) ∈ R2d ,

(4.3.5)

where h is the Planck function defined in (1.1.8), (see (4.2.4) for the notation | · |k ); (ii) for all z ∈ C+ we have z

A = (a#z )w |S(Rd ) .

(4.3.6)

We shall prove this theorem in Section 4.3.2. We observe that the power az0 is well defined by taking the principal branch of the logarithm, by (4.3.7) below. Notice also that from Theorems 4.3.5 and 4.3.6 it follows that a#k = a# · · · #a, when k ∈ N, k = 0. 2 34 5 k−times z z We can also give a simple description of the domain Dom A of A . Corollary 4.3.7. Under the hypotheses of Theorem 4.3.6, we have w + z * Dom A = u ∈ L2 (Rd ) : a#z u ∈ L2 (Rd ) ,

z ∈ C+ ,

w is here regarded as an operator on S (Rd ). where a#z Proof. Because a#z is hypoelliptic, the result follows from Theorems 4.3.6 and 4.2.4.

4.3.1 The Resolvent Operator z

As formula (4.3.2) suggests, to study the complex powers A we have to understand the structure of the resolvent operator of A. To to this, we need some lemmata. The proof of the first lemma is elementary and is left to the reader.

4.3. Complex Powers

167

Lemma 4.3.8. Let a0 be given in (4.3.4). Possibly for a greater R we have for all (x, ξ) ∈ R2d .

Re a0 (x, ξ) > −R |Im a0 (x, ξ)| , As a consequence,

6 |a0 (x, ξ)| ≤ 1 + R2 |a0 (x, ξ) + λ| , 6 λ ≤ 1 + R2 |a0 (x, ξ) + λ| ,

(4.3.7)

(4.3.8) (4.3.9)

for all (x, ξ) ∈ R2d and λ ≥ 0. Lemma 4.3.9. Let a0 be given in (4.3.4). Then, for all k ∈ N, it turns out that for all (x, ξ) ∈ R2d .

|a0 |k (x, ξ) |a0 (x, ξ)| , Moreover, for all k ∈ N we have 1 −1 , (a0 + λ) (x, ξ) |a0 (x, ξ) + λ| k

(4.3.10)

for all (x, ξ) ∈ R2d and λ > 0.

(4.3.11)

Proof. Since a is hypoelliptic, a0 − a has compact support and a0 never vanishes, it follows that (4.3.10) holds. This fact, together with (4.3.8) shows that a0 + λ satisfies the estimates |a0 + λ|k |a0 + λ| uniformly with respect to λ. Hence, arguing as in the proof of Lemma 1.3.5 gives (4.3.11), as desired. From the strong uncertainty principle and Lemma 4.3.8 one easily obtains the following result. Lemma 4.3.10. Assume the strong uncertainty principle (1.1.10) and let a0 be given in (4.3.4). Consider two real numbers ν, ν0 such that (1 + |x| + |ξ|)−ν0 |a0 (x, ξ)| (1 + |x| + |ξ|)ν .

(4.3.12)

Then we have h(x, ξ)ν0 /δ (1 + λ) |a0 (x, ξ) + λ| h(x, ξ)−ν/δ (1 + λ) ,

(4.3.13)

for all (x, ξ) ∈ R2d and λ ≥ 0. Lemma 4.3.11. Assume the strong uncertainty principle, and let a0 be given in (4.3.4). Assume that for all λ ∈ R+ two symbols φλ and ψλ are given, such that for all k ∈ N we have

|φλ |k |a0 |L |a0 + λ|J hN ,

|ψλ |k |a0 |

L

|a0 + λ|

J

hN ,

(4.3.14)

for λ > 0, with L , L , J , J , N , N ∈ R. Then for all N, k ∈ N we have |φλ #ψλ |k |a0 | and

L +L

|a0 + λ|

|RN (φλ , ψλ )|k |a0 |L +L |a0 + λ|J

J +J

+J

for λ > 0, where RN (φλ , ψλ ) is defined in (4.2.1).

hN

hN

+N

,

+N +N +1

(4.3.15) ,

(4.3.16)

168


Proof. From (4.3.14) and (4.3.13), for all k ∈ N we obtain J

φλ k,S(hγ ,g) (1 + λ) , J

ψλ k,S(hγ ,g) (1 + λ)

λ > 0, ,

λ > 0,

with 2 γ = L− + J− ν0 /δ − L+ + J+ ν/δ + N , γ = L− + J− ν0 /δ − L+ + J+ ν/δ + N . From Lemma 4.2.2 and (4.2.3) for all N0 , k ∈ N there exists an integer l = lN0 ,k ≥ N0 + k such that |φλ #ψλ |k sup |φλ |j sup |ψλ |j + |RN0 (φλ , ψλ )|k 0≤j≤l

0≤j≤l

sup |φλ |j sup |ψλ |j + RN0 (φλ , ψλ ) k,S(hγ +γ +N0 +1 ) hγ 0≤j≤l

|a0 |

L +L

+γ +N0 +1

0≤j≤l

sup |φλ |j sup |ψλ |j + φλ l,S(hγ ) ψλ l,S(hγ ) hγ 0≤j≤l

+γ +N0 +1

0≤j≤l

J +J

|a0 + λ|

hN

+N

J +J

+ (1 + λ)

hγ

+γ +N0 +1

,

for λ > 0. Then (4.3.15) follows from (4.3.13), when we choose N0 ≥ L+ + L+ ν0 /δ − L− + L− ν/δ+ + J+ + J+ ν0 /δ − J− + J− ν/δ + N + N − γ − γ − 1 = (|L | + |L | + |J | + |J |) (ν0 + ν) /δ − 1.

The proof of (4.3.16) is similar.

Theorem 4.3.12. Assume the strong uncertainty principle. Let a ∈ Hypo(M, M0 ; Φ, Ψ) satisfying (4.3.3). Then for each λ > 0 there exists a hypoelliptic symbol −1 qλ ∈ Hypo M0−1 , (M + λ) ; Φ, Ψ , such that (a) for all k ∈ N we have qλ −

1 h , a0 + λ k |a0 + λ|

where a0 is defined in (4.3.4); 2

As usual we set x− = min{x, 0} and x+ = max{x, 0}.

for λ > 0,

(4.3.17)

4.3. Complex Powers

169

(b) for all k, N ∈ N we have |1 − qλ #(a + λ)|k hN , |1 − (a + λ)#qλ |k hN ,

(4.3.18)

for λ > 0. Proof. Define r1,λ = 1 − (a0 + λ)−1 # (a + λ) , a1 = a − a0 .

(4.3.19)

Because a1 has compact support, by (4.3.12) and (4.3.8), for all k ∈ N we have |a1 |k (x, ξ) (1 + |x| + |ξ|)−ν h(x, ξ) |a0 (x, ξ)| h(x, ξ) |a0 (x, ξ) + λ| h(x, ξ), (4.3.20) and |a + λ|k |a0 + λ| , for λ > 0. Then from Lemma 4.3.11 and (4.3.11) we may conclude that for all k ∈ N we have a1 + R0 (a0 + λ)−1 , a + λ h, |r1,λ |k (4.3.21) a0 + λ k k for λ > 0, where R0 is defined in (4.2.1). Define r0,λ = 1 and rj,λ = r1,λ # · · · #r1,λ , j times, when j ≥ 1. Then from Lemma 4.3.11, (4.3.21) and (4.3.11), for all j, k ∈ N we have |rj,λ |k hj , −1 rj,λ # (a0 + λ) k

(4.3.22) j

h , |a0 + λ|

(4.3.23)

for λ > 0. −1 Now we consider the asymptotic sum of the symbols rj,λ # (a0 + λ) . An application of Proposition 1.1.6 (or better, of its proof3 ) shows that there exists qλ such that N −1 rj,λ # (a0 + λ) ∈ S(|a0 + λ|−1 hN +1 ), (4.3.24) qλ − j=0

for all N ∈ N, uniformly with respect to λ > 0. Estimate (4.3.17) follows from (4.3.24) with N = 0, which also gives |qλ |k

1 , |a0 + λ|

for λ > 0.

(4.3.25)

3 Although |a + λ| need not be temperate, we can apply that result to the space S(|a + 0 0 λ|−1 hN +1 ), because in the proof of Proposition 1.1.6 we did not use that hypothesis.

170


Since h(x, ξ) → 0 as |x| + |ξ| → +∞, (4.3.17) and (4.3.25) imply that qλ ∈ Hypo(M0−1 , (M + λ)−1 ). It remains to prove estimates (4.3.18). Setting cλ = q λ −

N

rj,λ # (a0 + λ)

−1

,

j=0

by (4.3.24) and Lemma 4.3.11 we have 1 − qλ # (a + λ) = 1 −

N

rj,λ # (a0 + λ)

−1

# (a + λ) − cλ # (a + λ)

j=0

=1−

N

rj,λ #(1 − r1,λ ) − cλ #(a + λ)

j=1

= rN +1,λ − cλ #(a + λ) ∈ S(hN +1 ), uniformly with respect to λ, which implies the first one of the estimates (4.3.18). In order to prove the second estimate in (4.3.18), we observe that starting from −1 r˜1,λ = 1 − (a + λ) # (a0 + λ) , and r1,λ , r˜j,λ = r˜1,λ # · · · #˜ 2 34 5

j ≥ 1,

j−times

we may consider q˜λ ∼

∞

−1

(a0 + λ)

#˜ rj,λ ∈ S(|a0 + λ|−1 ),

j=0

such that for all k, N ∈ N we have |1 − (a + λ)#˜ qλ |k hN ,

for λ > 0.

Thus, from Lemma 4.3.11 and (4.3.25), for all k, N ∈ N we have |qλ − qλ #(a + λ)#˜ qλ |k

hN , |a0 + λ|

whereas from the first estimate in (4.3.18) and (4.3.26) we get qλ |k |˜ qλ − qλ #(a + λ)#˜ so that |qλ − q˜λ |k

hN , |a0 + λ|

hN . |a0 + λ|

(4.3.26)

4.3. Complex Powers

171

Hence |1 − (a + λ)#qλ |k ≤ |1 − (a + λ)#˜ qλ |k + |(a + λ) # (qλ − q˜λ )|k hN ,

for λ > 0.

Theorem 4.3.13. Under the hypotheses of Theorem 4.3.6, we have that for all λ > 0 −1 the operator aw + λI : S(Rd ) → S(Rd ) is invertible and aw (aw + λI) : S(Rd ) → d S(R ) is a hypoelliptic pseudo-differential operator with Weyl symbol bλ ∈ Hypo(1, M0 /(M + λ); Φ, Ψ) such that for all k ∈ N we have the estimates bλ − a0 a0 h , a0 + λ k a0 + λ

(4.3.27)

for λ > 0. Proof. We begin by observing that from Corollary 1.3.9 we have the following global regularity property: for all λ > 0 we have u ∈ S (Rd ) and (aw + λI) u ∈ S(Rd ) =⇒ u ∈ S(Rd ), where aw is here understood as an operator on S (Rd ). Now we show that aw +λI : S(Rd ) → S(Rd ) is invertible for all λ > 0. Let A be the unbounded operator in L2 (Rd ) with domain S(Rd ) defined by Au = aw u, u ∈ S(Rd ). Then aw + λI is one-to-one because, by hypothesis, A + λI is. On the other hand we know that the range of A+λI is L2 (Rd ). Therefore, given d 2 d any f ∈ S(R ) there exists u ∈ L (R ) such that A + λI u = f . But u ∈ S(Rd ) by the above mentioned global regularity, and therefore (aw + λI) u = f , that is aw + λI is onto. Then for all λ > 0 we may consider aw (aw + λI)−1 : S(Rd ) → S(Rd ). We want to show that this operator is pseudo-differential. We have −1

aw (aw + λI)

= aw qλw + qλw aw (1 − (a + λ) #qλ ) w

−1

+ (1 − qλ # (a + λ)) aw (aw + λI)

w w

(1 − (a + λ) #qλ ) .

For all r, s ∈ R, consider the Sobolev spaces H r,s := H(Φr Ψs ), cf. Definition 1.5.2. We have S(Rd ) =

r,s

H r,s ,

S (Rd ) =

r,s

H r,s ,

(4.3.28)

172


with the topologies of S(Rd ) and S (Rd ) equal to the initial and final topology of intersection and union. In particular it follows that an operator is continuous from S into S if and only if it is continuous from H r,s into H p,q for all r, s, p, q ∈ R. From (4.3.18), the strong uncertainty principle and Proposition 1.5.5 (a) we obtain easily that for all r, s, p, q ∈ R we have w

w

(1 − qλ # (a + λ)) u H r,s u H p,q , (1 − (a + λ) #qλ ) u H r,s u H p,q ,

(1 − qλ # (a + λ))w aw u H r,s u H p,q , for all u ∈ S(Rd ) and λ > 0. Let w

−1

Sλ = (1 − qλ # (a + λ)) aw (aw + λI)

w

(1 − (a + λ) #qλ ) .

By hypothesis A + λI is non-negative, so we have −1 −1 ≤ 1 + sup λ A + λI sup A A + λI λ>0

B(L2 )

λ>0

B(L2 )

(4.3.29)

< ∞.

It follows that for all r, s, p, q ∈ R we have

Sλ u H r,s u H p,q

and

Sλ u H r,s

1

u H p,q , λ

and therefore

1

u H p,q (1 + λ)−1 u H p,q ,

Sλ u H r,s min 1, λ

(4.3.30)

for all u ∈ S(Rd ) and λ > 0. These estimates imply that Sλ is a regularizing pseudo-differential operator, with Weyl symbol σλ satisfying for all α, β ∈ Nd and N ∈ N the estimates −N α β Dξ Dx σλ (x, ξ) (1 + |x| + |ξ|) , 1+λ

for all (x, ξ) ∈ R2d and λ > 0. −1

From (4.3.28) and (4.3.29) we obtain that aw (aw + λ) operator with Weyl symbol

is a pseudo-differential

bλ = a#qλ + qλ #a − qλ #a# (a + λ) #qλ + σλ . Then the result follows from Theorem 4.3.12, (4.3.19), (4.3.20), Lemmata 4.3.10 and 4.3.11. −1

Corollary 4.3.14. For all λ > 0 we have that (aw + λI) operator with hypoelliptic Weyl symbol a ˜λ =

1 (1 − bλ ) . λ

is a pseudo-differential

(4.3.31)

4.3. Complex Powers

173

Proof. We have (aw + λI)−1 =

1 1 I − aw (aw + λI)−1 = (I − bw λ). λ λ

Remark 4.3.15. Since a#˜ aλ = bλ , for all k ∈ N we have the estimates a0 h a0 , a#˜ a − λ a0 + λ k a0 + λ

(4.3.32)

for λ > 0.

4.3.2 Proof of Theorem 4.3.6 In all of this section we assume that the hypotheses of Theorem 4.3.6 are satisfied. Let a ˜λ be the symbol in (4.3.31). We set a ˜#0 λ = 1 and

a ˜#k a ˜λ # · · · #˜ aλ , λ =2 34 5

k ≥ 1.

k-times

Lemma 4.3.16. Given any symbol q ∈ S(M ; Φ, Ψ), for all (x, ξ) ∈ R2d and all k ∈ N, the function a#k λ ∈ R+ → q#˜ (x, ξ) (4.3.33) λ is smooth and ∂ #(k+1) (x, ξ) = −k q#˜ a (x, ξ). q#˜ a#k λ λ ∂λ

(4.3.34)

Proof. Because a ˜λ is the symbol of the resolvent (aw + λI)−1 , it must satisfy the resolvent identity: ˜λ0 = −(λ − λ0 )˜ aλ #˜ aλ0 , a ˜λ − a from which, thanks to Theorem 4.3.13, Corollary 4.3.14 and (4.3.8), we obtain for all k ∈ N the estimate ˜λ0 |k (x, ξ) |˜ aλ − a

|(λ − λ0 )| , λλ0

(x, ξ) ∈ R2d , λ > 0,

which implies that λ → (˜ aλ ) is continuous as a map valued in S(1; Φ, Ψ). It follows that aλ0 ) (x, ξ) (q#˜ aλ ) (x, ξ) − (q#˜ = − (q#˜ aλ #˜ aλ0 ) (x, ξ) λ − λ0 aλ0 ) (x, ξ), as λ → λ0 . This proves (4.3.34) for k = 1. converges to − (q#˜ aλ0 #˜ The case corresponding to k > 1 follows by induction. Identity (4.3.34) implies also that (4.3.33) is smooth.

174


Lemma 4.3.17. Given z ∈ C+ and (x, ξ) ∈ R2d , we have that the integral ∞ #k pa,z,k (x, ξ) = γk (z) λz−1 (a#˜ aλ ) (x, ξ) dλ 0

is convergent for all integers k > Re z, where γk (z) is defined in (4.3.1). Moreover, we have pa,z,k (x, ξ) = pa,z,k+1 (x, ξ),

k > Re z.

(4.3.35)

Proof. From (4.3.32) and Lemma 4.3.11 for all k, l ∈ N we obtain the estimate: a0 (x, ξ) k #k , aλ ) (x, ξ) (4.3.36) (a#˜ a0 (x, ξ) + λ l for all (x, ξ) ∈ R2d and λ > 0 . This implies (pointwise) integrability. So we have only to prove (4.3.35). −1 ˜w Because a ˜λ is the symbol of the resolvent (aw + λI) , aw and a λ commute: a#˜ aλ = a ˜λ #a. Therefore, thanks to Lemma 4.3.16, an integration by parts gives ∞ ∞ #k z−1 λ (a#˜ aλ ) dλ = λz−1 a#k #˜ a#k dλ λ 0 0 1 z #k #k λ=∞ k ∞ z #k #(k+1) dλ λ a #˜ aλ + λ a #˜ aλ = z z 0 λ=0 k ∞ z−1 k ∞ z−1 #k #(k+1) λ (a#˜ aλ ) dλ − λ (a#˜ aλ ) dλ, = z 0 z 0 because #(k+1) #(k+1) λ a#k #˜ aλ aλ a#k + a#(k+1) #˜ = (λ + a) #˜ aλ # a#k #˜ λ = (a#˜ aλ ) It follows that ∞ 0

λz−1 (a#˜ aλ )

#k

dλ =

k k−z

0

∞

#k

.

λz−1 (a#˜ aλ )

#(k+1)

dλ,

which implies (4.3.35) because from (4.3.1) we have γk (z)k/(k − z) = γk+1 (z). Thanks to Lemma 4.3.17, for all (x, ξ) ∈ R2d and z ∈ C+ , we may define a#z (x, ξ) = pa,z,k (x, ξ), where k is any integer greater than Re z. Now we show that for all l ∈ N and z ∈ C+ we have #z a − az0 |a0 |Re z h. l

(4.3.37)

4.3. Complex Powers

175

Because h(x, ξ) → 0 as |x| + |ξ| → +∞, (4.3.37), (4.3.10) and Lemma 1.3.5 imply in particular that a#z ∈ S M Re z , M0Re z . From (4.3.32), Lemma 4.3.11 and Lemma 4.3.8 for all k, l ∈ N we obtain the estimate

k

k a |a0 | 0 #k aλ ) − h, (4.3.38) (a#˜ a0 + λ |a0 | + λ l

for λ > 0. On the other side, by using the identity (see, e.g., Gradshteyn and Ryzhik [94, 3.194.3, page 285]) ∞ z−1 k λ w wz , dλ = k (w + λ) γk (z) 0 with k ∈ N, z ∈ C+ and w ∈ C \ 0 such that Re z < k and |arg w| < π, we obtain 7

k 8 ∞ a 0 #k a#z − az0 = γk (z) aλ ) − dλ. λz−1 (a#˜ a0 + λ 0 So from (4.3.38) we get #z a − az |γk (z)| 0 l =

∞

Re z−1

λ 0

|a0 | |a0 | + λ

k h dλ

|γk (z)| |a0 |Re z h, γk (Re z)

which is (4.3.37). In order to prove (4.3.6) it suffices to verify that [Re z]+1 w z , z ∈ C+ . A u = a#z u, u ∈ Dom A w z Indeed, this shows that A is a closed extension of a#z |S(Rd ) , and also that w w (a#z ) |S(Rd ) , which has domain {u ∈ L2 (Rd ) : a#z u ∈ L2 (Rd )} by Proposiz . tion 4.2.1, is a closed extension of JA [Re z]+1 Now, when u ∈ Dom A we have z

z A u = JA u = γk (z)

∞

−1 k λz−1 A A + λI u dλ,

0

for k > Re z > 0, with convergence in L2 (Rd ). On the other hand, w −1 k #k A A + λI = (a#˜ aλ ) ,

176


on L2 (Rd ), so we need only show that ∞ w #z w #k u = γk (z) λz−1 (a#˜ u dλ, a aλ )

(4.3.39)

0

[Re z]+1 . for all k > Re z > 0 and all u ∈ Dom A We need the following result. Lemma 4.3.18. Consider a family of symbols φλ ∈ S(M ), λ > 0, such that the map (a) λ → φλ (x, ξ) is continuous on R+ , for all (x, ξ) ∈ R2d , (b) λ → φλ k,S(M ) is integrable on R+ , for all k ∈ N.

Then

∞

φλ (x, ξ) dλ

ψ(x, ξ) = 0

exists and belongs to S(M ) and ψw u =

∞ 0

φw λ u dλ,

(4.3.40)

for all u ∈ L2 (Rd ) such that the integral on the right-hand side of (4.3.40) converges in L2 (Rd ) as an improper Riemann integral. Proof. If we let k = 0 in hypothesis (b), we have that λ → φλ (x, ξ) is integrable on R+ . By hypothesis (a), ψ(x, ξ) is the pointwise limit of a sequence of Riemann sums Jj=1 φλj (x, ξ)Δλj . By (b), these Riemann sums are bounded in the symbol space S(M ) and therefore it follows from Proposition 1.1.2 that they converge to ψ also in S (R2d ) and that ψ ∈ S(M ). J 2 d Now, the Riemann sums j=1 φw λj u(x)Δλj converge by hypothesis in L (R ), ∞ w d and therefore in S (R ), to 0 φλ u(x) dλ. Hence it suffices to show that they also converge in S (Rd ) to ψ w u. This is clear if u ∈ S(Rd ), because of Proposition 1.2.2. The same conclusion actually holds for all u ∈ L2 (Rd ) by a “3 -argument”, since by Proposition 1.5.5 (a) and the Closed Graph Theorem we have, for some l ∈ N, the estimate

φw u H(M −1 ) φ l,S(M ) u L2 ,

φ ∈ S(M ), u ∈ L2 (Rd ),

and the Sobolev space H(M −1 ) is continuously embedded in S (Rd ) by Proposition 1.5.4 (a). We now apply Lemma 4.3.18 to the family of symbols aλ ) φλ (x, ξ) := λz−1 (a#˜

#k

(x, ξ),

4.4. Hilbert-Schmidt and Trace-Class Operators

177

which verifies (a) in Lemma 4.3.18 by Lemma 4.3.16. It remains to show that there exists a weight M such that the seminorms #k

λz−1 (a#˜ aλ ) l,S(M )

are integrable with respect to λ over R+ , for k > Re z and l ∈ N. Now, by (4.3.36) and Lemma 4.3.10 we obtain #k aλ ) (x, ξ) (1 + λ)k (a#˜ l sup sup < ∞, k, l ∈ N. h(x, ξ)−(ν+ν0 )/δ λ>0 (x,ξ)∈R2d Hence, for all k, l ∈ N we have aλ )

λz−1 (a#˜

#k

l,S(h−(ν+ν0 )/δ ,g)

λRe z−1 k

(1 + λ)

,

λ > 0,

which is integrable for k > Re z. This concludes the proof of Theorem 4.3.6.

4.4 Hilbert-Schmidt and Trace-Class Operators This section is devoted to the abstract theory of Hilbert-Schmidt and trace-class operators in a Hilbert space H (of infinite dimension), and to some criteria for a pseudo-differential operator to belong to those classes. Denote by B(H) the space of bounded linear operators on a complex Hilbert space H. B(H) is complete with respect to the operator norm

T B(H) = sup T u H . uH =1

Since we are dealing in this section with bounded operators, we mention that such an operator T is self-adjoint if and only if it is symmetric: (T u, v)H = (u, T v)H for all u, v ∈ H. We now prove that any self-adjoint operator T ∈ B(H) which is non-negative, in the sense that (T u, u)H ≥ 0 for all u ∈ H (cf. Remark 4.3.2), admits a unique square root, i.e., a self-adjoint non-negative operator S ∈ B(H) such that S 2 = T . To this end, we need r the following lemma. for r ∈ R, k ∈ N, k = 0, Set, as usual, k = r(r − 1)(r − 2) · · · (r − k + 1)/k! and 0r = 1 for r ∈ R (observe that, in particular, kr = 0 if r ∈ N, k > r). Lemma 4.4.1. Given a bounded operator B with norm B B(H) ≤ 1, then for each r > 0 the series ∞ r Br = Bk (4.4.1) k k=0

178


converges absolutely in B(H) and therefore Br commutes with all the operators commuting with B. Moreover we have Br+s = Br Bs ,

r, s > 0.

(4.4.2)

Finally, if B is self-adjoint, then Br is self-adjoint and non-negative. Proof. The absolute convergence the series is a consequence of the ∞ rof convergence ∞ r n of the numerical series k=0 k . To prove this, observe that k=0 k x coincides, except for the first [r] + 2 terms, with the Taylor series of the function −(1 − x)r if [r] is even, or (1 − x)r if [r] is odd ([r] stands for the integer part of r). ∞ In any case, the sum of the series k=0 kr xk , which is an increasing function for 0 ≤ x < 1, is therefore bounded there. Hence, N N ∞ r r k r k = lim x ≤ lim k x→1− k k x < ∞. x→1− k=0

k=0

k=0

Taking the limit as N → ∞ gives the desired conclusion. Since the convergence is absolute, equality (4.4.2) follows from the identity

r+s n

n s r , = n−k k

for n ∈ N,

k=0

which in turn can be proved by expanding as binomial series both sides of the equation (1 + x)r+s = (1 + x)r (1 + x)s . The remaining part of the statement is clear, because the map T → T ∗ is an 2 . isometry in B(H) and Br = Br/2 Theorem 4.4.2. Given a non-negative self-adjoint operator T ∈ B(H) there exists a unique non-negative self-adjoint operator S ∈ B(H) such that S 2 = T . Moreover S commutes with all operators commuting with T . √ S is called the square root of T and is denoted by T . Proof. Let us prove uniqueness. If S0 is another non-negative self-adjoint operator in B(H) such that S02 = T , then we claim that S and S0 must commute. Indeed, the operator i(S0 S − SS0 ) is self-adjoint, so it suffices to show that ((S0 S − SS0 )u, u)H = 0 for all u ∈ H. Now since S and S0 commute with T = S02 = S 2 , we have ((S0 S − SS0 )u, u)H = 2T 2 − (S0 S)2 − (SS0 )2 = −((S0 S − SS0 )u, u)H , which gives the claim. Hence, (S − S0 )2 S + (S − S0 )2 S0 = (S − S0 )(S 2 − S02 ) = 0.


179

But (S − S0 )2 S and (S − S0 )2 S0 are non-negative. Thus they must vanish, as well as their difference, which is (S − S0 )3 . This implies (S − S0 )4 = (S − S0 )(S − S0 )3 = 0. But then (S − S0 )2 = 0 and in turn S − S0 = 0 because S − S0 is self-adjoint; in fact, 0 = (S − S0 )4 u, u H = (S − S0 )2 u, (S − S0 )2 u H , for u ∈ H, 0 = (S − S0 )2 u, u H = (S − S0 )u, (S − S0 )u H , for u ∈ H. √ Now we prove the existence of a square root. If T = 0 we let T = 0. If T = 0, set B = T −1 B(H) T − I. Then, if u H = 1, 2 (Bu, u)H = T −1 B(H) (T u, u)H − u H ∈ [−1, 0] ,

which implies B B(H) ≤ 1. Thus we can define 9 √ T = T B(H) B1/2 . We have 2

(B1/2 ) = B1 = because

1 k

∞ 1 k=0

k

B k = I + B,

= 0 for k > 1. It follows that √ ( T )2 = T B(H) (I + B) = T.

The last part of the statement follows from Lemma 4.4.1.

As a consequence of the existence of the square root of a non-negative selfadjoint operator we can prove the existence and uniqueness of the polar decomposition of a bounded linear operator. It relies on two ingredients we are going to introduce. First, given a linear operator T ∈ B(H) we define its absolute value as √ |T | := T ∗ T . Notice that |T | is a self-adjoint operator. It is also compact if T is compact. Moreover, for every u ∈ H,

|T |u 2H = (|T |u, |T |u)H = (|T |2 u, u)H = (T ∗ T u, u)H = T u 2H .

(4.4.3)

The second ingredient is the notion of partial isometry. A partial isometry is a linear operator U ∈ B(H) such that U is an isometry (Ker U )⊥ → H, i.e.,

180


U x H = x H for all u ∈ (Ker U )⊥ . Observe that, by the polarization identity, we have (U x, U y) = (x, y) for all x, y ∈ (Ker U )⊥ . Trivially the same equality holds for x ∈ (Ker U )⊥ , y ∈ Ker U , so that we get U ∗ U = I on (Ker U )⊥ . This implies that U ∗ U is the orthogonal projection on (Ker U )⊥ and also that U ∗ is an isometry U (H) = (Ker U ∗ )⊥ → H. Hence U ∗ is a partial isometry too. Then the result just obtained, applied to U ∗ , tells us that U U ∗ is the orthogonal projection on U (H). When, in addition, Ker U = {0} and U is onto, then we refer to U as a unitary operator. Theorem 4.4.3. Given an operator T ∈ B(H) there exists a partial isometry U and a non-negative self-adjoint operator S ∈ B(H) such that T = US

(4.4.4)

Ker U = Ker S = Ker T.

(4.4.5)

and Moreover U and S are uniquely determined by the condition (4.4.4) and Ker U = Ker S. In fact we have S = |T |. Thus (4.4.4) becomes T = U |T |,

(4.4.6)

which is called the polar decomposition of T . Proof. Let S = |T |. Let U u = 0, if u ∈ Ker S = S(H) ⊥ . If u ∈ S(H), let U u = T v, where v is any solution of Sv = u. Because of (4.4.3) we have that Ker S = Ker T . This shows that if Sw = Sv, then T w = T v, and the definition of U is consistent. Moreover from (4.4.3) we obtain also that U is an isometry from S(H) in H. We can extend U to S(H) by continuity. In this way we obtain a partial isometry such that (4.4.4) and (4.4.5) are satisfied. Let us prove uniqueness. From (4.4.4) and the condition Ker U = Ker S, therefore (Ker U )⊥ = S(H), we obtain S 2 = SU ∗ U S = (U S)∗ U S = T ∗ T. √ Then S = T ∗ T = |T | by uniqueness of the square root. Since, as we have already seen, Ker S = Ker T , we have that U is uniquely determined by (4.4.4) on S(H). So U is uniquely determined on S(H) = (Ker S)⊥ by continuity, and the proof is complete. We also recall the definition of the singular values of a compact operator. If T is a compact operator, the eigenvalues {μj }N j=0 , μj = 0, of |T | are called singular values of T . Here it is understood that N ∈ N or N = ∞. They are the square roots of the non-zero eigenvalues of T ∗ T . In the sequel we assume that they are arranged in decreasing order and repeated as many times as their multiplicity. An important property is Courant’s minimax principle.


181

Proposition 4.4.4. μj is the minumum of the norms of the restrictions of T to the orthogonal complement of a j-dimensional subspace of H, μj = min{ T |E ⊥ H : dim E = j},

(4.4.7)

and this minimum is attained by taking for E the eigenspace corresponding to the first j eigenvalues μ0 , μ1 , . . . , μj−1 of |T |. Proof. The operator T ∗ T is compact and self-adjoint, so there exists an orthonormal basis of Ker(T ∗ T )⊥ given by eigenvectors {ψk }N k=1 (N ∈ N or N = ∞) of T ∗ T , say T ∗ T ψk = μ2k ψk . Then, for all u ∈ H, μ2k |(u, ψk )H |2 .

T u 2H = (T ∗ T u, u)H = k

It is then clear that T u H ≤ μj u H if u is orthogonal to ψ0 , . . . , ψj−1 , and the equality holds for u = ψj . On the other hand, for any j-dimensional subspace E, E ⊥ contains a vector v ∈ span{ψ0 , . . . , ψj }, v = 0, so that T v H ≥ μj v H . Notice that, if T is self-adjoint, the minimax principle gives a variational characterization of the eigenvalues. Now we study the properties of Hilbert-Schmidt operators. Definition 4.4.5. A linear operator T on H is a Hilbert-Schmidt operator if there exists an orthonormal basis {ψj }j∈N of H such that ∞

T ψj 2H < ∞.

(4.4.8)

j=0

The set of Hilbert-Schmidt operators is denoted by B2 (H). The square root of 4.4.8 is called the Hilbert-Schmidt norm of T : ⎧ ⎫1/2 ∞ ⎨ ⎬

T ψj 2H .

T B2 (H) = ⎩ ⎭ j=0

This is justified by the following property. Proposition 4.4.6. The sum of the series in (4.4.8) is independent of the basis {ψj }. Proof. Let {ηj } be a second orthonormal basis of H. We have ∞

T ψj 2H =

j=0

∞ ∞

|(T ψj , ηk )H |2 =

j=0 k=0

=

∞ k=0

T ∗ ηk 2H .

∞ ∞

|(ψj , T ∗ ηk )H |2

k=0 j=0

(4.4.9)

182


An application of this formula with ψj = ηj gives as well, which concludes the proof.

∞ j=0

T ηj 2H =

∞ k=0

T ∗ ηk 2H

We now list some properties of Hilbert-Schmidt operators. Proposition 4.4.7. (a) T ∈ B2 (H) ⇐⇒ |T | ∈ B2 (H) and

T B2 (H) = |T |B (H) ;

(4.4.10)

2

(b) T ∈ B2 (H) ⇐⇒ T ∗ ∈ B2 (H) and

T B2 (H) = T ∗ B2 (H) ;

(4.4.11)

(c) Every T ∈ B2 (H) is compact, in particular bounded and

T B(H) ≤ T B2 (H) ;

(4.4.12)

(d) For each S ∈ B(H) and T ∈ B2 (H) we have ST ∈ B2 (H), T S ∈ B2 (H) and

ST B2 (H) ≤ S B(H) T B2 (H) ,

T S B2 (H) ≤ T B2 (H) S B(H) .

(4.4.13) (4.4.14)

Proof. Let {ψj } be an orthonormal basis of H. (a) By (4.4.3) we have

T 2B2 (H) =

∞

T ψj 2H =

j=0

∞ |T |ψj 2 = |T |2 H B

2 (H)

.

j=0

(b) It follows from (4.4.9) with ψj = ηj . (c) Given T ∈ B2 (H), we have

T u 2H =

∞

|(T u, ψj )H |2 =

j=0

≤ u 2H

∞

|(u, T ∗ ψj )H |2

j=0 ∞

T ∗ ψj 2H = u 2H T ∗ 2B2 (H) = u 2H T 2B2 (H) .

j=0

This proves that T ∈ B(H) and (4.4.12). Now for each k ∈ N, let Tk u =

k

(u, ψj )H T ψj ;

j=0

then Tk is compact because has finite rank. Moreover

T − Tk 2B2 (H) =

∞ j=k+1

T ψj 2H → 0,

as k → ∞.


183

In particular Tk → T in B(H) by (4.4.12) and so T is compact because it is the limit of a sequence of compact operators. (d) We have

ST 2B(H) =

∞

ST ψj 2H ≤ S 2B(H)

j=0

∞

T ψj 2H = S 2B(H) T 2B2 (H) .

j=0

Moreover

T S B2 (H) = S ∗ T ∗ B2 (H) ≤ S ∗ B(H) T ∗ B2 (H) = S B(H) T B2 (H) .

Proposition 4.4.8. B2 (H) is a Hilbert space with respect to the inner product (independent of the basis): (S, T )B2 (H) =

∞

(Sψj , T ψj )H .

j=0

Proof. It follows from Cauchy-Schwarz’ inequality and Definition 4.4.5 that the series which defines (S, T )B2 (H) converges. The independence of its sum of the choice of the basis is a consequence of Proposition 4.4.6 and the polarization identity. The axioms of inner product are also clearly satisfied. Let us now prove completeness. Let Tn be a Cauchy sequence in B2 (H). Then, by (4.4.12), Tn is a Cauchy sequence in B(H), and hence converges in B(H), say Tn → T . We claim that T ∈ B2 (H) and Tn → T in B2 (H). Indeed, observe that for every > 0 we have J

(Tn − Tk )ψj 2H ≤

j=0

for n, k large enough, and every J ∈ N. Taking the limits as k → ∞ and then as J → ∞ yields Tn − T 2B2 (H) ≤ , which gives the claim. We now give a characterization of the Hilbert-Schmidt operators in terms of singular values. Proposition 4.4.9. T ∈ B2 (H) if and only if it is compact and μ2j < ∞, j

where μj is the sequence of the singular values of T . Moreover,

T B2 (H) =

j

μ2j

1/2 .

(4.4.15)

184


Proof. If T ∈ B2 (H), then it is compact by Proposition 4.4.7. If T is compact and {ψj } is an orthonormal basis of H given by the eigenvectors of T ∗ T , then

T 2B2 (H) =

∞

T ψj 2H =

j=0

∞

(T ∗ T ψj , ψj )H =

j=0

μ2j .

j

This completes the proof. Now we turn to trace-class operators.

Definition 4.4.10. A linear operator T ∈ B(H) is called a trace-class operator if 6 |T | is a Hilbert-Schmidt operator. The set of the trace-class operators is denoted by B1 (H). For each T ∈ B1 (H) we define the trace of T as Tr T =

∞

(T ψj , ψj )H ,

(4.4.16)

j=0

for any orthonormal basis {ψj } of H. The convergence of the series in (4.4.16) and the independence of its sum of the basis follow from Proposition 4.4.8, since one has 6 6 Tr T = ( |T |, |T |U ∗ )B2 (H) , where T = U |T | is the polar decomposition of T , cf. (4.4.6). Proposition 4.4.11. B1 (H) is a linear space and T ∈ B1 (H) if and only if T is compact and μj < ∞, (4.4.17) j

where μj is the sequence of singular values of T . Moreover, if T ∈ B1 (H), μj . (4.4.18) Tr |T | = j

Finally, if T ∈ B1 (H) is self-adjoint with eigenvalues λj , then Tr T =

∞

λj .

(4.4.19)

j=0

6 Proof. Let T = U |T | be the polar decomposition of T . By Proposition 4.4.7, |T | 6 2 |T | is compact. The condition T ∈ B2 (H) reads is compact, hence T = U 6 ∞ 2

|T |ψ

< ∞, which is revealed to be equivalent to (4.4.17) by choosing j H j=0 as ψj an orthonormal basis given by eigenfunctions of |T |. The remaining part of the statement is also immediate from (4.4.16), except the fact that B1 (H) is a linear space. To see this, observe that if E1 and E2 are


185

subpaces of dimensions j and k respectively, the space E1 + E2 is contained in a space of dimension j + k. Therefore by (4.4.7) we have μj+k (T1 + T2 ) ≤ μj (T1 ) + μk (T2 ) for any couple of compact operators T1 , T2 , and j, k ∈ N, where μj (T ) stands for the j-th singular value of T . Hence by the characterization (4.4.17) we see that if T1 , T2 ∈ B1 (H), then T1 + T2 ∈ B1 (H). On the other hand, trivially T ∈ B1 (H) and z ∈ C imply zT ∈ B1 (H). Formula (4.4.19) actually holds for all trace-class operators (Lidskii’s Theorem), but the proof is trickier. Proposition 4.4.12. (a) T ∈ B1 (H) ⇐⇒ |T | ∈ B1 (H); (b) T ∈ B1 (H) ⇐⇒ T ∗ ∈ B1 (H) and Tr T ∗ = Tr T ;

(4.4.20)

(c) if S ∈ B(H) and T ∈ B1 (H), then ST, T S ∈ B1 (H) and Tr (ST ) = Tr (T S).

(4.4.21)

Proof. (a) It follows immediately from the definition. (b) Let T = U |T | be the polar decomposition of T . Then |T ∗ | = U |T |U ∗ ,

(4.4.22) 6 because T ∗ = |T |U ∗ . By using the fact that U ∗ U = I on the range of |T |, this implies that 6 6 6 (U |T |U ∗ )2 = U |T |U ∗ U |T |U ∗ = U |T |U ∗ , i.e.,

6

|T ∗ | = U

6 |T |U ∗ .

(4.4.23) 6 ∗ ∗ Thus by Proposition 4.4.7, |T | ∈ B2 (H), that is T ∈ B1 (H). Because T ∗∗ = T this shows that T ∈ B1 (H) ⇐⇒ T ∗ ∈ B1 (H). Finally, from (4.4.16) we get at once (4.4.20). (c) Consider first the case in which S is unitary. Then √ √ |ST | = T ∗ S ∗ ST = T ∗ T = |T |. Then ST ∈ B1 (H). Moreover from (c) we have T ∗ ∈ B1 (H) and (ST )∗ = T ∗ S ∗ ∈ B1 (H). Thus T S ∈ B1 (H). Now let {ψj } be an orthonormal basis of H. As S is unitary, also {Sψj } is an orthonormal basis of H. Thanks to (4.4.16) this implies that Tr (ST ) =

∞ j=0

(ST ψj , ψj )H =

∞ j=0

(ST Sψj , Sψj )H =

∞

(T Sψj , ψj )H = Tr (T S).

j=0

If S is not unitary, the result follows from the following lemma.

186


Lemma 4.4.13. Every bounded operator is a linear combination of four unitary operators. Proof. If S ∈ B(H), then S = S1 + iS2 with S1 =

1 (S + S ∗ ), 2

S2 =

1 (S − S ∗ ), 2i

and S1 and S2 are self-adjoint. On the other hand, if S = 0 is self-adjoint, let T = S −1 B(H) S. Then T B(H) = 1 and T =

1 6 6 1 T + i I − T2 + T − i I − T2 . 2 2

Because and

∗ 6 6 T ± i I − T2 = T ∓ i I − T2

6 6 T + i I − T 2 T − i I − T 2 = I, 6 6 T − i I − T 2 T + i I − T 2 = I,

the proof is complete. Proposition 4.4.14. If S, T ∈ B2 (H), then ST ∈ B1 (H) and Tr (ST ) = (T, S ∗ )B2 (H) .

(4.4.24)

Proof. Let {ψj } be an orthonormal basis of H and let ST = U |ST | be the polar decomposition of ST . Then 6 |ST |2

B2 (H)

=

=

∞ 6 6 |ST |ψj , |ST |ψj j=0 ∞

(U ∗ ST ψj , ψj )H =

∞

j=0

= H

∞

(|ST |ψj , ψj )H

j=0

(T ψj , S ∗ U ψj )H = (T, S ∗ U )B2 (H) ,

j=0

because U ∗ S ∈ B2 (H) by Proposition 4.4.7. Thus ST ∈ B1 (H). Moreover Tr (ST ) =

∞ j=0

(ST ψj , ψj )H =

∞

(T ψj , S ∗ ψj )H = (T, S ∗ )B2 (H) .

j=0

Definition 4.4.15. We define the trace-class norm of an operator T ∈ B1 (H) as

T B1 (H) = Tr |T |.


187

In Proposition 4.4.17 below we shall prove that in fact the trace-class norm satisfies the norm-axioms. Proposition 4.4.16. If T ∈ B1 (H), we have T ∈ B2 (H) and

T B2 (H) ≤ T B1 (H) .

(4.4.25)

|Tr T | ≤ T B1 (H) ,

T ∗ B1 (H) = T B1 (H) ,

(4.4.26) (4.4.27)

Moreover

and

ST B1 (H) ≤ S B(H) T B1 (H) ,

T S B1 (H) ≤ T B1 (H) S B(H) ,

(4.4.28) (4.4.29)

for all S ∈ B(H). Finally * +

T B1 (H) = sup |Tr (ST )| : S ∈ B(H), S B(H) = 1 .

(4.4.30)

6 2 Proof. Let T = U |T | be the polar decomposition of T . As T = U |T | and 6 |T | ∈ B2 (H), by Propositions 4.4.7 and 4.4.14 we have that T ∈ B2 (H), |T |2 ∈ B1 (H), and 2

T 2B2 (H) = |T |B (H) = Tr |T |2 . 2

μ2j

Let μj be the sequence of singular values of |T |. Then |T |2 has singular values and by (4.4.18) 2 2 2 μj ≤ μj = (Tr |T |) = T 2B1 (H) . Tr |T |2 = j

j

This proves (4.4.25). Let {ψj } be an orthonormal basis of H made of eigenvectors of |T |. Then by (4.4.18) we have ∞ |Tr T | = (U |T |ψj , ψj )H ≤ μj = Tr |T | = T B1 (H) , j=0

j

because (U ψj , ψj )H ≤ 1, for all j ∈ N. Moreover, by (4.4.22) and (4.4.24), we get 6 6 6 6 Tr |T ∗ | = Tr (U |T |U ∗ ) = U |T |, U |T | B (H) = |T |, |T | B2 (H) = Tr |T |, 2 from which (4.4.27) follows.

188

Chapter 4. Spectral Theory Now, if S ∈ B(H) and ST = V |ST | is the polar decomposition of ST , we

have Tr |ST | =

∞

(V ∗ ST ψj , ψj )H =

j=0

≤

∞

(V ∗ SU ∗ |T |ψj , ψj )H

j=0

μj S B(H) = S B(H) Tr |T |.

j

This gives (4.4.28). Formula (4.4.29) follows from (4.4.28) and (4.4.27). It remains to prove (4.4.30). But, if S B(H) = 1, we have |Tr ST | ≤ ST B1 (H) ≤ T B1 (H) . On the other hand

Tr |T | = Tr (U ∗ T )

with U ∗ B(H) = 1, which completes the proof of (4.4.30).

Proposition 4.4.17. B1 (H) is a Banach space with respect to the trace-norm. Proof. First we have to prove that the trace-norm satisfies the norm-axioms. Everything is straightforward but the triangular inequality. We have

T1 + T2 B1 (H) = ≤ ≤

sup SB(H) =1

sup SB(H) =1

sup SB(H) =1

|Tr S(T1 + T2 )| {|Tr (ST1 )| + |Tr (ST2 )|} |Tr (ST1 )| +

sup SB(H) =1

|Tr (ST2 )|

= T1 B1 (H) + T2 B1 (H) . Now we prove completeness. Let Tn ∈ B1 (H) be a Cauchy sequence. By (4.4.25), Tn is a Cauchy sequence in B2 (H), which is complete. Thus Tn converges to T in B2 (H). We have to prove that T ∈ B1 (H) and that Tn → T in B1 (H). To this end, observe first that, if S ∈ B1 (H) and ψj is an orthonormal basis of H, ∞ |(Sψj , ψj )H | ≤ S B1 (H) . (4.4.31) j=0

Indeed, let θj be such that (Sψj , ψj )H = eiθj |(Sψj , ψj )H | and define the unitary operator V ψj = e−iθj ψj . Then we have (V Sψj , ψj )H = (Sψj , V ∗ ψj )H = (Sψj , eiθj ψj )H = e−iθj (Sψj , ψj )H = |(Sψj , ψj )H |.


189

Hence, by (4.4.26) and (4.4.28) we get ∞

|(Sψj , ψj )H | =

j=0

∞ (V Sψj , ψj )H = Tr (V S) ≤ S B1 (H) , j=0

as desired. Now, since Tn is a Cauchy sequence, by (4.4.28) and (4.4.31) for every > 0 we have J |S(Tn − Tk )ψj , ψj )H | ≤ , j=0

for n, k large enough and every J ∈ N, S ∈ B(H), S B(H) = 1. Taking the limit as k → ∞ and then as J → ∞ yields ∞

|(S(Tn − T )ψj , ψj )H | ≤ .

j=0

If we choose S as the partial isometry satisfying S(Tn − T ) = |Tn − T | we get

Tn − T B1 (H) ≤ provided n is large enough, and also that T = (T − Tn ) + Tn belongs to B1 (H). Remark 4.4.18. There is a strong analogy between the theory of Hilbert-Schmidt and trace-class operators and that of the functions in Lebesgue spaces L2 (Rd ) and L1 (Rd ) respectively. The trace for the operators corresponds to the integral for functions. In fact one could define, using (4.4.16), the trace for any non-negative self-adjoint operator (allowing that series to diverge); then an operator T ∈ B(H) belongs to Br (H), r = 1, 2 if and only if Tr |T |r < ∞. The Banach spaces Br (H), defined in this way for all 1 ≤ r < ∞, are two-sided ideals in B(H) and consist of compact operators. They are called Schatten-von Neumann classes. This analogy is one of the main sources of inspiration in the development of Non-Commutative Geometry. Now we characterize the Hilbert-Schmidt operators on L2 (Rd ). Proposition 4.4.19. T ∈ B2 (L2 (Rd )) if and only if there exists (a unique) KT ∈ L2 (R2d ) such that (4.4.32) T u (x) = KT (x, y)u(y) dy, for u ∈ L2 (Rd ). Moreover we have

T B2 (L2 (Rd )) = KT L2 (R2d ) . Proof. Let {ψj }j∈N be an orthonormal basis of L2 (Rd ). Then {ψj (x)ψk (y)}(j,k)∈N×N

(4.4.33)

190


is an orthonormal basis of L2 (R2d ). If T ∈ B2 (L2 (Rd )) we let KT (x, y) =

∞

(T ψj , ψk )L2 ψk (x)ψj (y).

(4.4.34)

j, k=0

Then

KT 2L2 =

∞

|(T ψj , ψk )L2 |2 =

∞

T ψj 2L2 = T B2 (L2 ) ,

j=0

j,k=0

and KT (x, y)u(y) dy = =

∞

(T ψj , ψk )L2 (u, ψj )L2 ψk (x)

j, k=0 ∞

(T u, ψk )L2 ψk (x) = T u(x).

k=0

Vice-versa, if KT ∈ L2 (R2d ), (4.4.32) defines an operator T ∈ B2 L2 (Rd ) satisfying (4.4.34) and so (4.4.33). This completes the proof. It is also easy to characterize the pseudo-differential operators which are Hilbert-Schmidt. Proposition 4.4.20. A pseudo-differential operator in Rd extends to a Hilbert-Schmidt operator on L2 (Rd ) if and only if its Weyl symbol a belongs to L2 (R2d ), and

aw B2 (L2 (Rd )) = (2π)−d/2 a L2 (R2d ) . Proof. The desired result is a consequence of Proposition 1.2.1 with τ = particular (1.2.4), and Proposition 4.4.19.

1 , 2

in

Now we examine under which conditions a pseudo-differential operator in Rd extends to a trace-class operator on L2 (Rd ). Theorem 4.4.21. Every regularizing operator in Rd (i.e., having symbol in S(R2d )) extends to a trace-class operator on L2 (Rd ). More generally, there exists N ∈ N depending only on d such that, for every symbol a having distribution partial derivatives ∂ α a ∈ L1 (R2d ), |α| ≤ N , A = aw extends to a trace-class operator on L2 (Rd ), with the uniform estimate

A B1 (L2 ) |∂ α a(z)| dz. (4.4.35) R2d

|α|≤N

For such operators we further have Tr A = (2π)−d

a(z) dz. R2d

(4.4.36)


191

Proof. Let A be a regularizing operator with Weyl symbol a ∈ S(R2d ), and let P = −Δ + |x|2 be the harmonic oscillator. We know from Proposition 4.4.11 and Theorem 2.2.3 that P −k is trace-class if the integer k is large enough. Moreover, P k belongs to OPΓ2k (Rd ), see Definition 2.1.2. We can write A = AP k P −k , so that by Proposition 4.4.16,

A B1 (L2 ) ≤ AP k B(L2 ) P −k B1 (L2 ) sup

sup (1 + |z|)N0 |∂ α a(z)|

|α|≤N0 z∈R2d

(4.4.37)

for some N0 ∈ N, where we also used Theorems 1.4.1 and 1.2.16 (the left symbol 1 is obtained by applying to a the operator e 2 Dx ·Dξ , which is continuous on S(R2d ), cf. (1.2.12)). Now, let a satisfy + * supp a ⊂ B := z ∈ R2d : |z| ≤ 1 . By (4.4.37) we have

A B1 (L2 ) sup

sup |∂ α a(z)|.

|α|≤N0 z∈R2d

Because a has compact support, we have z1 z2d α ··· ∂w1 · · · ∂w2d ∂w a(w) dw2d · · · dw1 , ∂ α a(z) = −∞

−∞

hence sup

sup |∂ α a(z)| ≤

|α|≤N0 z∈R2d

sup |α|≤N0 +2d

|∂ α a(w)| dw.

This implies that (4.4.35) holds for smooth a supported in B, with N = N0 + 2d. Let us now observe that if A0 is the pseudo-differential operator with Weyl symbol a0 (z) = a(z − z0 ), z0 = (x0 , ξ0 ), we have

x+y ¯ dξ ¯ − x0 , ξ − ξ0 u(y) dy 2

x − x0 + y i(x−x0 −y )ξ0 u(x0 + y ) dy ¯ dξ ¯ = ei(x−x0 −y )ξ a ,ξ e 2 = U −1 AU u (x),

A0 u(x) =

where

ei(x−y)ξ a

U u(x) = e−ixξ0 u(x0 + x).

192


Because U is unitary, we have

A0 B1 (L2 ) = U −1 AU B1 (L2 ) = A B1 (L2 ) . On the other hand, also |∂ α a(z)| dz = |∂ α a(z − z0 )| dz = |∂ α a0 (z)| dz is invariant, thus we obtain that (4.4.35) holds for every smooth function a which is supported in a ball of radius 1. Now we get rid of this last condition. Let {θj (z)}j∈N be a partition of unity of functions supported in balls of radius 1, and such that each supp θj intersects at most a fixed number of supp θk , k ∈ N. We can easily construct {θj } in such a way that for each α ∈ N2d there exists Cα such that (4.4.38) sup |∂ α θj (z)| ≤ Cα , for j ∈ N. z∈R2d

Let Aj be the operator with Weyl symbol θj a. By what we just proved we have ∞ ∞

A B1 (L2 ) ≤

Aj B1 (L2 ) sup |∂ α (θj a)(z)| dz. j=0 |α|≤N

j=0

Now, by (4.4.38) and Leibniz’ formula, we have α |∂ α (θj a)(z)| ≤ Cβ |∂ α−β a(z)|. β β≤α

Thus sup |α|≤N

|∂ α (θj a)(z)| dz sup

|α|≤N

supp θj

|∂ α a(z)| dz.

Hence

A B1 (H) sup

|α|≤N

|∂ α a(z)| dz,

which proves (4.4.35) for Schwartz symbols. The same estimate for a general symbol a as in the statement follows at once from a limiting argument, which uses (4.4.35) for Schwartz symbols, the completeness of B(L2 ) and Proposition 1.2.2. Let us now consider the estimate (4.4.36). Again, by using (4.4.26) and a limiting argument, we may prove it just for Schwartz symbols. In terms of the corresponding integral kernel K(x, y) the equality (4.4.36) then reads K(x, x) dx, (4.4.39) Tr A = Rd

cf. (1.2.2). We may also prove this equality only for finite linear combinations of kernels of the type ϕ1 (x)ϕ2 (y), ϕ1 , ϕ2 ∈ S(Rd ). By linearity, we can further consider just a kernel K(x, y) = ϕ1 (x)ϕ2 (y), ϕ1 L2 = 1, for which it is clear that (4.4.39) holds: take, in the definition of Tr A, an orthonormal basis which contains ϕ1 as an element.

4.5. Heat Kernel

193

4.5 Heat Kernel Let A be a pseudo-differential operator with real-valued, positive, elliptic Weyl symbol a ∈ S(M ; Φ, Ψ), with M (x, ξ) (1 + |x| + |ξ|)δ , for some δ > 0, cf. Definition 1.3.1. Then, we know from Theorem 4.2.9 that its spectrum consists of a sequence of eigenvalues λj , j ∈ N, diverging to +∞ as j → ∞, with the corresponding eigenfunctions ψj defining an orthonormal basis of L2 (Rd ). We can then define, for t ≥ 0, the operator e−tA f =

∞

e−tλj (f, ψj )L2 (Rd ) ψj ,

f ∈ L2 (Rd ),

j=0

with unconditional convergence in L2 (Rd ). Hence e−tA turns out to be a bounded operator on L2 (Rd ). The aim of the section is to show that this operator is in fact a regularizing operator for t > 0. The main result is the following one. Theorem 4.5.1. Let a ∈ S(M ; Φ, Ψ) be real-valued, with a ∼ ∞ j=0 aj , where aj ∈ S(M hj ; Φ, Ψ) are real-valued, and a0 (x, ξ) M (x, ξ) (1 + |x| + |ξ|)δ , for some δ > 0. Let A = aw . Then the operator e−tA is a pseudo-differential operator with Weyl symbol u(t, x, ξ) satisfying, for every k, l, N ∈ N, J ≥ 1, T > 0, the estimates ⎛ ⎞ J−1 N l⎝ uj (t, ·)⎠ M l−N hJ , t ∈ [0, T ], (4.5.1) t ∂t u(t, ·) − k

j=0

where u0 (t, x, ξ) = e−ta0 (x,ξ) and, for j ≥ 1, uj (t, x, ξ) = e−ta0 (x,ξ)

2j

tl ul,j (x, ξ),

ul,j ∈ S(M l hj ; Φ, Ψ),

(4.5.2)

l=1

(see (4.2.4) for the notation | · |k ). Proof. We first look for a symbol v(t, x, ξ) such that (∂t + aw )v w = K(t), vw |t=0 = I,

(4.5.3)

where K(t) is an operator with smooth kernel Kt (x, y) satisfying ∂tl Kt ∈ S(R2d ) uniformly ∞ for t in compact sets of [0, +∞). Precisely, we look for v in the form v ∼ j=0 vj , where vj have the same form as uj in the statement, i.e., v0 (t, x, ξ) = e−ta0 (x,ξ) and, for j ≥ 1, vj (t, x, ξ) = e−ta0 (x,ξ)

2j l=1

tl vl,j (x, ξ), vl,j ∈ S(M l hj ; Φ, Ψ).

(4.5.4)

194


The formulas (4.5.3) and (4.2.1) yield the following transport equations: ⎧ {as ,vk }l ⎪ ⎨∂t vj + s+k+l=j (2i)l l! = 0, v0 (0, x, ξ) = 1, ⎪ ⎩ vj (0, x, ξ) = 0,

j ≥ 1.

For j = 0 we obtain v0 (t, x, ξ) = e−ta0 (x,ξ) . It is easy to show by induction that the solutions vj , j ≥ 1, have the required form, as well. By using the assumption a0 M , one also verifies from (4.5.4) that |tN ∂tl vj (t, ·)|k M l−N hj , uniformly for t ∈ [0, +∞). Moreover, by arguing as in the proof of Proposition ∞ 1.1.6 we can construct v ∼ j=0 vj which satisfies, for every T > 0, J−1 N l ∂ vj (t, ·) M l−N hJ v(t, ·) − t t k

j=0

uniformly for t ∈ [0, T ]. This construction shows that v w satisfies (4.5.3), for a suitable operator K(t), regularizing for every t ≥ 0, with a kernel satisfying the desired estimates. We now look at the operator e−tA . Since it solves the homogeneous version of (4.5.3), we have t e−(t−s)A K(s)f ds, f ∈ S(Rd ). v w f − e−tA f = 0

It is therefore sufficient to prove that the operator v w −e−tA has a symbol in S(R2d ), together with its derivatives with respect to t of any order N ∈ N, uniformly for t in compact subsets of [0, +∞). To this end, observe that, since M (x, ξ) (1 + |x| + |ξ|)δ , we have S(Rd ) = ∩j∈N H(M j ), and the norm f H(M j ) in the Sobolev space H(M j ) can be defined as (A + cI)j f L2 , c being a large constant so that A + cI is invertible (see Section 1.5). Moreover t → e−tA is a strongly continuous map of bounded operators on L2 (Rd ), and e−tA commutes with (A + cI)j . Taking into account the properties of the kernel of K(s), this gives the desired estimate dn −tA f = for N = 0. The estimates for the derivatives follow similarly, for dt ne (−A)n e−tA f for every f ∈ S(Rd ), n ∈ N. It follows from Theorem 4.5.1 that the operator e−tA is regularizing for t > 0, hence trace-class by Theorem 4.4.21. Its trace is given by the sum of the eigenvalues, cf. (4.4.19), i.e., Tr e−tA =

∞ j=0

e−tλj .

4.6. Weyl Asymptotics

195

Proposition 4.5.2. Assume the hypotheses in Theorem 4.5.1, and let u(t, x, ξ) be the symbol of e−tA . Then, for t > 0, ∞

e−tλj =

u(t, x, ξ) dx ¯ dξ. ¯

(4.5.5)

j=0

Proof. The desired result follows at once from Theorem 4.4.21 because u(t, x, ξ) ∈ S(R2d ) for every fixed t > 0. An alternative and direct proof goes as follows. Denote by Kt (x, y) the distribution kernel of e−tA . We have Kt (x, y) =

∞

e−tλj ψj (x)ψj (y),

j=0

with convergence in L2 (R2d ). We have in fact Kt ∈ S(R2d ) for t > 0. t t Since e−tA = e− 2 A e− 2 A , we can write Kt (x, y) = Kt/2 (x, z)Kt/2 (z, y) dz = Kt/2 (x, z)Kt/2 (y, z) dz, where we also used that e−tA is self-adjoint. Setting x = y and integrating with respect to x yields Kt (x, x) dx = |Kt/2 (x, y)|2 dx dy. The right-hand side of this equality by Parseval’s formula reads ∞

e−tλj ψj 4L2 (Rd ) =

j=0

On the other hand,

∞

e−tλj .

j=0

Kt (x, x) dx =

by (1.2.2). This concludes the proof.

u(t, x, ξ) dx ¯ dξ ¯

4.6 Weyl Asymptotics We now study the asymptotic distribution of the eigenvalues of some special classes of elliptic self-adjoint operators A satisfying the assumptions of Theorem 4.5.1. d Precisely, we consider the classes OPΓm ρ (R ), m > 0, 0 < ρ ≤ 1, studied in

196


d Chapter 2 (see Definition 2.1.2) and the classes OPGm,n cl(ξ,x) (R ), m > 0, n > 0, of Chapter 3 (see Definition 3.2.5). We use the notation in the previous section. In particular we denote by λj the eigenvalues (counted with multiplicity) of A. Moreover, we define the so-called counting function N (λ) = #{j : λj ≤ λ}. d Theorem 4.6.1. Let a ∈ Γm ρ (R ), m > 0, 0 < ρ ≤ 1 be real-valued, a(x, ξ) = am (x, ξ) + am−ρ (x, ξ) for |x| + |ξ| large, where am (x, ξ) is real-valued and satisfies 0 < am (tx, tξ) = tm am (x, ξ), for t > 0, (x, ξ) ∈ Rd , and am−ρ ∈ Γm−ρ (Rd ). Then ρ w the trace of the heat kernel of A = a has the asymptotic behaviour ∞ j=0

where

e

−tλj

2d ∼ CΓ 1 + m

(2π)−d C= 2d

2d

t− m , as t → 0+ ,

(4.6.1)

2d

S2d−1

am (Θ)− m dΘ.

(4.6.2)

Proof. Let a ˜m (x, ξ) be a smooth positive function which coincides with am for d |x|+|ξ| large. Hence a ˜m ∈ Γm 1 (R ). Let u and uj be as in the statement of Theorem 4.5.1 (now Φ(x, ξ) = Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )ρ/2 , M (x, ξ) = (1 + |x|2 + |ξ|2 )m/2 , h(x, ξ) = (1 + |x|2 + |ξ|2 )−ρ ). By(4.5.5) and Theorem 4.5.1 we have ∞

e−tλj =

e−tãm (x,ξ) dx ¯ dξ ¯ +

J−1

R2d

j=0

j=1

+ R2d

⎛

R2d

uj (t, x, ξ) dx ¯ dξ ¯

⎝u(t, x, ξ) −

J

⎞ uj (t, x, ξ)⎠ dx ¯ dξ. ¯ (4.6.3)

j=0

Using (4.5.1) with N = l = 0 and J large enough, the last integral is easily seen to be O(1) as t → 0+ . We now study the first integral. Since a and am coincide away from a compact set, we have −t˜ am (x,ξ) e dx ¯ dξ ¯ = e−tam (x,ξ) dx ¯ dξ ¯ + O(1), as t → 0+ . R2d

R2d

On the other hand, using polar coordinates r, Θ in R2d and the further change of variable r → σ = tr m am (Θ) give 2d 2d − 2d (2π)−d −tam (x,ξ) m Γ t e dx ¯ dξ ¯ = am (Θ)− m dΘ m m R2d S2d−1

2d 2d − 2d (2π)−d Γ 1+ t m am (Θ)− m dΘ. = 2d m S2d−1

4.6. Weyl Asymptotics

197

Similarly one treats the integrals of uj , j = 1, . . . , J − 1. Namely, using the expressions (4.5.2), since a ˜m (x, ξ) (1 + |x|2 + |ξ|2 )m/2 = M (x, ξ), we have 2j

R2d

|uj (t, x, ξ)| dx ¯ dξ ¯

e−tãm (x,ξ) tl a ˜m (x, ξ)l (1 + |x|2 + |ξ|2 )−ρj dx ¯ dξ. ¯

l=1

We can replace in this formula a ˜m by am , with an error which is O(1). The same changes of variables as above and the Dominated Convergence Theorem then show 2d that the last integral is o(t− m ), as t → 0+ . This concludes the proof. We need the following Tauberian theorem (see, e.g., Taylor [187, Vol. II, Proposition 3.2, page 89] for the proof). Theorem 4.6.2. (Karamata’s Tauberian Theorem) Let σ be a positive Borel measure on the real semi-axis (0, ∞). Suppose that, for some α > 0,

+∞ 0

e−λt dσ(λ) ∼ Ct−α ,

Then

C λα , Γ(α + 1)

σ(λ) ∼

as t → 0+ .

as λ → +∞.

Theorem 4.6.3. Under the same hypotheses as in Theorem 4.6.1, the counting function N (λ) of the operator A = aw has the asymptotic behaviour 2d

N (λ) ∼ Cλ m , as λ → +∞,

(4.6.4)

where C is given in (4.6.2). Proof. The desired result follows at once from Theorem 4.6.1 and Theorem 4.6.2 applied to the measure σ(λ) = σ([0, λ]) = N (λ). Indeed, ∞

−tλj

e

=

e−tλ dN (λ).

j=0

Notice that, by using the homogeneity of am we can re-write the formula (4.6.4) as dx ¯ dξ, ¯ as λ → +∞ N (λ) ∼ am (x,ξ)≤λ

which is, up to the factor (2π)−d , the volume of the set {(x, ξ) ∈ R2d : am (x, ξ) ≤ λ} (one can easily see that the same formula also holds with the whole symbol a in place of am ). Heuristically this formula states that, on average, every eigenfunction occupies in phase space a set of volume comparable with 1, which agrees with the uncertainty principle in Harmonic Analysis. We can also deduce the asymptotic behaviour of the eigenvalues themselves.

198


Proposition 4.6.4. Under the hypotheses of Theorem 4.6.3 we have λj ∼ C − 2d j 2d , as j → ∞, m

m

where the constant C is given in (4.6.2). Proof. It follows from (4.6.4) that for every > 0 there exists λ0 such that 2d

1 − ≤ N (λ)C −1 λ− m ≤ 1 + ,

(4.6.5)

for λ > λ0 . We now choose an integer j0 > 0 such that λj0 > λ0 and λj0 +1 > λj0 . We claim that − 2d (4.6.6) 1 − ≤ jC −1 λj m ≤ 1 + , for j > j0 . Indeed, for every j > j0 there exist integers j1 and j2 such that j0 ≤ j1 < j ≤ j2 and λj1 < λj1 +1 = λj2 < λj2 +1 . In particular, we have N (λj1 ) = j1 and N (λj2 ) = j2 , so that from (4.6.5) we have − 2d

1 − ≤ j1 C −1 λj1 m ≤ 1 + , and

(4.6.7)

− 2d

1 − ≤ j2 C −1 λj2 m ≤ 1 + .

(4.6.8)

Moreover, N (λ) = j1 for λj1 ≤ λ < λj2 , so that for these values of λ it turns out that 2d 1 − ≤ j1 C −1 λ− m ≤ 1 + , and, by continuity,

− 2d

1 − ≤ j1 C −1 λj2 m ≤ 1 + .

(4.6.9)

It follows from (4.6.8) and (4.6.9) that − 2d

1 − ≤ jC −1 λj2 m ≤ 1 + , and it suffices to observe that λj = λj2 . Hence (4.6.6) is proved. As a consequence of that formula we have (1 + )− 2d C − 2d j 2d ≤ λj ≤ (1 − )− 2d C − 2d j 2d , m

m

m

m

m

m

which implies the desired result.

Example 4.6.5. For the eigenvalues of the harmonic oscillator, with Weyl symbol a2 (x, ξ) = |ξ|2 + |x|2 , we obtain from (4.6.4) N (λ) ∼ Cλd ,

as λ → +∞,

with C given by (4.6.2) C=

(2π)−d 2d

dΘ = S2d−1

1 , 2d d!

Notes

199

corresponding to (2.2.32) directly computed on the eigenvalues in Chapter 2. Moreover, Proposition 4.6.4 gives for C as before 1

1

λj ∼ C − d j d ,

as j → ∞.

d Finally, we consider operators in the classes OPGm,n cl(ξ,x) (R ), m > 0, n > 0. The main result is the following one. d Theorem 4.6.6. Consider a real-valued symbol a ∈ Gm,n cl(ξ,x) (R ), m, n > 0, with m d d n σψ (a)(x, ξ) > 0 for all x ∈ R and ξ ∈ R \{0}, σe (a)(x, ξ) > 0 for all x ∈ Rd \{0} m,n (a)(x, ξ) > 0 for all x ∈ Rd \ {0} and ξ ∈ Rd \ {0}, cf. (3.2.2), and ξ ∈ Rd , σψ,e (3.2.4) and (3.2.7). Then the counting function N (λ) of the operator A = aw has the asymptotic behaviour ⎧ d ⎪ ⎨Cm λ m log λ, for m = n, d N (λ) ∼ Cm λm , for m < n, ⎪ ⎩ nd Cn λ , for m > n,

where d (2π)−d m,m σψ,e (a)− m dθ dθ , Cm = dm d−1 d−1 S S −d d (2π) Cm = σψm (a)− m dθ dx, d d d−1 Rx S −d d (2π) σen (a)− n dθ dξ. Cn = d d d−1 Rξ S The proof follows the same pattern as that of Theorem 4.6.3. We omit it and refer the interested reader to Maniccia and Panarese [138].

Notes The results collected in Section 4.1 are standard and can be found in most books of Functional Analysis, see, e.g., Reed and Simon [168, Chapter VIII] for a detailed account. The characterization in Theorem 4.2.4 of the domain of A, for a pseudodifferential operator A with hypoelliptic symbol, is a slight generalization of an argument of Hörmander [117] and is extracted from Buzano and Nicola [29]. The proof of Theorem 4.2.9 is inspired by the pattern in Boggiatto, Buzano and Rodino [19]. The brackets {·, ·}j in (4.2.2) can be regarded as a type of higher order Poisson brackets, and in fact reduce to them for j = 1; they enjoy an invariance property with respect to linear symplectic changes of variables (see, e.g., Buzano

200


and Nicola [30]). Their invariance, up to a certain extent, with respect to nonlinear symplectomorphisms is discussed in Mughetti and Nicola [150]. Complex powers of pseudo-differential operators have been studied by several authors, starting from the work of Seeley [179], [181], [180], where the ζ-function ζ(z) = Tr Az for boundary value problems was introduced. Generalizations have been then considered, among others, by Kumano-go and Tsutsumi [130], Kumanogo [129], Beals [9], [10], Robert [169], Helffer [109]. Indeed, the study of poles of the zeta function has important applications to index theory, as shown in the celebrated paper by Atiyah, Bott and Patodi [7], and Weyl asymptotics, for which we refer to Duistermaat and Guillemin [75] and also to Shubin [183]. Among other applications, we point out that the study of bounded imaginary powers of pseudo-differential operators also gives information on the maximal regularity for evolution equations, in view of the theorem of Dore and Venni [74]; see for example Coriasco, Schrohe and Seiler [66] and the references therein. Recently attention has been mostly fixed on complex powers of pseudo-differential operators on manifolds with boundary with a given boundary fibration structure; the relevant operators are then elliptic in a calculus which is not, in general, temperate (such as, e.g., Melrose’s b-calculus [143]). In this context one is interested in the relationships with the geometric properties of the underlying manifold; we refer, for example, to the contributions by Schrohe [174], [176], Loya [135], [136], Melrose and Nistor [146], and Lauter and Moroianu [131]. In the last two papers the complex powers were also used to define various Wodzicki-type residues as generators of the Hochschild cohomology in dimension 0, cf. the next chapter. The presentation in Section 4.3 follows the approach in [30], where more general pseudo-differential operators in Rd , with Weyl symbol in Hörmander’s classes S(m, g), were considered. In this context complex powers were already treated by Robert [169] for (globally) elliptic symbols diverging at infinity, and consequently with compact resolvent. As already mentioned, here we allow the symbol to tend to zero in some direction and, also, the spectrum is allowed to have zero as an accumulation point. Applications to Schatten von-Neumann properties of pseudo-differential operators were given in [29], [30]; see also Buzano and Toft [32], where more general trace-class pseudo-differential operators are studied. The material on Hilbert-Schmidt and trace-class operators in Section 4.4 is standard. Classical references are the books of Gohberg and Krein [93] and Schatten [173]. See also Connes [50] for applications in Non-Commutative Geometry and [19] for other applications to the Spectral Theory of pseudo-differential operators. The construction of the heat semigroup e−tA is classical too, see, e.g., Treves [191] and Taylor [187], but our presentation is also inspired by Maniccia [137], Maniccia and Panarese [138]. See also Buzano [26] for analytic semigroups generated by globally regular operators. Applications of the heat method to index theory can be found in [7]. The remainder term in the Weyl formula (4.6.4) can be estimated by other methods, see Tulovskiˇı and Shubin [193], [183], Hörmander [118], [19], the optimal formula being obtained by an analysis of the wave group

Notes 2/m

201

eitA ; see [109] and also the preceding work by Hörmander [116] for operators on compact manifolds. The results for G-operators in Theorem 4.6.6 appeared in [137], [138]. Estimates for the remainder term were given in Nicola [152]; further improvements were announced by Coriasco and Maniccia (personal communication). For a comprehensive study of Weyl asymptotics in several other contexts we refer the reader to Ivrii [122] and the references therein. The case of the multi-quasi-elliptic operators considered in Chapter 2 is treated in great detail in Boggiatto and Buzano [18] and in [19]. Finally we refer the reader interested in spectral asymptotics for systems of operators of Γ-type, generalizing in particular the scalar harmonic oscillator in Section 2.2, to the detailed study by Parmeggiani and Wakayama [160] and Parmeggiani [159].

Chapter 5

Non-Commutative Residue and Dixmier Trace Summary Let us recall the following definition, valid for any algebra A over C. Definition 5.0.1. The linear map τ : A → C is called a trace if it vanishes on commutators, i.e., if τ ([P, Q]) = τ (P Q − QP ) = 0 for all P, Q ∈ A.

(5.0.1)

If τ is a trace, then λτ is a trace for all λ ∈ C. As a basic example, take the algebra A of r × r matrices A = (Ajk ) over C; there is a unique non-trivial trace, up to a multiplicative constant, given by Tr A =

r

Ajj .

(5.0.2)

j=1

We want to study traces on algebras of pseudo-differential operators in Rd . Take first the algebra R(Rd ) of all regularizing operators, i.e., operators K with kernel in S(R2d ); there is a unique trace functional: Tr K = K(x, x) dx, (5.0.3) where we write K(x, y) for the kernel of K. After choosing a basis for L2 (Rd ) given by functions in S(Rd ), we may regard K as an infinite matrix, and we see that (5.0.3) extends to all the trace-class pseudo-differential operators considered in Theorem 4.4.21; in fact, in terms of the Weyl symbol a(x, ξ) of A = aw we have Tr A = a(x, ξ) dx ¯ dξ. ¯ (5.0.4)

204

Chapter 5. Non-Commutative Residue and Dixmier Trace

∞ d To fix ideas, consider a ∈ Γm cl (R ), that is a(z) ∼ k=0 am−k (z), with z = (x, ξ) and am−k (z) positively homogeneous in z of degree m − k, cf. Definition 2.1.3. We assume m ∈ Z. Then for m < −2d the operator aw is trace-class, and (5.0.4) makes sense, whereas for m ≥ −2d the integral is divergent in general. We would like to define a trace, according to Definition 5.0.1, on the whole algebra d OPΓ∞ cl (R ) =

d OPΓm cl (R ).

(5.0.5)

m∈Z d To this end we set for A ∈ OPΓ∞ cl (R ) with arbitrary order m ∈ Z:

Res A = S2d−1

a−2d (Θ) dΘ,

(5.0.6)

where dΘ is the usual surface measure on S2d−1 and a−2d (z) is the term homogeneous of degree −2d in the asymptotic expansion of the symbol a(z). Following the terminology of Wodzicki, who gave a similar definition for classical pseudodifferential operators on compact manifolds, we shall call Res A in (5.0.6) the non-commutative residue of the operator A. In Section 5.1 we shall prove that d Res is a trace on the algebra OPΓ∞ cl (R ). This trace vanishes if the order of the operator is less than −2d. For one thing this shows that the non-commutative residue is not an extension of the usual trace functional; it also implies that it is zero on the ideal of regularizing operators R(Rd ) and therefore yields a trace on d d the quotient algebra A = OPΓ∞ cl (R )/R(R ). In fact we shall prove that Res turns out to be the unique trace on A up to a multiplicative constant. An interesting property of the non-commutative residue is that it coincides d with Dixmier’s trace on operators A in OPΓ−2d cl (R ). For classical pseudo-differential operators on compact manifolds, this was observed by Connes, in his work on non-commutative geometry. Let us recall the definition of Dixmier trace on an (infinite dimensional) Hilbert space H. Let T belong to K(H), the ideal of the compact operators in B(H). Consider |T | = (T ∗ T )1/2 , cf. Theorem 4.4.2, and let μ1 (T ) ≥ μ2 (T ) ≥ . . . be the sequence of the eigenvalues of |T |, repeated according to their multiplicity. Denoted by N σN (T ) = μn (T ), (5.0.7) n=1

we define L(1,∞) (H) = {T ∈ K(H) : σN (T ) = O(log N )},

(5.0.8)

endowed with the norm T 1,∞ = supN ≥2 σN (T )/ log N . The space L(1,∞) (H) is an ideal of B(H). The following preliminary definition will be sufficient for our practical use.

Summary

205

Definition 5.0.2. Let T ∈ L(1,∞) (H) be a non-negative operator. We define Dixmier’s trace as N 1 μn (T ) (5.0.9) Dixmier − Tr (T ) = lim N →∞ log N n=1 provided the limit exists in R. To deal with the case when the limit in (5.0.9) does not exist, we consider a linear form ω on Cb (1, ∞), with ω ≥ 0, ω(1) = 1 and ω(f ) = 0 if limx→+∞ f (x) = 0. Given a bounded sequence a = (aN )N ≥1 , we construct the function fa = ∞ N ≥1 aN χ[N −1,N ) ∈ L (R+ ) and define the ω-limit limω aN = ω(M fa ), where, t ∞ for g ∈ L (R+ ), M g(t) = log1 t 1 g(s) s ds is the Cesàro mean of g. In the case of convergent sequences the ω-limit coincides with the usual limit. Hence in general one substitutes (5.0.9) with N 1 μn (T ), Trω (T ) = lim ω log N n=1

(5.0.10)

depending on ω. We have that Trω is additive on positive operators, so it can be extended to a linear map on L(1,∞) (H). Finally, observe that Trω vanishes on trace-class operators and commutators, hence it is a trace in the sense of Definition 5.0.1. Let us return to pseudo-differential operators. In Section 5.1 we shall prove d (1,∞) that any A ∈ OPΓ−2d (L2 (Rd )) and cl (R ) belongs to L Res A = 2d(2π)d Trω (A).

(5.0.11)

The limit (5.0.9) exists in the present case, hence (5.0.11) is independent of ω. As an example consider the inverse of the d-th power of the harmonic oscillator (5.0.12) H −d = (−Δ + |x|2 )−d . d −2d We have H −d ∈ OPΓ−2d , hence cl (R ) with principal symbol a−2d (z) = |z| dΘ = Ω2d , Res H −d = S2d−1

where Ω2d denotes the measure of S2d−1 . On the other hand from Weyl’s formula, and precisely Proposition 4.6.4, we have for the eigenvalues λj of H d = (−Δ + |x|2 )d , j λj ∼ , j → ∞, C with C = (2π)−d (2d)−1 Res H −d .

206


Passing then to the eigenvalues μj = λ−1 of H −d , we deduce j μj ∼ Since

N

1 j=1 j

Res H −d , 2d(2π)d j

j → ∞.

∼ log N , from (5.0.10) we actually obtain Res H −d = Ω2d = 2d(2π)d Trω H −d .

Section 5.2 is devoted to similar results for the G-classes considered in Chapter 3. d In short: for A ∈ OPGm,n cl(ξ,x) (R ), m ∈ Z, n ∈ Z, we define −d,−d Trψ,e A = σψ,e (a)(x, ξ) dθ dθ (5.0.13) Sd−1

Sd−1

−d,−d (a)(x, ξ) is the term bi-homogeneous of orders −d, −d in the douwhere σψ,e ble asymptotic expansion of the symbol a(x, ξ) of A, cf. (3.2.7). We shall obtain (5.0.13) as coefficient in the Laurent expansion of a bi-holomorphic family of G-operators; the proceeding will provide in a natural way two other trace functionals on sub-algebras of the G-operators. We shall also obtain coincidence with Dixmier’s trace, after a slight change in the definition (5.0.10), for d A ∈ OPG−d,−d cl(ξ,x) (R ). Finally in Section 5.3 we shall discuss Dixmier’s traceability of the operators in the general classes of Chapter 1.

5.1 Non-Commutative Residue for Γ-Operators We write in this section Hm (R2d \ {0}), m ∈ R, for the class of functions a(z) ∈ C ∞ (R2d \ {0}), z = (x, ξ), which are positively homogeneous of degree m in R2d , i.e., (5.1.1) a(tz) = tm a(z), for t > 0, z ∈ R2d , z = 0. If a ∈ Hm (R2d \ {0}), then the Euler identity holds: 2d

zj ∂zj a = ma.

(5.1.2)

j=1 d Then recall that the symbol a(z) belongs to Γm cl (R ), the subspace of the classical symbols in Γm (Rd ), if it admits an asymptotic expansion

a(z) ∼

∞

am−k (z)

(5.1.3)

k=0

where am−k ∈ Hm−k (R2d \ {0}). In the following we shall assume m ∈ Z. We recall that we may define elliptic symbols by assuming am (z) = 0 for z ∈ R2d , z = 0.

(5.1.4)

5.1. Non-Commutative Residue for Γ-Operators

207

In the first part of this section we shall consider pseudo-differential operators d A = Op0 (a) ∈ OPΓm cl (R ) in the standard quantization form: u(ξ) dξ ¯ (5.1.5) Au(x) = eixξ a(x, ξ) (the results will be actually independent of the quantization, cf. Remark 5.1.5 below). We write here d Γ∞ cl (R ) =

d Γm cl (R ),

d OPΓ∞ cl (R ) =

m∈Z

d OPΓm cl (R ).

(5.1.6)

m∈Z

−∞ d m d 2d d Note that Γ−∞ cl (R ) = ∩m∈Z Γcl (R ) coincides with S(R ) and OPΓcl (R ) = m d d ∩m∈Z OPΓcl (R ) coincides with the class R(R ) of the regularizing operators. Also note that every classical symbol ∞ determinesm itsd asymptotic expansion in a unique way. Indeed, if a ∼ k=0 am−k ∈ Γcl (R ), we can recover am (z) from limλ→+∞ λ−m a(λz). Take then an excision function χ ∈ C ∞ (R2d ), with χ(z) = 0 for |z| ≤ 1, χ(z) = 1 for |z| ≥ 2. By applying the same procedure (Rd ) we obtain am−1 and so on. Therefore it easily to a(z) − χ(z)am (z) ∈ Γm−1 cl follows that there is an isomorphism of algebras = d d A = OPΓ∞ Hk (R2d \ {0}) (5.1.7) cl (R )/R(R ) m∈Z k≤m

where on the right the product is induced by the symbol product (α!)−1 ∂ξα aDxβ b. a◦b∼

(5.1.8)

α

The first result of this section is the existence and the uniqueness of a trace on the algebra A defined in (5.1.7), according to Definition 5.0.1. Consider the (2d − 1)-form on R2d given by σ(z) =

2d

j ∧ . . . ∧ dz2n , (−1)j+1 zj dz1 ∧ . . . ∧ dz

(5.1.9)

j=1

j means that dzj is omitted. where dz

d Definition 5.1.1. Let A = Op0 (a) ∈ OPΓm cl (R ) with a ∼ k≥0 am−k , m ∈ Z, and let j : S2d−1 → R2d be the canonical injection. We define the non-commutative residue of A as Res A = S2d−1

where σ is defined in (5.1.9).

a−2d (z) j ∗ σ,

(5.1.10)

208


Note that j ∗ σ, restriction of σ in (5.1.9) to the unit sphere S2d−1 of R2d , coincides with the usual surface measure dΘ, so we recapture the definition of Res A in (5.0.6). However the present form (5.1.10) will be more suitable for the computations which follow. Clearly Res vanishes for operators of order m < −2d, in particular for regularizing operators. This, together with the linearity, implies that it is well defined on A in (5.1.7). If p ∈ H−2d (R2d \ {0}), then Euler’s identity (5.1.2) implies that the form pσ 2d on R \ {0} is closed. Indeed, d(pσ) = (dp) ∧ σ + p dσ = −2dp dz1 ∧ . . . ∧ dz2d + 2dp dz1 ∧ . . . ∧ dz2d = 0. Hence in (5.1.10) we may replace S2d−1 with any (2d − 1)-cycle homologous to S2d−1 in R2d \ {0}. Now we establish the main result. Theorem 5.1.2. The non-commutative residue Res defined in (5.1.10) is a trace on the algebra A in (5.1.7) and any other trace is a multiple of Res. We will need the following lemmata. Lemma 5.1.3. Let g ∈ H−2d+1 (R2d \ {0}). Then 1, . . . , 2d.

S2d−1

∂zk g(z) j ∗ σ = 0, k =

Proof. The statement follows by observing that the form (∂zk g)σ is exact. In fact we shall prove that (5.1.11) (∂zk g)σ = d(gσk ), where σk is given by the contraction between

∂ ∂zk

and σ, i.e.,

∂ i ∧ . . . ∧ dz k ∧ . . . ∧ dz2d "σ = (−1)i+k+1 zi dz1 ∧ . . . ∧ dz ∂zk i=1 k−1

σk =

+

2d

k ∧ . . . ∧ dz i ∧ . . . ∧ dz2d . (5.1.12) (−1)i+k zi dz1 ∧ . . . ∧ dz

i=k+1

We have d(gσk ) = dg ∧ σk + g dσk ,

(5.1.13)

k ∧ . . . ∧ dz2d dσk = (−1)k (2d − 1) dz1 ∧ . . . ∧ dz

(5.1.14)

and we easily compute

and

∂ dg ∧ σk = dg ∧ "σ ∂zk

∂ ∂ = "dg σ − "(dg ∧ σ) ∂zk ∂zk

5.1. Non-Commutative Residue for Γ-Operators ∂ " ((−2d + 1)g dz1 ∧ . . . ∧ dz2d ) ∂zk k ∧ . . . ∧ dz2d . = (∂zk g) σ − (−1)k (2d − 1)g dz1 ∧ . . . ∧ dz

209

= (∂zk g) σ −

(5.1.15)

Substituting the expressions obtained in (5.1.14) and (5.1.15) in (5.1.13) we get (5.1.11). Lemma 5.1.4. Let f ∈ Hm (R2d \ {0}) and suppose one of the following conditions is satisfied: (i) m = −2d; (ii) m = −2d and

S2d−1

f (z) j ∗ σ = 0.

Then there exist functions hk ∈ Hm+1 (R2d \ {0}), k = 1, . . . , 2d, such that f (z) = 2d k=1 ∂zk hk (z). Proof. (i) If m = −2d, then 2d k=1

∂zk (zk f ) =

2d

zk ∂zk f + 2df = (m + 2d)f,

k=1

by Euler’s identity (5.1.2). (ii) By hypothesis, the form f σ on R2d \ {0} is exact. So also j ∗ (f σ) on S2d−1 is exact. On the other hand it is easy to verify that for every x ∈ S2d−1 the (2d − 2)-linear forms {j ∗ (σk )(x)}2d k=1 (where σk is defined in (5.1.12)) span the space ∧2d−2 Tx∗ S2d−1 (which has dimension 2d − 1); hence we can write 2d ∗ ∗ gk j (σk ) (5.1.16) j (f σ) = d k=1

for suitable functions gk ∈ C ∞ (S2d−1 ). Now, switching to polar coordinates (ρ, Θ), ρ > 0, Θ ∈ S2d−1 , (5.1.16) becomes 2d ∗ ∗ gk (Θ)j (σk ) f (z(1, Θ))j (σ) = d k=1

or also

2d ρ−2d+1 gk (Θ) ρ2d−1 j ∗ (σk ) , ρ−2d f (z(1, Θ)) ρ2d j ∗ (σ) = d

k=1 2d

as an equality between forms on R \ {0}. Since σ and therefore σk in polar coordinates do not contain dρ, by homogeneity we have ρ2d j ∗ (σ) = σ and ρ2d−1 j ∗ (σk ) = σk . Then, setting hk (z) := |z|−2d+1 gk (Θ(z)) ∈ H−2d+1 (R2d \ {0}),

210


we obtain fσ = d

2d

h k σk

.

k=1

2d

In view of (5.1.11) we get f =

k=1

∂zk hk .

Proof of Theorem 5.1.2. The first assertion is proved if we show that Res([A, B])= d m d 0 for A, B ∈ A. Let a ∈ Γm cl (R ), b ∈ Γcl (R ) be the classical symbols of A and B respectively; then the symbol of the commutator has asymptotic expansion, cf. (5.1.8), (α!)−1 ∂ξα aDxα b − ∂ξα bDxα a , (x, ξ) = z ∈ R2d . α

d We may rewrite this expression as j=1 ∂ξj Aj + ∂xj Bj for suitable asymptotic expansions Aj , Bj . Then the integrals over S2d−1 of (∂ξj Aj )−2d and (∂xj Bj )−2d are zero by Lemma 5.1.3. We now prove uniqueness. Let us suppose τ is another trace on A. Consider d A = Op0 (a) ∈ OPΓm cl (R ), m ∈ Z, a ∼ j≥0 am−j ; then the functions Dzk a for 1 ≤ k ≤ d and −Dzk a for d + 1 ≤ k ≤ 2d are the symbols of the operators [A, Op0 (zk )], but τ vanishes on commutators, so τ (Op0 (∂zα a)) = 0 for all multiindices α = 0. Applying Lemma 5.1.4 (i) to the function am−j (z) for m − j = −2d, we can 2d m−j+1 write am−j (z) = k=1 ∂zk bk,m−j (z) for suitable functions bk,m−j (z) ∈ H 2d (R \ {0}). If we set bk (z) ∼ bk,m−j (z), cf. Proposition 1.1.6, then we j≥0,j=m+2d

have a(z) ∼

2d

∂zk bk (z) + χ(z)a−2d (z),

(5.1.17)

k=1

where χ ∈ C0∞ (R2d ), χ(z) = 0 for |z| ≤ 1, χ(z) = 1 for |z| ≥ 2. Set −1 r = Ω2d a−2d (z) j ∗ σ, S2d−1

where Ω2d is the measure of S2d−1 . We may rewrite (5.1.17) as a(z) ∼ rχ(z)|z|−2d +

2d

∂zk bk (z) + χ(z) a−2d (z) − r|z|−2d .

k=1

Since a−2d (z) − r|z|−2d is homogeneous of degree −2d and

S2d−1

a−2d (z) − r|z|−2d j ∗ σ = 0,

(5.1.18)

5.1. Non-Commutative Residue for Γ-Operators

211

by Lemma 5.1.4 (ii) it is a finite sum of derivatives of homogeneous functions. From (5.1.18) we therefore obtain τ (Op0 (a)) =τ Op0 (rχ(z)|z|−2d ) τ Op0 (χ(z)|z|−2d ) Res A, = Ω2d

which proves the theorem.

Remark 5.1.5. (a) From symbolic calculus, cf. Remark 1.2.6, it turns out that, if A = Op0 (a) has τ -symbol bτ , the difference a − bτ is a formal series of derivatives and therefore, by Lemma 5.1.3, S2d−1 (a − bτ )−2d j ∗ σ = 0. Hence in Definition 5.1.1 we may replace a and its asymptotic expansion by bτ with the corresponding expansion. In particular, we may use the Weyl symbol. (b) Theorem 5.1.2 remains valid for pseudo-differential operators with symbols d M M m d M in the classes Γm ⊗ CM . In that case the cl (R , C , C ) := Γcl (R ) ⊗ C definition of the non-commutative residue will read Tr a−2d (z) j ∗ σ, Res A = S2d−1

where Tr is the matrix trace. (c) Theorem 5.1.2 tells us that Res spans the vector space (A/[A, A]) C of all traces on A. More precisely, introduced in H−2d (R2d \ {0}) the equivalence relation ∼ defined by p ∼ q if (p − q)σ is exact, we have the following isomorphisms of vector spaces: C (A/[A, A]) → H−2d (R2d \ {0})/ ∼ → H 2d−1 (R2d \ {0}) given by λRes → λ S2d−1 (·)j ∗ σ → λ S2d−1 (·). Let us now show that the non-commutative residue coincides with Dixmier’s d trace for operators A ∈ OPΓ−2d cl (R ). Observe that such operators are compact 2 d on L (R ) by Theorem 1.4.2. Let L(1,∞) (L2 (Rd )) be defined according to (5.0.8) and for T ∈ L(1,∞) 2 (L (Rd )) define Trω (T ) as in (5.0.10). We also need to recall some basic facts on the spectrum of operators of positive order (see Chapter 4). Namely, let d A = aw ∈ OPΓm cl (R ), m > 0, be elliptic with real Weyl symbol a ∼ j≥0 am−j . Then A has a self-adjoint realization in L2 (Rd ) and its spectrum is an unbounded sequence of real isolated eigenvalues of finite multiplicity. This sequence diverges to +∞ or to −∞. Let (λk )k∈N be the sequence of the eigenvalues repeated according to their multiplicity. Modulo a change of sign we can suppose here that it diverges to +∞. By Theorem 4.6.3 we have the following asymptotic formula for the counting function N (λ): 2d

N (λ) ∼ Cλ m

as λ → +∞,

(5.1.19)

212


where C=

(2π)−d 2d

2d

S2d−1

am (z)− m j ∗ σ,

(5.1.20)

as k → ∞,

(5.1.21)

whereas for the eigenvalues we have λk ∼

k C

m

2d

cf. Proposition 4.6.4. d 2 d Theorem 5.1.6. Let A ∈ OPΓ−2d cl (R ), regarded as a compact operator on L (R ). (1,∞) 2 d Then A ∈ L (L (R )) and

Res A = 2d(2π)d Trω (A),

(5.1.22)

independently of ω. d Proof. We first verify the statement of the theorem when A ∈ OPΓ−2d cl (R ) is 2 d d non-negative as an operator on L (R ), injective on S(R ) and has a real elliptic Weyl symbol. Under these assumptions it follows that A = B −1 for an operator w 2d d B = b ∈ OPΓcl (R ) with real Weyl symbol b ∼ j≥0 b2d−j , semibounded from below, cf. Remark 1.7.13 and Theorem 4.2.9. Note that b2d (z) = a−2d (z)−1 . From (5.1.21), (5.1.20) we have the following asymptotic behaviour for the eigenvalues λk of B:

λk ∼

2d(2π)d k k = = 2d(2π)d (Res A)−1 k −1 j ∗ σ C b (z) 2d 2d−1 S

as k → ∞.

This gives for the eigenvalues of A, that are λ−1 k , the formula λ−1 k ∼ Then

Res A −1 k 2d(2π)d

σN (A) Res A ∼ log N 2d(2π)d

as k → ∞.

as N → ∞,

and (5.1.22) holds for A. d Now, if we fix an operator B as above, given any A ∈ OPΓ−2d cl (R ) we can write A = (AB)B −1 , with AB ∈ OPΓ0cl (Rd ) ⊂ B(L2 (Rd )), cf. Theorem 1.4.1, and therefore A ∈ L(1,∞) (L2 (Rd )), since L(1,∞) (L2 (Rd )) is an ideal of B(L2 (Rd )). To prove (5.1.22) in the general case, observe that by linearity it suffices to prove the result for operators with real Weyl symbol. Now if A = aw is such an operator we may write a = (a + Cq) − Cq where q(x, ξ) = (1 + |x|2 + |ξ|2 )−d and C = − inf a/q + 1. So we can assume that A has an elliptic real Weyl symbol. By arguing as in the proof of Lemma 4.2.8 we can further suppose that A is nonnegative, modulo operators in OPΓ−2d−1 (Rd ), which are trace-class by Theorem cl

5.2. Trace Functionals for G-Operators

213

4.4.21 and on which both the traces Res and Trω vanish. Finally, as by Fredholm theory and global regularity V = Ker A is a finite dimensional subspace of S(Rd ), cf. Theorem 2.1.14, the orthogonal projection PV on V is regularizing. Now A = (A + PV ) − PV and A + PV falls in the class of operators considered at the beginning of the present proof. Hence Theorem 5.1.6 is proved.

5.2 Trace Functionals for G-Operators In this section we consider classical pseudo-differential operators of type G, namely d A ∈ OPGm,n cl(ξ,x) (R ), m ∈ Z, n ∈ Z. We recall from Section 3.2 that the cord responding class of symbols Gm,n cl(ξ,x) (R ) consists of all a(x, ξ) which belong to Gm,n (Rd ), i.e., satisfying for every α ∈ Nd , β ∈ Nd the estimates |∂ξα ∂xβ a(x, ξ)| ξm−|α| xn−|β|

for all x ∈ Rd , ξ ∈ Rd ,

(5.2.1)

and admit a double asymptotic expansion a(x, ξ) ∼ a(x, ξ) ∼

∞ j=0 ∞

σψm−j (a)(x, ξ),

(5.2.2)

σen−k (a)(x, ξ),

(5.2.3)

k=0

where σψm−j (a)(x, ξ) is homogeneous of degree m − j with respect to ξ ∈ Rd \ {0} and σen−k (a)(x, ξ) is homogeneous of degree n − k with respect to x ∈ Rd \ {0}; moreover: σψm−j (a)(x, ξ)

∼

σen−k (a)(x, ξ) ∼

∞ k=0 ∞

m−j,n−k σψ,e (a)(x, ξ),

(5.2.4)

m−j,n−k σψ,e (a)(x, ξ),

(5.2.5)

j=0 m−j,n−k (a)(x, ξ) are the same in (5.2.4), (5.2.5), homogeneous separately where σψ,e with respect to ξ of degree m − j, with respect to x of degree n − k. We finally recall that elliptic symbols a(x, ξ) can be characterized by imposing simultaneously σψm (a)(x, ξ) = 0 for all x ∈ Rd , ξ ∈ Rd \ {0}; σen (a)(x, ξ) for all m,n x ∈ Rd \ {0}, ξ ∈ Rd ; σψ,e (a)(x, ξ) = 0 for all x ∈ Rd \ {0}, ξ ∈ Rd \ {0}. For other details, in particular concerning the symbolic calculus, we refer to Chapter 3. In the following we shall adopt Weyl quantization, denoting A = aw the Weyl

214


m,n operator with symbol a. We shall write for short Gm,n cl(ξ,x) , OPGcl(ξ,x) instead of m,n m,n Gcl(ξ,x) (Rd ) and OPGcl(ξ,x) (Rd ). Observe also that

OPG−∞,−∞ := cl(ξ,x)

OPGm,n cl(ξ,x)

m∈Z n∈Z

coincides with the class R(Rd ) of the regularizing operators. Let us now define the following operator algebras. Definition 5.2.1. Let A = ∪n∈Z ∪m∈Z OPGm,n cl(ξ,x) /R. We define the two-sided ideals of A OPGm,n Ie = OPGm,n Iψ = cl(ξ,x) /R, cl(ξ,x) /R, m∈Z n∈Z

n∈Z m∈Z

and the quotient algebras Aψ = A/Ie ,

Ae = A/Iψ ,

Aψ,e = A/(Iψ + Ie ).

The first result of this section is the explicit construction of trace functionals for each of the algebras in Definition 5.2.1. These traces come from residues of the trace of holomorphic operator families. As we observed in the summary, on the ideal of regularizing operators every trace is a multiple of the functional a(x, ξ) dx ¯ dξ, ¯ (5.2.6) Tr(aw ) = i.e., the usual operator trace. That formula still holds for any a ∈ Gm,n cl(ξ,x) provided m < −d, n < −d, see Theorem 4.4.21. In order to extend it further, we need to regularize the resultant divergent integral, and we do this by means of holomorphic families of symbols. Namely, let a(x, ξ) ∈ Gm,n cl(ξ,x) , and consider the family of symbols m+Re z,n+Re τ a ˜(τ, z) = a ˜(τ, z; x, ξ) = [x]τ [ξ]z a(x, ξ) ∈ Gcl(ξ,x) ,

τ, z ∈ C,

(5.2.7)

where [·] denotes an arbitrary strictly positive C ∞ function on Rd with [y] = |y| for |y| ≥ 1. Notice that a ˜(0, 0) = a. Lemma 5.2.2. Let a ∈ Gm,n ˜ be as in (5.2.7). Then the function cl(ξ,x) and let a t(τ, z) := Tr(˜ a(τ, z)w ) is defined and holomorphic for Re z < −m − d, Re τ < −n − d, and extends to a meromorphic function of τ, z with at most simple poles on the lines z = −m − d + j, τ = −n − d + k, j, k ∈ N. Proof. By Theorem 4.4.21 we have ¯ dξ, ¯ t(τ, z) = a ˜(τ, z; x, ξ) dx ¯ dξ ¯ = [x]τ [ξ]z a(x, ξ) dx


215

provided Re z < −m − d, Re τ < −n − d, and the function t(τ, z) is then holomorphic there. In order to show the desired meromorphic extension, write t(τ, z) = t1 (τ, z) + t2 (τ, z) + t3 (τ, z) + t4 (τ, z) where t1 , t2 , t3 , t4 are the integrals respectively on A1 = {|x| ≤ , |ξ| ≤ 1}, A2 = {|x| ≤ , |ξ| ≥ 1}, A3 = {|x| ≥ , |ξ| ≤ 1}, A4 = {|x| ≥ , |ξ| ≥ 1}. In fact, it would suffice to set = 1, but in view of future developments it is useful to work with an arbitrary ≥ 1. Clearly t1 (τ, z) is an entire function. As t2 is concerned, we note that for |ξ| ≥ 1 and every p ∈ N, p ≥ 1, we have a ˜(τ, z; x, ξ) =

p−1

σψm−j (a)(x, ξ/|ξ|)|ξ|z+m−j [x]τ + rp (x, ξ)[x]τ |ξ|z ,

j=0

˜(τ, z) in the with a remainder rp ∈ Gm−p,n cl(ξ,x) . Substituting this expression for a integral a ˜(τ, z; x, ξ) dx ¯ dξ ¯ t2 (τ, z) = |x|≤

|ξ|≥1

and introducing polar coordinates for the integration in the variables ξ yield −d

t2 (τ, z) = −(2π)

p−1 j=0

1 z+m+d−j

Sd−1

|x|≤

[x]τ σψm−j (a) dθ dx + Rp, (τ, z), (5.2.8)

where Rp, (τ, z) is holomorphic for Re z < −m − d + p and all τ ∈ C. Interchanging the roles of the variables x, ξ we obtain t3 (τ, z) = −(2π)−d

q−1 k=0

τ +n+d−k τ +n+d−k

|ξ|≤1

Sd−1

[ξ]z σen−k (a) dθ dξ (τ, z), (5.2.9) + Rq,

where Rq, (τ, z) is holomorphic for Re τ < −n − d + q and all z ∈ C. Finally, repeating the same argument twice, we get

t4 (τ, z) = (2π)−d

q−1 p−1 j=0 k=0

×

Sd−1

Sd−1

1 τ +n+d−k z+m+d−j τ +n+d−k

m−j,n−k σψ,e (a) dθ dθ +

p−1 j=0

+

q−1 k=0

1 R (τ ) z + m + d − j q,j,

τ +n+d−k (τ, z). (5.2.10) R (z) + Rp,q, τ + n + d − k p,k

216


Rq,j, (τ )

Here we set =

|x|≥

Sd−1

[x]τ rq,j (x, θ) dθ dx,

Hence Rq,j, (τ ) is holomorphic for Re τ < −n − d + q. with rq,j ∈ (τ ) → 0 as → +∞ uniformly for τ in Notice, for future reference, that Rq,j, compact subsets of {τ ∈ C : Re τ < −n − d + q}. (z) is holomorphic for Re z < −m − d + p, whereas Rp,q, (τ, z) Similarly, Rp,k is holomorphic for Re τ < −n − d + q, Re z < −m − d + p. So, we have verified that t(τ, z) extends to a meromorphic function on Re τ < −n−d+q, Re z < −m−d+p. As p and q are arbitrary, this concludes the proof.

Gm−j,n−q . cl(ξ,x)

˜(τ, z) as in (5.2.7), we now consider the For a ∈ Gm,n cl(ξ,x) , m ∈ Z, n ∈ Z, and a functionals defined by ψ (aw )−z Tr e (aw )+τ 2 V +τ zV +z 2 V , (5.2.11) τ zTr(˜ a(τ, z)w ) = Trψ,e (aw )−τ Tr where V, V , V are holomorphic near (0, 0). ψ , Tr e defined in (5.2.11) have the folProposition 5.2.3. The functionals Trψ,e , Tr lowing explicit expressions: −d,−d w −d Trψ,e (a ) = (2π) σψ,e (a) dθ dθ , (5.2.12) Sd−1

ψ (a ) = (2π) Tr w

−d

σψ−d (a) dθ dx − log Trψ,e (aw )

lim

→+∞

|x|≤

−

Sd−1

n+d i i=1

e (aw ) = (2π)−d lim Tr

→+∞

i

|ξ|≤

−

Sd−1

i=1

Sd−1

Sd−1

−d,i−d σψ,e (a) dθ dθ

, (5.2.13)

σe−d (a) dθ dξ − log Trψ,e (aw )

m+d i

i

Sd−1

Sd−1

Sd−1

i−d,−d σψ,e (a) dθ dθ

. (5.2.14)

Proof. We refer to the proof of Lemma 5.2.2, where now we take p > m + d, q > n + d. τ zTr(a(τ, z)w ) using Formula (5.2.12) follows at once as the limit lim (τ,z)→(0,0)

the expressions (5.2.8), (5.2.9), (5.2.10). ψ as To prove (5.2.13), we observe that we can obtain Tr ψ (aw ) = − lim τ −1 lim (τ zTr(˜ a(τ, z)w ) − Trψ,e (aw )). Tr τ →0

z→0

(5.2.15)


217

4 We use the decomposition τ zTr(˜ a(τ, z)w ) = i=1 τ zti (τ, z) as in the proof of Lemma 5.2.2, and we perform the limit as z → 0: the expressions τ zt1 (τ, z) and τ zt3 (τ, z) tend to 0, as well as all the terms in (5.2.8) and (5.2.10), except possibly those corresponding to j = m + d. What remains is, on the whole, independent of (τ ) and the terms of but, on the other hand, as → +∞ the expression Rq,m+d, the first sum in (5.2.10) with k > n + d tend to zero uniformly for small τ . Then we have lim (τ zTr(˜ a(τ, z)w ) − Trψ,e (aw ))

z→0

= (2π)

+

−d

n+d−1 k=0

−

τ

τ

[x] |x|≤

Sd−1

τ +n+d−k τ +n+d−k

σψ−d (a) dθ dx

Sd−1

Sd−1

τ + τ

Sd−1

−d,n−k σψ,e (a) dθ dθ

Sd−1

−τ

−d,−d σψ,e (a) dθ dθ

−1

w

Trψ,e (a )

+ o(1),

where o(1) stands for a function of τ which tends to 0 as → +∞ uniformly for small τ . Hence (5.2.13) follows from (5.2.15). In the same way one proves (5.2.14). ψ and Tr e to Remark 5.2.4. Let us note that the restrictions Trψ and Tre of Tr > > m,−d−1 −d−1,n and n∈Z OPGcl(ξ,x) are given by m∈Z OPGcl(ξ,x) Trψ (aw ) = (2π)−d w

−d

Tre (a ) = (2π)

Rd x

Rd ξ

Sd−1

Sd−1

σψ−d (a) dθ dx,

a∈

Gm,−d−1 cl(ξ,x) ,

(5.2.16)

m∈Z

σe−d (a) dθ dξ,

a∈

G−d−1,n cl(ξ,x) ,

(5.2.17)

n∈Z

e turn out just the finite parts of the integrals in (5.2.16) and ψ and Tr and Tr > (5.2.17) when a ∈ m∈Z,n∈Z Gm,n cl(ξ,x) . Furthermore, the functionals Trψ and Tre vanish on Ie and Iψ respectively, so that they are well defined on Aψ and Ae as extensions of Trψ and Tre . Theorem 5.2.5. The functional Trψ,e defines a trace on the algebra A which vanishes on Iψ and Ie and therefore it induces traces on Aψ , Ae and Aψ,e . On Iψ and Ie trace functionals are given respectively by Trψ and Tre defined in (5.2.16) and (5.2.17). When n ≥ 2, for all these algebras the above functionals are the unique traces up to multiplication by a constant. Proof. In all cases the desired conclusion easily follows by the same arguments as in the proof of Theorem 5.1.2. To avoid an overweight of this section, we prefer then to omit any detail.

218


We pass now to consider Dixmier traces. With respect to the discussion in the Summary, cf. Definition 5.0.2 and (5.0.10), we use here a somewhat more general notion. We consider traces whose natural domain is contained in the ideal K(H) of compact operators on the Hilbert space H. For T ∈ K(H), let μn (T ), n = 1, 2, . . ., be the sequence of the eigenvalues of |T |, counted with their multiplicity and labelled in decreasing order and let N σN (T ) = n=1 μn (T ), N = 1, 2, . . ., as in (5.0.7). For a fixed sequence α of positive numbers αN such that (i) αN → +∞; (ii) α0 > α1 − α0 and αN +1 − αN ≥ αN +2 − αN +1 for N ∈ N; −1 α2N → 1; (iii) αN −1 σN (T ) ∈ l∞ (N)}. Repeating the we define the ideal Iα (H) := {T ∈ K(H) : αN arguments in the summary, we then consider a linear form ω on Cb (1, ∞), the space of the continuous bounded functions on [1, ∞], with ω ≥ 0, ω(1) = 1 and ω(f ) = 0 if limx→+∞ f (x) =0. Given a bounded sequence a = (an )n≥1 , we construct the function fa = n≥1 an χ[n−1,n) ∈ L∞ (R+ ) and define the ω-limit t limω an = ω(M fa ) where, for g ∈ L∞ (R+ ), M g(t) := log1 t 1 g(s) s ds is the Cesàro mean of g. In the case of convergent sequences the ω-limit coincides with the usual limit.

Definition 5.2.6. Let α = (αN ) be a sequence as above and T ∈ Iα (H), T ≥ 0. We define the Dixmier trace of T as −1 σN (T ). Trα,ω (T ) = lim αN ω

Dixmier’s trace extends to a linear map on Iα (H). In the case of the sequence αN = log N we recapture the definition in (5.0.10); we shall continue to use the notation Trω for the Dixmier trace associated with that sequence and to denote by L(1,∞) (H) its domain, cf. the following more general definition. Definition 5.2.7. For 1 < p < ∞ we define the subspace L(p,∞) (H) ⊂ K(H) as the set of all compact operators T with σN (T ) = O(N 1−1/p ). Similarly we define L(1,∞) (H) ⊂ K(H) by the condition σN (T ) = O(log N ). (p,∞) For 1 < p < ∞, we define the subspace Llog (H) ⊂ K(H) as the set of all (1,∞)

compact operators T with σN (T ) = O(N 1−1/p (log N )−1/p ); Llog will be defined by the condition σN (T ) = O((log N )2 ).

(H) ⊂ K(H)

All these spaces are normed ideals contained in K(H), containing the ideal B1 (H) of trace-class operators. (1,∞)

Remark 5.2.8. Let us observe that for p = 1 the ideal Llog (H) is the natural domain of the Dixmier trace associated with the sequence αN = (log N )2 . In short we shall denote it by Trω .


219

2 d All the spaces OPGm,n cl(ξ,x) with m < 0, n < 0 are contained in K(L (R )). In order to establish relations between these spaces and the ideals in Definition 5.2.1, we have to study the asymptotic behaviour of the spectrum of such operators. We recall some basic facts from Chapter 4. Let a ∈ Gm,n cl(ξ,x) , m > 0, n > 0, be a real elliptic symbol, semibounded from below. Then the corresponding operator aw has a self-adjoint realization L2 (Rd ); it is bounded from below and has discrete spectrum {λk }k∈N diverging to +∞. Denote by N (λ) the counting function associated with the operator aw . Then ⎧ d ⎪ ⎨Cm λ m log λ for m = n, d (5.2.18) N (λ) ∼ Cm λm for m < n, ⎪ ⎩ nd for m > n, Cn λ

where

d (2π)−d m,m σψ,e (a)− m dθ dθ , dm d−1 d−1 S S (2π)−d d σψm (a)− m dθ dx, Cm = d d d−1 Rx S −d d (2π) Cn = σen (a)− n dθ dξ, d Rd Sd−1 ξ

Cm =

(5.2.19) (5.2.20) (5.2.21)

cf. Theorem 4.6.6. We shall need the following simple lemma. Lemma 5.2.9. For 1 ≤ p < ∞, let gp be the inverse function of fp : (1, ∞) → R+ , fp (x) = xp log x. Then (a) if (an ) and (bn ) are positive sequences with an ∼ bn we have gp (an ) ∼ gp (bn ); (b) for every positive sequence (kn ) diverging to +∞ we have gp (kn ) ∼ (pkn / log kn )1/p . Proof. (a) The statement follows by observing that, for 0 < x < x , we have 0
n, cl(ξ,x) ⊂ ⎪L ⎩ (−d/n,∞) 2 d L (L (R )) if m < n.

220


Furthermore we have for a ∈ G−d,−d cl(ξ,x) ,

Trψ,e (aw ) = 2d2 Trω (aw ) Trψ (aw ) = d Trω (aw ) w

w

Tre (a ) = d Trω (a )

(5.2.23)

for a ∈

G−d,n cl(ξ,x)

with n ∈ Z, n < −d,

(5.2.24)

for a ∈

Gm,−d cl(ξ,x)

with m ∈ Z, m < −d,

(5.2.25)

independently of ω. Proof. We verify the first inclusion in (5.2.22). The other cases can be proved in the same way. Consider first the case of an operator A ∈ OPGm,m cl(ξ,x) , non-negative as an 2 d d operator on L (R ), injective on S(R ) and with a real elliptic Weyl symbol. Then A is invertible on S(Rd ) and S (Rd ) and A−1 ∈ OPG−m,−m cl(ξ,x) has real elliptic Weyl symbol, semibounded from below, cf. Remark 1.7.13 and Theorem 4.2.9. Hence from (5.2.18) and (5.2.19) (with −m in place of m) we have the formula NA−1 (λ) ∼ C˜m λ− m log λ, d

with

−d

(2π) C˜m = − dm

Sd−1

(5.2.26)

m,m σψ,e (a)− m dθ dθ . d

Sd−1

On the other hand, (5.2.26) is equivalent to the following formula for the eigenvalues λk of A−1 : −d −1 λk m log λk ∼ C˜m k, cf. the proof of Proposition 4.6.4, which by Lemma 5.2.9 implies −1 k) ∼ (−dk/(mC˜m log k))−m/d . λk ∼ g− d (C˜m m

For the eigenvalues of A, that are λ−1 k , we obtain the formula m/d ˜ λ−1 . k ∼ (−dk/(mCm log k))

(5.2.27)

From (5.2.27) it follows that N k=1

m

− md d ˜ −1 d N log x Cm dx m x 1 ⎧ m m m ⎨ d d ˜ −1 d N 1+ d (log N )− d C − m m ∼ d+m ⎩ 1 C˜ (log N )2 −d

λ−1 k ∼

−

2

(−d/m,∞)

(−d/m,∞)

for − d < m < 0, for m = −d.

(5.2.28)

(L2 (Rd )). As Llog (L2 (Rd )) is an ideal of B(L2 (Rd )) Hence A ∈ Llog the first inclusion in (5.2.22) follows, since one can write P ∈ OPGm,m cl(ξ,x) as P = (P A−1 )A where P A−1 is bounded in L2 (Rd ).

5.3. Dixmier Traceability for General Pseudo-Differential Operators

221

Now we come to the relations (5.2.23), (5.2.24), (5.2.25) between the traces Trψ,e , Trψ , Tre and the Dixmier traces. We limit ourselves to consider (5.2.23). It follows from (5.2.28) that (5.2.22) holds for an operator A = aw as in the first part of the present proof (with m = −d). The extension to every symbol a ∈ G−d,−d cl(ξ,x) goes exactly as in the last part of the proof of Theorem 5.1.6.

5.3 Dixmier Traceability for General Pseudo-Differential Operators A natural question is whether the results of Sections 5.1 and 5.2 can be extended to Weyl operators A = aw with symbols a in the classes S(M ; Φ, Ψ) considered in Chapter 1, i.e., satisfying the estimates |∂ξα ∂xβ a(x, ξ)| M (x, ξ)Ψ(x, ξ)−|α| Φ(x, ξ)−|β| ,

x ∈ Rd , ξ ∈ Rd .

(5.3.1)

The definition of non-commutative residue cannot be reproduced in this general setting, because of the lack of homogeneity, however we may investigate Dixmier traceability, i.e., whether such general pseudo-differential operators belong to L(1,∞) (L2 (Rd )) = {A ∈ K(L2 (Rd )) : σN (A) = O(log N )},

(5.3.2)

N with σN (A) = n=1 μn (A), where μn (A), n = 1, 2, . . ., is the sequence of the eigenvalues of |A|. We will assume the strong uncertainty principle: h(x, ξ) := Φ(x, ξ)−1 Ψ(x, ξ)−1 (1 + |x| + |ξ|)−δ ,

x ∈ Rd , ξ ∈ Rd ,

(5.3.3)

for some δ > 0. We also suppose that M is a regular weight, cf. Definition 1.5.1, satisfying M (x, ξ) → 0 as (x, ξ) → ∞, (5.3.4) so that aw ∈ K(L2 (Rd )) by Theorem 1.4.2. We shall now express a sufficient condition on the weight M in (5.3.1) to have aw ∈ L(1,∞) (L2 (Rd )) for all a ∈ S(M ; Φ, Ψ). Before stating the result, we recall the definition of L1w (Rn ), the Lorentz-Marcinkiewicz space of the L1 -weak functions in Rn . Namely, L1w (Rn ) = {f : Rn → C measurable: sup s · measure({|f | > s}) < ∞}.

(5.3.5)

s>0

For example, the functions |z|−n and (1 + |z|2 )−n/2 , z ∈ Rn , are in L1w (Rn ). Theorem 5.3.1. Let the strong uncertainty principle (5.3.3) be satisfied and let M be a regular weight fulfilling (5.3.4). Then, if M ∈ L1w (R2d ), a ∈ S(M ; Φ, Ψ) ⇒ aw ∈ L(1,∞) (L2 (Rd )).

(5.3.6)

222


As examples, we may recapture in part the results of Sections 5.1 and 5.2. In fact, for Φ(x, ξ) = Ψ(x, ξ) = (1 + |x|2 + |ξ|2 )1/2 , M (x, ξ) = (1 + |x|2 + |ξ|2 )−d we have M (x, ξ) ∈ L1w (R2d ) and we obtain OPΓ−2d (Rd ) ⊂ L(1,∞) (L2 (Rd )), cf. Theorem 5.1.6. In the case Φ(x, ξ) = x, Ψ(x, ξ) = ξ, M (x, ξ) = x−d ξ−d− , or M (x, ξ) = x−d− ξ−d for some > 0, we have M ∈ L1w (R2d ) and we obtain OPG−d−,−d (Rd ) ⊂ L(1,∞) (L2 (Rd )), OPG−d,−d− (Rd ) ⊂ L(1,∞) (L2 (Rd )), cf. Theorem 5.2.10. Theorem 5.3.1 will be obtained as a consequence of some auxiliary propositions. First observe that, since M is a regular weight, M ∈ S(M ; Φ, Ψ). Consider (M −1 )w , the operator with Weyl symbol M −1 . In view of (5.3.4), it is a selfadjoint operator in L2 (Rd ) with a spectrum made of a sequence of eigenvalues bounded from below. It follows that (M −1 + c)w is bounded from below, say, by 1 if c is large enough, and the inverse A−1 on S(Rd ), S (Rd ) is well defined as a pseudo-differential operator with Weyl symbol in S(M ; Φ, Ψ), cf. Remark 1.7.13. Let us now write a = M −1 + c, so that A = aw . Notice that we have as well a−1 ∈ L1w (R2d ).

(5.3.7)

We observe that Theorem 5.1.3 is proved if we verify that A−1 ∈ L(1,∞) (L2 (Rd )). Indeed, given any pseudo-differential operator P with Weyl symbol in S(M ; Φ, Ψ) we can write P = P AA−1 . Since P A has a symbol in S(1; Φ, Ψ), it is bounded in L2 (Rn ) by Theorem 1.4.1, and L(1,∞) is an ideal in the space of bounded operators. Hence we deduce that P ∈ L(1,∞) (L2 (Rd )). Thus we are reduced to prove that the eigenvalues λj of A satisfy λ−1 j = O(1/j)

as j → ∞.

(5.3.8)

Theorem 5.3.2. The operator e−tA , t ≥ 0 can be written as e−tA = bw t + S(t), where bt is a bounded family of symbols in S(1; Φ, Ψ) for t ≥ 0, satisfying for all α ∈ Nd , β ∈ Nd , |∂ξα ∂xβ bt (x, ξ)| e−ta(x,ξ)/2

for all t ≥ 0, (x, ξ) ∈ R2d ,

(5.3.9)

and S(t) is a trace-class operator with

S(t) B1 (L2 ) t

for all t ≥ 0.

(5.3.10)

Proof. The first part of the proof of Theorem 4.5.1 shows that there exist symbols vj (t, x, ξ), j = 0, 1, . . ., of the form v0 (t, x, ξ) = e−ta0 (x,ξ) and, for j ≥ 1, vj (t, x, ξ) = e−ta0 (x,ξ)

2j l=1

tl vl,j (x, ξ), vl,j ∈ S(M l hj ; Φ, Ψ),

(5.3.11)

5.3. Dixmier Traceability for General Pseudo-Differential Operators such that bt (x, ξ) :=

N j=0

vj (t, x, ξ) satisfies (∂t + aw )bw t = K(t), w b0 = I,

223

(5.3.12)

for some operator K(t) with Weyl symbol belonging to a bounded subset of S(M −1 hN +1 , g) when t ≥ 0. It follows therefore from Theorem 4.4.21 that, if N is chosen so that M −1 hN +1 ∈ L1 (R2d ) (which is possible in view of (5.3.3)), K(t) turns out to be trace-class and K(t) B1 (L2 ) ≤ C for every t ≥ 0. In order to verify (5.3.10) we observe that, since the operator e−tA solves (5.3.12) with K = 0, we have t −tA − e = e−(t−s)A K(s) ds. (5.3.13) bw t 0

Then, by (4.4.28),

bw t

−tA

−e

B1 (L2 ) ≤

t 0

≤

0

t

e−(t−s)A K(s) B1 (L2 ) ds

e−(t−s)A B(L2 ) K(s) B1 (L2 ) ds t.

This concludes the proof. Proposition 5.3.3. We have ∞

e−tλj = O(t−1 ),

as t # 0.

j=1

Proof. By Theorem 5.3.2 we have ∞

e−tλj = Tr e−tA = e−tA B1 (L2 ) ≤ bw t B1 (L2 ) + S(t) B1 (L2 ) ,

(5.3.14)

j=1

with S(t) B1 t, whereas it follows from Theorem 4.4.21 that, for N large enough,

∂ γ bt L1 . (5.3.15)

bw t B1 (L2 ) |γ|≤N

On the other hand, by (5.3.9),

∂ γ bt L1 e−ta(x,ξ)/2 dx dξ =

+∞ 0

where we set λ(s) = measure({a/2 ≤ s}).

e−ts dλ(s),

(5.3.16)

224


Now we have λ(s) = 0 for s in a right neighbourhood of 0 and λ(s) ≤ Cs, since a−1 ∈ L1w (R2d ), see (5.3.7). Thus, integrating by parts in (5.3.16) yields

+∞

e−ts dλ(s) = t

0

+∞

0

e−ts λ(s)ds ≤ Ct

+∞

e−ts s ds = Ct−1 .

0

−1 This shows that bw ) as t # 0, which together with (5.3.14) and t B1 (L2 ) = O(t (5.3.10) concludes the proof.

Let now N (λ) be the counting function of A. Proposition 5.3.4. We have N (x) = O(x) as x → +∞. Proof. Since N (t−1 ) =

t−1

+∞

dN (λ) = 0

0

we get N (t−1 ) ≤ e

+∞

χ[0,t−1 ] (λ)dN (λ),

e−tλ dN (λ).

0

On the other hand, the right-hand side of (5.3.17) is exactly e is O(t−1 ) as t # 0 in view of Proposition 5.3.3.

(5.3.17) ∞ j=1

e−tλj , which

We finally prove (5.3.8). Proposition 5.3.5. We have λ−1 j = O(1/j) as j → ∞. Proof. We know from Proposition 5.3.4 that N (x) ≤ Cx for x ≥ 0. Now, given any j, take j1 ≥ j such that λj = λj1 < λj1 +1 . Then N (λj1 ) = j1 so that j ≤ j1 ≤ Cλj1 = Cλj . This concludes the proof.

Theorem 5.3.1 is therefore proved.

Notes We first review related results for pseudo-differential operators on compact manifolds. The non-commutative residue had been used initially in the one-dimensional case by Manin [139] and Adler [2] in their work on algebraic aspects of the Korteweg-de Vries equation. In 1987 Wodzicki gave a more detailed account of the non-commutative residue and related topics, cf. [196] and also the survey by Kassel [126]. Guillemin discovered the non-commutative residue independently in the context of a new proof of Weyl’s formula [106]. Connes obtained (5.0.11) on compact manifolds in his work on Non-Commutative Geometry. Different variants

Notes

225

and generalizations of the non-commutative residue of Wodzicki are presented in Guillemin [107] concerning Fourier integral operators, Fedosov, Golse, Leichtman and Schrohe [76] concerning operators on manifolds with boundary, Schrohe [177] about manifolds with conical singularities, Nicola [151] about anisotropic operators on foliated manifolds. We refer also to Grubb [101], Melrose and Nistor [146], Paycha and Scott [155] for connections with the index problem and Laurent expansions of holomorphic families of pseudo-differential operators. About pseudo-differential operators in Rd , we mention first the contribution of Boggiatto and Nicola [20]; the contents in Section 5.1 correspond to a particular case of their results, concerning a more general class of anisotropic Γ-operators. The main reference for Section 5.2 is Nicola [152], which in turn follows the lines of [146]; see also Lauter and Moroianu [131]. For the non-normal traces used in Section 5.2, we refer to the original paper of Dixmier [73]. Finally, for Section 5.3 we refer to Nicola and Rodino [154], where a more general result was given in terms of the Weyl-Hörmander classes. Indeed the problem of the Dixmier traceability in Rd , treated in Section 5.3, is richer than on compact manifolds, see Gayral, GraciaBondía, Iochum, Schücker and Vàrilly [89] for related arguments in non-compact Non-Commutative Geometry.

Chapter 6

Exponential Decay and Holomorphic Extension of Solutions Summary In the preceding chapters we proved, under different assumptions of global hypoellipticity on the symbol in Rd , that the solutions in S (Rd ) of the homogeneous equation belong to S(Rd ). In particular, for the self-adjoint operators discussed in Chapter 4, all the eigenfunctions belong to S(Rd ). In the present chapter we show that this information can be strongly improved for G and Γ operators, namely we may give precise results of exponential decay and holomorphic extension of solutions. The next Section 6.1 introduces the spaces Sνμ (Rd ), μ > 0, ν > 0, which provide a precise language to describe the above mentioned properties. In short: the index ν expresses an exponential decay of order 1/ν, i.e., |f (x)| e−|x|

1/ν

,

x ∈ Rd ,

(6.0.1)

for some > 0; the index μ corresponds to exponential decay of order 1/μ of the Fourier transform: 1/μ (6.0.2) |f(ξ)| e−|ξ| , ξ ∈ Rd . If μ < 1, then (6.0.1), (6.0.2) imply extension of f to an entire function satisfying |f (x + iy)| e−|x|

1/ν

+δ|y|1/1−μ

,

x ∈ R d , y ∈ Rd ,

(6.0.3)

for some > 0, δ > 0; if μ = 1 the extension is limited to a strip of the complex domain.

228

Chapter 6. Exponential Decay and Holomorphic Extension of Solutions

Let us clarify our objectives on some model operators, on which we can test Sνμ -regularity of the solutions. Consider first the Schrödinger harmonic oscillator: (6.0.4) H = −Δ + |x|2 . d Corresponding to the eigenvalue λ = λk = j=1 (2kj + 1), k = (k1 , . . . , kd ) ∈ Nd , we have the eigenfunction, cf. Section 2.2: u(x) = uk (x) =

d

Pkj (xj )e−

|x|2 2

(6.0.5)

j=1

where Pr (t) stands for the r-th Hermite polynomial, cf. (2.2.26). Observe that the decay at infinity of u(x) in (6.0.5) is super-exponential of order 2, i.e., for a certain constant > 0: 2 (6.0.6) |u(x)| e−|x| , x ∈ Rd . The second aspect is holomorphic extension, namely u extends to an entire function, satisfying in the complex domain the following improved version of (6.0.6), for suitable positive constants and δ: |u(x + iy)| e−|x|

2

+δ|y|2

,

x ∈ Rd , y ∈ Rd .

(6.0.7)

Note also that the eigenfunctions (6.0.5) are invariant under Fourier transformation. In terms of the spaces Sνμ (Rd ), we therefore have ν = 1/2, cf. (6.0.1), and μ = 1/2, cf. (6.0.2), (6.0.3). Our aim in Section 6.2 will be to extend the properties (6.0.6), (6.0.7) to the eigenfunctions u(x) of all higher order Γ-elliptic equations with polynomial 1

coefficients, namely we shall prove u ∈ S 12 (Rd ). We shall also consider semilinear 2 perturbations of the harmonic oscillator: Hu − λu = −Δu + |x|2 u − λu = uk

(6.0.8)

where λ ∈ R and k ≥ 2. For the eigenfunctions of (6.0.8), i.e., homoclinics, we still have super-exponential decay of order 2, cf. (6.0.6), however (6.0.7) fails in general, and the extension to the complex domain u(x + iy) is limited to a strip {x + iy ∈ Cd : |y| < T } for some T > 0, as we shall show by a counterexample. In other terms, we have now u ∈ S 11 (Rd ). 2 The subsequent Sections 6.3, 6.4, 6.5 are devoted to similar results for Gelliptic equations. It is clear that in the G-case the optimal result is u ∈ S11 (Rd ), that is the estimate for > 0, |u(x)| e−|x| ,

x ∈ Rd ,

(6.0.9)

combined with analytic extension in a strip. Consider for example the G-elliptic ordinary differential operator Lu = −(1 + x2 )u − 2xu + x2 u,

x ∈ R.

(6.0.10)

6.1. The Function Spaces Sνμ (Rd )

229

The operator L is self-adjoint with compact resolvent and then there exists a sequence λj ∈ R, j = 1, 2, . . . , such that Luj = λj uj for some non-trivial uj ∈ S(Rd ). From the theory of the asymptotic integration, cf. Section 3.3, we have uj (x) = Cx−1 e−|x| + O(x−2 e−|x| )

for |x| → +∞,

and from Fuchs theory we may expect singularities at x = ±i. Initially in Section 6.3 we shall present a version of the G-pseudo-differential calculus for Sνμ (Rd ) classes, μ ≥ 1, ν ≥ 1, arguing on symbols p(x, ξ) satisfying, for some C > 0, |∂ξα ∂xβ p(x, ξ)| C |α|+|β| (α!)μ (β!)ν ξm−|α| xn−|β| for all x ∈ Rd , ξ ∈ Rd , α ∈ Nd , β ∈ Nd . (6.0.11) In particular, we shall construct parametrices for the G-elliptic operators. Because of the technical difficulties coming from the analytic case, our regularity results will be limited here to u ∈ Sθθ (Rd ), θ > 1, θ ≥ μ + ν − 1. For symbols satisfying (6.0.11) with μ = ν = 1, we shall reach the expected optimal regularity S11 (Rd ) in the conclusive Section 6.5 by a technique of a priori estimates and iterative methods, providing results also for semilinear operators. Applications mainly concern travelling wave equations, with G-elliptic linear part and semilinear perturbation. On the subject, we give in Section 6.4 a short survey for readers interested in the Mathematical Physics aspects. Relevant examples are: u − V u + u2 = 0 in R (6.0.12) for the solitary waves v(t, x) = u(x − V t), V > 0, of the Korteweg-de Vries equation, and higher order generalizations; the d-dimensional extension −Δu + u = uk ,

k ≥ 2,

(6.0.13)

appearing in Plasma Physics and Nonlinear Optics; the intermediate-long-wave equation in R (6.0.14) N (D)u + γu = u2 , γ > −1, with N (ξ) = ξCtgh ξ, which is a symbol of type (6.0.11) with μ = 1. Note that the linear parts of (6.0.12), (6.0.13), (6.0.14) have no eigenfunctions in S(Rd ), whereas the semilinearity produces homoclinics u ∈ S11 (Rd ).

6.1 The Function Spaces Sνμ (Rd ) The asymptotic information given by the Schwartz space S(Rd ) is not so satisfactory in Applied Mathematics, in particular one would like to know more precisely how fast the decay of f ∈ S(Rd ) is at infinity. To this end, it is convenient to use

230


the spaces Sνμ (Rd ), subspaces of S(Rd ). We shall define them in terms of simultaneous estimates of exponential type for f (x) and f(ξ). This seems the appropriate approach for applications to the operators of the preceding chapters, because of the symmetrical role of the variables x and ξ. Definition 6.1.1. The function f (x) is in Sνμ (Rd ), μ > 0, ν > 0, if f (x) ∈ S(Rd ) and there exists a constant > 0 such that 1

|f (x)| e−|x| ν ,

x ∈ Rd ;

(6.1.1)

1 μ f (ξ) e−|ξ| ,

ξ ∈ Rd .

(6.1.2)

We have the obvious inclusions Sνμ (Rd ) ⊂ Sνμ (Rd ) for μ ≤ μ , ν ≤ ν . It is also evident that application of the Fourier transform interchanges the indices μ and ν in Definition 6.1.1, namely: Theorem 6.1.2. For f ∈ S(Rd ), we have f ∈ Sνμ (Rd ) if and only if f ∈ Sμν (Rd ). In particular the spaces Sμμ (Rd ), μ > 0, are invariant under the action of the Fourier transform. It will be clear from the arguments in the sequel that Definition 6.1.1 does not change meaning, if referred to f ∈ L2 (Rd ), or even f ∈ S (Rd ), provided (6.1.1), (6.1.2) make sense. So we shall keep S(Rd ) as an universe-set, in the whole section. Example 6.1.3. The Gaussian function f (x) = e−

|x|2 2

1

belongs to S 12 (Rd ). In 2

2

|ξ| fact (6.1.1), (6.1.2) are satisfied with μ = ν = 12 , since f(ξ) = e− 2 . More 2 generally, all the functions in Rd of the type P (x)e−a|x| , with a > 0 and P (x) 1

any polynomial, belong to S 12 (Rd ). 2

Example 6.1.4. Consider then in S(R), for a fixed integer k > 0, the function 2k

g(t) = e−t , which satisfies (6.1.1) with ν =

t ∈ R,

(6.1.3)

1 . 2k

The Fourier transform 2k h(τ ) = g(τ ) = e−itτ −t dt, ¯ τ ∈ R,

(6.1.4)

cannot be computed in elementary terms. Note however that g(t) solves g + 2kt2k−1 g = 0,

(6.1.5)

hence h(τ ) is, modulo a multiplicative constant, the unique solution in S(R) of the transformed equation (−1)k+1 h(2k−1) + τ h = 0. (6.1.6) 2k


231

From the classical theory of asymptotic integration, we know that the bounded solutions of (6.1.6) must satisfy, for some > 0, |h(τ )| e−|τ | 1 hence (6.1.2) holds with μ = 1− 2k and g ∈ S 1 2k

h(τ ) in (6.1.4) belongs to S1−

1 2k

2k 2k−1

1 1− 2k 1 2k

,

(R). Symmetrically, the function

(R).

We would like now to pass from the estimates (6.1.1), (6.1.2) to estimates involving only f (x). The first step is to convert exponential bounds into factorial bounds. Proposition 6.1.5. The following conditions are equivalent: (1) the condition (6.1.1) holds, i.e., there exists a constant > 0 such that 1

|f (x)| e−|x| ν ,

x ∈ Rd ;

(6.1.7)

(2) there exists a constant C > 0 such that |xα f (x)| C |α| (α!)ν ,

x ∈ Rd , α ∈ Nd .

(6.1.8)

Proof. It will be convenient to re-write (6.1.7) in the form 1

1

|f (x)| ν e−|x| ν ,

x ∈ Rd ,

(6.1.9)

for a new constant > 0. In turn, (6.1.9) can be re-written as sup x∈Rd

∞

1

n

n (n!)−1 |x| ν |f (x)| ν < ∞.

(6.1.10)

n=0

Hence the sequence of the terms of the series is uniformly bounded, as well as the sequence of the ν-th powers: νn (n!)−ν |x| |f (x)| , n

n = 0, 1, . . . ,

and we obtain |x| |f (x)| −νn (n!)ν , n

x ∈ Rd , n = 0, 1, . . . .

Hence, writing |α| = n and applying (0.3.3): |xα f (x)| −νn (n!)ν C |α| (α!)ν ,

x ∈ Rd , α ∈ N d ,

for a suitable constant C > 0. Therefore (6.1.8) is proved.

232


In the opposite direction, let (6.1.8) be satisfied. Since |x|n ≤ kn |α|=n |xα | for a constant k depending only on the dimension d, cf. (0.3.1) and (0.3.3), then |x|n |f (x)| ≤ kn |xα f (x)| (kC)n (α!)ν , x ∈ Rd , n = 0, 1, . . . . |α|=n

|α|=n

Since α! ≤ n! and the number of the multi-indices α in the sum does not exceed 2d+n−1 , cf. (0.3.9), (0.3.16), we may further estimate with a new constant C > 0 n

|x| |f (x)| C n (n!)ν ,

x ∈ Rd , n = 0, 1, . . . .

Therefore the sequence C −n (n!)−ν |x| |f (x)| , n

n = 0, 1, . . .

is uniformly bounded for x ∈ Rd , as well as the sequence n

1

C − ν (n!)−1 |x| ν |f (x)| ν , n

n = 0, 1, . . . .

1

If we choose =

C− ν 2 1

, we conclude that 1

e|x| ν |f (x)| ν =

n ∞ ∞ n 1 C− ν 1 −1 ν ν (n!) |x| |f (x)| . n n 2 2 n=0 n=0

We obtain (6.1.9), hence (6.1.7). The proof of Proposition 6.1.5 is complete. The following result gives some equivalent definitions of the class

Sνμ (Rd ).

Theorem 6.1.6. Assume μ > 0, ν > 0, μ + ν ≥ 1. For f ∈ S(Rd ) the following conditions are equivalent: (i) f ∈ Sνμ (Rd ). (ii) There exists C > 0 such that |xα f (x)| C |α| (α!)ν , β ξ f (ξ) C |β| (β!)μ ,

x ∈ Rd , α ∈ Nd ; ξ ∈ Rd , β ∈ Nd .

(iii) There exists C > 0 such that

xα f (x) L2 C |α| (α!)ν , β ξ f (ξ) 2 C |β| (β!)μ , L

α ∈ Nd ; β ∈ Nd .

(iv) There exists C > 0 such that

xα f (x) L2 C |α| (α!)ν , β ∂ f (x) 2 C |β| (β!)μ , L

α ∈ Nd ; β ∈ Nd .


233

(v) There exists C > 0 such that α β x ∂ f (x) 2 C |α|+|β| (α!)ν (β!)μ , L (vi) There exists C > 0 such that α β x ∂ f (x) C |α|+|β| (α!)ν (β!)μ ,

α ∈ Nd , β ∈ Nd .

x ∈ Rd , α ∈ Nd , β ∈ Nd .

Proof. The preceding Proposition 6.1.5 shows that (i) is equivalent to (ii). We shall prove (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (vi) ⇒ (ii). First, we assume (ii) and prove 2 −M d (iii). Fixing an integer M > 4 so that (1 + |x| ) 2 < ∞, we have L

2

xα f (x) L2 sup (1 + |x| )M |xα f (x)| ,

α ∈ Nd .

x∈Rd

Write then

(1 + |x|2 )M =

cγ x2γ

|γ|≤M

where the cγ are positive integers, which we can estimate in terms of the fixed integer M . Therefore: sup xα+2γ f (x) , α ∈ Nd .

xα f (x) 2 L

d |γ|≤M x∈R

At this moment we apply the first estimate in (ii) and we obtain

xα f (x) L2 C |α+2γ| ((α + 2γ)!)ν , α ∈ Nd . |γ|≤M

In view of (0.3.6) we have (α + 2γ)! ≤ 2|α+2γ| α!(2γ)!, where |γ| ≤ M . Note also that the number of the terms in the sum in the right-hand side is estimate by 2M +d , cf. (0.3.9), (0.3.15) hence the first inequality in (iii) follows, for a new constant C > 0. Arguing similarly in the ξ variables, we obtain the second inequality. By Plancherel’s formula, we have (iii) ⇔ (iv). Let us prove (iv) ⇒ (v). Integrating by parts and using Leibniz’ formula we have: α β x ∂ f (x)2 2 = (∂ β f, x2α ∂ β f )L2 = (f, ∂ β (x2α ∂ β f ))L2 L β 2α γ! (x2α−γ f, ∂ 2β−γ f )L2 . ≤ γ γ γ≤β,γ≤2α

β 2α

Let us observe that γ γ ≤ 2|β|+2|α| , in view of (0.3.9). Applying then the Cauchy-Schwarz inequality, we obtain α β x ∂ f (x)2 2 ≤ 2|β|+2|α| γ! x2α−γ f (x)L2 ∂ 2β−γ f (x)L2 . (6.1.11) L γ≤β,γ≤2α

234


Using the assumptions (iv), where we may assume C ≥ 1, we have for γ ≤ β, γ ≤ 2α: γ! x2α−γ f (x)L2 ∂ 2β−γ f (x)L2 C 2|α|+2|β| γ!(2α − γ)!ν (2β − γ)!μ C 2|α|+2|β| (2α)!ν (2β)!μ ,

α ∈ Nd , β ∈ Nd .

(6.1.12)

In fact, the last estimate follows by writing γ! ≤ (γ!)μ (γ!)ν

(6.1.13)

in view of the hypothesis μ + ν ≥ 1. We now return to (6.1.11), where we apply (6.1.12) and we observe that the number of the terms in the sum does not exceed 22|α|+|β|+2d , cf. (0.3.15). We then conclude, for a new constant C > 0, α β x ∂ f (x)2 2 C |α|+|β| (2α)!ν (2β)!μ , L

α ∈ N d , β ∈ Nd .

Since (2α)! ≤ 22|α| (α!)2 and (2β)! ≤ 22|β| (β!)2 in view of (0.3.6), we obtain (v). To prove (v) ⇒ (vi) we use the Sobolev embedding theorem (0.2.6). Namely, fixing an integer s > d2 , we have α β x ∂ f (x) xα ∂ β f (x) s H γ α β ∂ (x ∂ f (x))L2 , x ∈ Rd , α ∈ Nd , β ∈ Nd . (6.1.14) |γ|≤s

Applying Leibniz’ rule we may further estimate the right-hand side of (6.1.14) by γ α (6.1.15) δ! xα−δ ∂ β+γ−δ f (x)L2 . δ δ δ≤γ |γ|≤s

δ≤α

From (0.3.9) we have γδ αδ δ! ≤ Cs 2|α| , where the constant Cs does not depend on α. Moreover, the number of the terms of the sums in (6.1.15) can be estimated by an integer independent of α. On the other hand, using the assumption (v) and (0.3.6) we obtain α−δ β+γ−δ x ∂ f (x)L2 C |α|+|β|−2|δ|+|γ| (α − δ)!ν (β + γ − δ)!μ Cs|α|+|β| (α!)ν (β!)μ ,

α ∈ N d , β ∈ Nd ,

(6.1.16)

for a constant Cs depending on s. Combining (6.1.14), (6.1.15) and (6.1.16) we obtain, for a new constant C > 0, α β x ∂ f (x) C |α|+|β| (α!)ν (β!)μ , x ∈ Rd , α ∈ Nd , β ∈ Nd , that is (vi).


235

It remains to prove (vi) ⇒ (ii). To this end, first observe that (vi) with β = 0 gives the first inequality of (ii). On the other hand d β (6.1.17) ξ f (ξ) = ∂ β f (ξ) ≤ (2π)− 2 ∂ β f L1 , ξ ∈ Rd . 2 Fixing an integer M > d2 , so that (1 + |x| )−M < ∞, we obtain β ∂ f

L1

2 M

L1

sup (1 + |x| )

β ∂ f (x) .

(6.1.18)

x∈Rd

From (vi) we have

2 sup (1 + |x| )M ∂ β f (x) C |β| (β!)μ ,

β ∈ Nd ,

(6.1.19)

x∈Rd

and combining (6.1.17), (6.1.18) and (6.1.19) we get the second inequality in (ii). The proof of Theorem 6.1.6 is therefore complete. The assumption μ+ν ≥ 1 in the statement of Theorem 6.1.6 is not restrictive. In fact we shall now prove that for μ+ν < 1 the spaces Sνμ (Rd ) are trivial, i.e., they contain only the zero function. To this end, we first give the following propositions, deserving independent interest. Proposition 6.1.7. Assume f ∈ S(Rd ), μ > 0, ν > 0. Then the estimates (vi) in Theorem 6.1.6 are valid if and only if there exist positive constants C and such that 1 β ∂ f (x) C |β| (β!)μ e−|x| ν , x ∈ Rd , β ∈ Nd . (6.1.20) Hence, when μ + ν ≥ 1 the estimates (6.1.20) give an equivalent definition of Sνμ (Rd ). Proof. Just apply Proposition 6.1.5 to the function C −|β| (β!)−μ ∂ β f (x). An inspection of the proof shows that the bounds are independent of β, hence (6.1.20) is equivalent for a new constant C > 0 to C −|β| (β!)−μ xα ∂ β f (x) C |α| (α!)ν ,

which gives the estimates (vi) in Theorem 6.1.6.

Proposition 6.1.8. Assume f ∈ S(Rd ), 0 < μ < 1, ν > 0. Let (6.1.20) be satisfied for suitable constants C > 0, > 0. Then f extends to an entire analytic function f (x + iy) in Cd , with 1

|f (x + iy)| e−|x| ν +δ|y|

1 1−μ

,

x ∈ R d , y ∈ Rd ,

(6.1.21)

where δ is a suitable positive constant. In the case μ = 1, ν > 0, f extends to an analytic function f (x + iy) in the strip {x + iy ∈ Cd : |y| < T } with 1

|f (x + iy)| e−|x| ν , for a suitable T > 0.

x ∈ Rd , |y| < T,

(6.1.22)

236


Proof. For ν > 0 and 0 < μ ≤ 1 the estimates (6.1.20) imply β ∂ f (x) C |β| β!, x ∈ Rd , β ∈ Nd . This is sufficient to conclude analyticity in Rd and analytic extension to the strip {x + iy ∈ Cd : |y| < T } with T given by C1 . If μ < 1, then f extends further to the entire function defined by the Taylor expansion f (z) =

∂ β f (0) β!

β

z = x + iy ∈ Cd .

zβ ,

(6.1.23)

|∂ β f (0)| C |β| (β!)μ−1 in view of (6.1.20) and the radius of convergence of In fact β! the power series (6.1.23) is ∞. To prove (6.1.21), we now evaluate f (x + iy) by Taylor expanding at the points x ∈ Rd : f (x + iy) =

∂ β f (x) β!

β

(iy)β .

Hence from (6.1.20) we have 1

|f (x + iy)| e−|x| ν

C |β| (β!)μ−1 y β .

(6.1.24)

β

Write the sum in the right-hand side of (6.1.24) as

d C |β| (β!)μ−1 y β = M (|yj |)

(6.1.25)

j=1

β

with M (t) =

∞

C n (n!)μ−1 tn ,

t > 0.

(6.1.26)

n=0

To estimate M (t), set 1

λ = (2Ct) 1−μ , and re-write

t > 0,

1−μ ∞ 1 λn M (t) = . 2n n! n=0

(6.1.27)

(6.1.28)

In view of (0.3.13) we have

λn n!

1−μ

≤ e(1−μ)λ ,

(6.1.29)


237

and applying (6.1.27), (6.1.29) in (6.1.28) we conclude, for some δ > 0, M (t) eδt

1 1−μ

,

t > 0.

Returning to (6.1.25) and then to (6.1.24), we obtain (6.1.21). The same arguments give (6.1.22). Proposition 6.1.8 is therefore proved. Hence, in the case μ + ν ≥ 1, every f ∈ Sνμ (Rd ) extends to the complex domain, as an entire function satisfying (6.1.21) if μ < 1, as a holomorphic function in a strip satisfying (6.1.22) if μ = 1. On the other hand, in the case μ + ν < 1 we may apply the following Liouville-type proposition. Proposition 6.1.9. Assume 0 < λ < θ. Let f (x + iy) be an entire function in Cd satisfying for positive constants and δ: |f (x + iy)| e−|x|

θ

+δ|y|λ

,

x ∈ Rd , y ∈ Rd .

(6.1.30)

Then f ≡ 0. Proof. Consider the entire function f (iz) = f (ix − y). For the product f (z)f (iz) we obtain, from (6.1.30), |f (z)f (iz)| e−|x|

θ

+δ|y|λ −|y|θ +δ|x|λ

e

,

x ∈ Rd , y ∈ Rd .

Since we assume λ < θ, the right-hand side tends to zero for x + iy → ∞. Hence, according to the Liouville theorem, the entire function f (z)f (iz) is identically zero. But then f ≡ 0, and Proposition 6.1.9 is proved. We may now return to the question of the triviality of the classes Sνμ (Rd ), μ + ν < 1. The result can be read as a version of the uncertainty principle of Heisenberg, generically expressed by the statement that a function f (x) and its Fourier transform f(ξ) cannot both be small at infinity. Theorem 6.1.10. Let f ∈ S(Rd ) satisfy 1

|f (x)| e−|x| ν

for x ∈ Rd ,

|f(ξ)| e−|ξ|

1 μ

for ξ ∈ Rd ,

(6.1.31)

for some > 0, μ > 0, ν > 0 and μ + ν < 1. Then f ≡ 0. In other words, according to Definition 6.1.1, the classes Sνμ (Rd ) are trivial if μ + ν < 1. Moreover, if μ + ν < 1, each of the conditions (ii), (iii), (iv), (v), (vi) in Theorem 6.1.6, as well as (6.1.20), implies f ≡ 0. Proof. First, let us go back to the proof of Theorem 6.1.6 and observe that the assumption μ + ν ≥ 1 was only used to prove (iv) ⇒ (v), namely to obtain the key estimate (6.1.13). In the case μ + ν < 1 we may replace (6.1.13) by the obvious identity γ! = (γ!)μ (γ!)ν (γ!)1−(μ+ν) . (6.1.32)

238


Since γ ≤ β in (6.1.12), we can estimate in (6.1.32): (γ!)1−(μ+ν) ≤ (β!)1−(μ+ν) ≤ (2β!) hence γ! ≤ (γ!)μ (γ!)ν (2β!)

1−(μ+ν) 2

1−(μ+ν) 2

,

.

(6.1.33)

Using (6.1.33) instead of (6.1.13), we obtain a weaker version of (6.1.12), namely γ! x2α−γ f (x)L2 ∂ 2β−γ f (x)L2 ∗

C 2|α|+2|β| (2α)!ν (2β)!μ ,

α ∈ N d , β ∈ Nd ,

(6.1.34)

with

1 − (μ + ν) . (6.1.35) 2 Continuing to argue as in the proof of Theorem 6.1.6, for μ + ν < 1 we then obtain (v) for a new couple of indices μ∗ , ν, with μ∗ = μ +

μ + ν < μ∗ + ν < 1.

(6.1.36)

This is sufficient for our purposes. Namely, starting from (6.1.31), that is f ∈ Sνμ (Rd ) with μ + ν < 1, we obtain (ii), (iii), (iv) in Theorem 6.1.6 with same μ, ν, and (v), (vi), hence (6.1.20), for the new couple μ∗ , ν, in view of Proposition 6.1.7. From Proposition 6.1.8 we obtain analytic extension to an entire function with bounds 1 1 1−μ∗ |f (x + iy)| e−|x| ν +δ|y| , x ∈ Rd , y ∈ Rd . 1 1 Finally we apply Proposition 6.1.9 with λ = 1−μ In fact then, from ∗, θ = ν. (6.1.36), or directly from (6.1.35), we have λ < θ in (6.1.30). Hence f ≡ 0. Theorem 6.1.10 is proved.

Finally, we give other equivalent definitions of the classes Sνμ (Rd ), μ + ν ≥ 1, which we shall use in the applications of the next sections. Let us introduce the notation |α| f s,ν; =

xα f (x) H s , (6.1.37) ν (α!) d α∈N

f

{s,μ;δ}

=

δ |β| ∂ β f (x) s , μ H (β!) d

(6.1.38)

β∈N

f

s,μ,ν;δ,

=

α,β∈Nd

|α| δ |β| xα ∂ β f (x) s , ν μ H (α!) (β!)

(6.1.39)

where μ > 0, ν > 0, > 0, δ > 0 and we consider Sobolev norms with s ≥ 0. Proposition 6.1.11. Let μ > 0, ν > 0, μ + ν ≥ 1. For f ∈ S(Rd ) the following conditions are equivalent:


239

(a) f ∈ Sνμ (Rd ). (b) There exist > 0, δ > 0, s ≥ 0 such that

f

(c) There exist > 0, δ > 0, s ≥ 0 such that

s,ν;

f

< ∞ and

s,μ,ν;δ,

f

{s,μ;δ}

< ∞.

< ∞.

Proof. Note that (c) implies (b), since both f s,ν; and f {s,μ;δ} are estimated by f s,μ,ν;δ, . In turn, (b) implies (a). In fact, if f s,ν; < ∞, then each term of the sum in (6.1.37) is uniformly bounded, hence for s ≥ 0,

xα f (x) L2 ≤ xα f (x) H s −|α| (α!)ν ,

α ∈ Nd ,

and similarly from (6.1.38) β ∂ f (x) 2 ≤ ∂ β f (x) s δ −|β| (β!)μ , L H

β ∈ Nd .

Therefore we have proved (iv) in Theorem 6.1.6, which implies (a). Finally, we prove that (a) implies (c). To this end, we may start from (v) in Theorem 6.1.6. Assuming without loss of generality that s ≥ 0 is an integer, we may estimate the Sobolev norm of xα ∂ β f (x) as in (6.1.14), and then apply (6.1.15), (6.1.16). We obtain α β x ∂ f (x) s C |α|+|β| (α!)ν (β!)μ , α ∈ Nd , β ∈ Nd . H At this moment in (6.1.39) we choose = δ = (2C)−1 and then f

s,μ,ν;δ,

=

α β x ∂ f

1 |α|+|β| α,β

2

Hs

C |α|+|β| (α!)ν (β!)μ

< ∞.

The proof of Proposition 6.1.11 is concluded.

The case μ = ν plays an important role in the applications, because of the invariance under Fourier transform. It is convenient for Sμμ (Rd ) to reformulate (v), Theorem 6.1.6, in the following form. Proposition 6.1.12. A function f ∈ S(Rd ) belongs to Sμμ (Rd ), μ ≥ 12 , if and only if there exists a constant C > 0 such that α β x ∂ f (x) 2 C N N N μ f or |α| + |β| ≤ N, N = 0, 1, 2, . . . . (6.1.40) L Proof. From (6.1.40) we have α β x ∂ f (x) 2 C |α|+|β| (|α| + |β|)(|α|+|β|)μ , L

α ∈ Nd , β ∈ Nd .

On the other hand, from (0.3.12), (0.3.5), (0.3.3) we obtain (|α| + |β|)|α|+|β| ≤ (2ed)|α|+|β| α!β!.

(6.1.41)

240


Applying this to the right-hand side of (6.1.41), we obtain (v) in Theorem 6.1.6, hence f ∈ Sμμ (Rd ). In the opposite direction, starting from (v) in Theorem 6.1.6, for |α| + |β| ≤ N we have α β x ∂ f (x) 2 C |α|+|β| (α!β!)μ C N N N μ , N = 0, 1, 2, . . . , L where we assume C ≥ 1 and we apply (0.3.7). Proposition 6.1.12 is proved.

Starting from Sνμ (Rd ) one can define spaces of temperate ultradistributions containing the Schwartz temperate distributions as a subclass. They will have a minor role in the following, and we treat them briefly. To be precise, limiting attention to the case μ = ν > 1, we first define a topology in Sμμ (Rd ). Denote by μ Sμ,C (Rd ), C > 0, the space of all functions f ∈ S(Rd ) such that sup sup C −|α|−|β| (α!β!)−μ xα ∂ β f (x) < ∞.

(6.1.42)

α,β x∈Rd

From (vi) in Theorem 6.1.6 we have Sμμ (Rd ) =

μ Sμ,C (Rd ).

C>0 μ (Rd ) is a Banach space endowed with the norm given For any C > 0, the space Sμ,C by the left-hand side of (6.1.42). Therefore we can consider the space Sμμ (Rd ) as an inductive limit of an increasing sequence of Banach spaces. Equivalent topologies come from equivalent definitions of Sμμ (Rd ), so for example we may refer to (6.1.20) and define as norms, depending on the two parameters C > 0, > 0: 1 μ sup sup C −|β| (β!)−μ e|x| ∂ β f (x) .

β

(6.1.43)

x∈Rd

We shall denote by Sμμ (Rd ) the dual space, i.e., the space of all linear continuous forms on Sμμ (Rd ). To be definite: a linear form u on Sμμ (Rd ) belongs to Sμμ (Rd ) if and only if for every C > 0 we have |u(f )| sup sup C −|α|−|β| (α!β!)−μ xα ∂ β f (x) α,β x∈Rd

for all f ∈ Sμμ (Rd ). Given u ∈ S (Rd ), the restriction of u on Sμμ (Rd ) ⊂ S(Rd ) is an element of Sμμ (Rd ). In this sense, we have S (Rd ) ⊂ Sμμ (Rd ). Under the preceding assumption μ > 1, the theory of the temperate ultradistributions follows closely the theory of Schwartz. We do not give details, but limit ourselves to the following kernel theorem, which is a consequence of the nuclearity of the topology in Sμμ (Rd ), and to the definition of the Fourier transform.

6.2. Γ-Operators and Semilinear Harmonic Oscillators

241

Theorem 6.1.13. There exists an isomorphism between the space L(Sμμ (Rd ), Sμμ (Rd )) of all linear continuous maps from Sμμ (Rd ) to Sμμ (Rd ), and Sμμ (R2d ), which associates to every T ∈ L(Sμμ (Rd ), Sμμ (Rd )) a distribution KT ∈ Sμμ (R2d ) such that T f, g = KT , g ⊗ f for every f, g ∈ Sμμ (Rd ). The temperate ultradistribution KT is called the kernel of T . The Fourier transform, defined as standard by duality u (f ) = u(f), u ∈ μ d μ d μ d Sμ (R ), f ∈ Sμ (R ), extends to an isomorphism of Sμ (R ). We finally observe that the definition of Sνμ (Rd ) extends in a natural way to the cases when μ and ν μ take the values 0 or ∞. Namely S∞ (Rd ), denoted also by S μ (Rd ), is the space of d all f ∈ S(R ) such that, for a suitable C > 0 and all α, α β x ∂ f (x) C |β| (β!)μ , x ∈ Rd , β ∈ Nd , (6.1.44) whereas Sν∞ (Rd ), denoted also by Sν (Rd ), consists of all f ∈ S(Rd ) such that, for all β, α β x ∂ f (x) C |α| (α!)ν , x ∈ Rd , α ∈ Nd . (6.1.45) The class S0μ (Rd ), μ > 1, is defined by replacing (6.1.1) with the assumption that f has compact support. It coincides with Gμ0 (Rd ), the space of the so-called compactly supported Gevrey functions. This is equivalent to saying that f belongs to C0∞ (Rd ) and satisfies for a suitable C > 0 the estimates β ∂ f (x) C |β| (β!)μ , x ∈ Rd , β ∈ Nd . (6.1.46) Evidently, this implies f ≡ 0 if μ ≤ 1. The Gevrey spaces Gμ0 (Rd ), μ > 1, are non-empty and one can construct in them cut-off functions and partitions of unity. Similarly we define Sν0 (Rd ), which we may obtain by Fourier transforming Gν0 (Rd ).

6.2 Γ-Operators and Semilinear Harmonic Oscillators Let us consider the linear partial differential operators with polynomial coefficients in Rd , P = cαβ xβ Dxα (6.2.1) |α|+|β|≤m

where m is a positive integer and cαβ are given constants in C. Consider the Γ-principal symbol: pm (x, ξ) = cαβ xβ ξ α (6.2.2) |α|+|β|=m

and assume Γ-ellipticity pm (x, ξ) = 0

for (x, ξ) = (0, 0).

(6.2.3)

242


We know from the results of Chapter 2 that u ∈ S (Rd ) and P u = f ∈ S(Rd ) imply u ∈ S(Rd ), in particular all the solutions u ∈ S (Rd ) of the homogeneous equation P u = 0 belong to S(Rd ), see Theorem 2.1.6. It is also useful to recall, cf. (2.1.27), that there exists a positive constant C ∗ such that, for every u ∈ S(Rd ), xβ Dα u 2 ≤ C ∗ ( P u 2 + u 2 ). (6.2.4) x L L L |α|+|β|≤m

Theorem 6.2.1. Let P in (6.2.1) satisfy the Γ-ellipticity condition (6.2.3). If u ∈ S (Rd ) is a solution P u = f with f ∈ Sμμ (Rd ), μ ≥ 12 , then also u ∈ Sμμ (Rd ). In 1

particular, P u = 0 and u ∈ S (Rd ) imply u ∈ S 12 (Rd ). 2

Proof. In view of Proposition 6.1.12, it will be sufficient to prove the estimates α β x ∂ u(x) 2 ≤ C N +1 N N μ , |α| + |β| ≤ N, N = 0, 1, 2, . . . , (6.2.5) L for some C > 0. We know that P u = f ∈ Sμμ (Rd ) ⊂ S(Rd ) implies u ∈ S(Rd ). Then, choosing C > 1 sufficiently large, we have (6.2.5) for N ≤ 2m. Arguing by induction, assume that (6.2.5) is valid for all N < M , M > 2m, and prove it for N = M . For α, β satisfying |α| + |β| = M we write xβ Dxα u = xβ−δ xδ Dxα−γ Dxγ u, where we choose γ ≤ α, δ ≤ β so that |γ| + |δ| = M − m and |α − γ| + |β − δ| = m. To be definite, in the case |β| ≥ m, we may take θ ≤ β such that |θ| = m, and define consequently γ = α, δ = β − θ. Otherwise, we have |α| ≥ m, since we are assuming |α| + |β| = M > 2m; we may then take ρ ≤ α with |ρ| = m and define γ = α − ρ, δ = β. We have β α β−δ δ α−γ γ x Dx u 2 ≤ xβ−δ Dxα−γ (xδ Dxγ u) 2 + , D u x D x 2 x x L L L

δ γ ∗ δ γ β−δ δ P (x Dx u)L2 + x Dx uL2 + x x , Dxα−γ Dxγ u 2 ≤C L

≤ C ∗ xδ Dxγ (P u) 2 + P, xδ Dxγ u + xδ Dxγ u 2 L

+ xβ−δ xδ , Dxα−γ Dxγ u

L2

L2

L

,

where we have used (6.2.4). Since P u = f ∈ Sμμ (Rd ) and |γ| + |δ| = M − m, from Proposition 6.1.12, we have, for some constant C1 > 1, δ γ x D (P u) 2 ≤ C M −m+1 (M − m)(M −m)μ ≤ C M +1 M M μ . x 1 1 L Write explicitly, by using (6.2.1) P, xδ Dxγ =

|α|+ ˜ |β˜|≤m

˜ cα˜ β˜ xβ Dxα˜ , xδ Dxγ .


243

˜ ≤ m, we have Therefore, given C2 > 0 such that |cα˜ β˜| ≤ C2 for |˜ α| + |β| β α β˜ α˜ δ γ x Dx u 2 ≤ C ∗ C M +1 M M μ + C ∗ C2 D , x D x x x u 2 1 L L ˜ |α|+ ˜ |β |≤m + C ∗ xδ Dxγ uL2 + xβ−δ xδ , Dxα−γ Dxγ u . (6.2.6) L2

In (6.2.6) let us develop ˜ ˜ ˜ δ+β−σ ˜ δ+β−σ ˜ c1αδσ Dxγ+α−σ − c2βγσ Dxγ+α−σ , xβ Dxα˜ , xδ Dxγ = ˜ x ˜ x ˜ 0=σ≤β σ≤γ

0=σ≤α ˜ σ≤δ

where, with the help of (1.2.18) and Leibniz’ formula, we may compute = c1αδσ ˜

(−i)|σ| α! ˜ δ! , σ! (˜ α − σ)! (δ − σ)!

c2βγσ = ˜

˜ γ! β! (−i)|σ| . σ! (β˜ − σ)! (γ − σ)!

and

2 The constants c1αδσ can be roughly bounded from above by C3 M |σ| for some ˜ ˜ , cβγσ constant C3 depending only on the order m and the dimension d. Therefore, ˜ β˜ α˜ δ γ ˜ + M |σ| xδ+β−σ Dxγ+α−σ u 2 . x Dx , x Dx u 2 ≤ C3 L

˜ 0=σ≤β,σ≤γ

0=σ≤α,σ≤δ ˜

˜ ≤ m imply Observe at this moment that |γ| + |δ| = M − m and |˜ α| + |β|

L

(6.2.7)

|γ + α ˜ − σ| + |δ + β˜ − σ| ≤ M − 2 |σ| . Then, from the inductive hypothesis we have ˜ δ+β−σ ˜ Dxγ+α−σ u ≤ C M −2|σ|+1 (M − 2 |σ|)(M −2|σ|)μ ≤ C M M M μ M −2|σ|μ . x 2 L

Combining this last estimate with (6.2.7) and estimating the number of the terms in the sums by a constant C4 , depending only on m and on the dimension d, by the condition μ ≥ 12 , we conclude that β˜ α˜ δ γ (6.2.8) x Dx , x Dx u 2 ≤ C3 C4 C M M M μ . L

By similar arguments, observing that |α − γ| ≤ m and |β − δ| ≤ m, we may estimate the last term in the right-hand side of (6.2.6) as follows: β−δ δ α−γ γ x , Dx Dx u 2 ≤ C5 C M M M μ . (6.2.9) x L

244


In (6.2.6), we also have δ γ x D u ≤ C M −m+1 (M − m)(M −m)μ ≤ C M M M μ . x

(6.2.10)

Inserting (6.2.8), (6.2.9) and (6.2.10) in (6.2.6) and denoting by C6 the number of ˜ ≤ m, we finally get terms in the sum with |˜ α| + |β| β α x D u 2 ≤ C ∗ C M +1 + C ∗ C2 C3 C4 C6 C M + C ∗ C M + C5 C M M M μ . x 1 L Hence, assuming the inductive constant C sufficiently large, we obtain β α x Dx u 2 ≤ C M +1 M M μ for |α| + |β| ≤ M, L which gives the conclusion, i.e., u ∈ Sμμ (Rd ).

We have in particular from Proposition 6.1.8 that all the solutions u ∈ S (Rd ) of the Γ-elliptic equation P u = 0 extend as entire functions u(x + iy) in Cd , satisfying for suitable > 0, δ > 0: |u(x + iy)| e−|x|

2

+δ|y|2

x ∈ Rd , y ∈ Rd .

,

(6.2.11)

A natural question, in view of the applications to Mathematical Physics, is whether this is valid for the solutions of the corresponding semilinear equations. Let us limit attention to the case of the Schrödinger harmonic oscillator: −Δu + |x|2 u − λu = G[u],

(6.2.12)

where λ ∈ R and the nonlinear term is uk , k ≥ 2, or more generally k ≥ 2,

G[u] = L(uk ), with L=

cαβ xα Dβ ,

cαβ ∈ C.

(6.2.13) (6.2.14)

|α|+|β|≤1

As usual in the nonlinear case, we shall argue on solutions which already possess a certain regularity, expressed here in terms of the standard Sobolev spaces H s (Rd ). Theorem 6.2.2. Let u ∈ H s (Rd ), s > (6.2.14). Then u ∈ S 11 (Rd ).

d 2,

be a solution of (6.2.12), (6.2.13),

2

Therefore, from Proposition 6.1.7 and (6.1.8) we have that u(x) keeps the super-exponential decay of order 2, i.e., for a constant > 0, |u(x)| e−|x|

2

for x ∈ Rd ,

(6.2.15)

whereas the extension to the complex domain u(x + iy) is analytic only in a strip {x + iy ∈ Cd : |y| < T }, for some T > 0, not entire in general.


245

To prove Theorem 6.2.2 we need some preliminary considerations. In the following identities and estimates, assume u ∈ S(Rd ) and when possible extend by density to u ∈ H s (Rd ). We denote for short · s the norm in H s (Rd ). Write H = −Δ + |x|2

(6.2.16)

and consider the inverse H −1 on S(Rd ). Appealing to the results of Section 2.2 we have H −1 ∈ OPΓ−2 (Rd ), hence for p ∈ Nd , q ∈ Nd with |p| + |q| ≤ 2, H −1 ◦ xp Dq ∈ OPΓ0 (Rd ). On the other hand, operators in OPΓ0 (Rd ) are bounded on the standard Sobolev spaces, cf. Theorem 2.1.10, so we have proved the following smoothing property. Proposition 6.2.3. For every p ∈ Nd , q ∈ Nd , with |p| + |q| ≤ 2, we have −1 H ◦ xp Dq u ≤ C u

s s with C depending on s ≥ 0. We then introduce the notation, for μ > 0, ν > 0, α ∈ Nd , β ∈ Nd , > 0, δ > 0: |α| δ |β| α β x ∂x u(x). (6.2.17) [u(α,β) ]μ,ν ,δ (x) = α!ν β!μ We shall refer in the sequel to the norms (6.1.39); with the notation (6.2.17) they can be re-written (α,β) μ,ν u s,μ,ν;,δ = ],δ . (6.2.18) [u s

α∈Nd ,β∈Nd

In view of Proposition 6.1.11, we have that u belongs to Sνμ (Rd ) if and only if u s,μ,ν;,δ < ∞ for some > 0, δ > 0. We also define the partial sums ,δ [u] = SN

(α,β) μ,ν ],δ , [u

N = 0, 1, . . . ,

s

|α|+|β|≤N

(6.2.19)

where the dependence on μ, ν and s is omitted in the notation for shortness. ,δ [u] To prove that u ∈ Sνμ (Rd ), it is sufficient to prove that the sequence SN remains bounded for some > 0, δ > 0, s ≥ 0. Proof of Theorem 6.2.2. First part. Write α,β (α,β) μ,ν H[u(α,β) ]μ,ν ],δ + Mμ,ν,,δ [u] ,δ = [(Hu)

with α,β [u] = Mμ,ν,,δ

|α| δ |β| [H, xα ∂ β ]u. α!ν β!μ

(6.2.20)

(6.2.21)

246


Let u be a solution of (6.2.12), that is Hu = λu + G[u].

(6.2.22)

Inserting this into (6.2.20) we obtain the identity α,β (α,β) μ,ν H[u(α,β) ]μ,ν ],δ + [G[u](α,β) ]μ,ν ,δ = Mμ,ν,,δ [u] + λ[u ,δ .

(6.2.23)

Finally we apply H −1 to both sides, and we have α,β −1 −1 [u(α,β) ]μ,ν Mμ,ν,,δ [u] + λH −1 [u(α,β) ]μ,ν [G[u](α,β) ]μ,ν ,δ = H ,δ + H ,δ .

(6.2.24)

Assuming now u ∈ H s (Rd ) for a fixed s > d2 , we estimate the H s -norm of the lefthand side of (6.2.24) by the sum of the H s -norms of the terms in the right-hand side. From (6.2.19), (6.2.21) we then obtain ,δ ,δ ,δ SN [u] ≤ A,δ N [u] + BN [u] + CN [u],

where

A,δ N [u] =

0 0 such that ,δ A,δ N [u] ≤ C0 ( + δ)SN −1 [u],

for all N = 1, 2, . . . , and all , δ with 0 < < 1, 0 < δ < 1. Lemma 6.2.5. Assume μ ≥ 12 , ν ≥ 12 . We can find C1 > 0 such that ,δ ,δ BN [u] ≤ C1 u s + ( + δ)SN −1 [u] , for all N = 1, 2, . . . , and all , δ with 0 < < 1, 0 < δ < 1.


247

Lemma 6.2.6. Assume μ ≥ 1, ν ≥ 12 . Let k be the integer in the nonlinearity (6.2.13). We can find C2 > 0 such that k ,δ ,δ k [u] ≤ C2 u s + ( + δ)(SN CN −1 [u]) , for all N = 1, 2, . . . , and all , δ with 0 < < 1, 0 < δ < 1. For sake of simplicity, we limit the proof of the previous lemmata to the one-dimensional case, the generalization to the d-dimensional case involving only notational complications. So in the sequel assume α ∈ N, β ∈ N, x ∈ R, etc.; also d . write here D for dx Proof of Lemma 6.2.4. In view of Proposition 6.2.3, the estimates are easy for a fixed N , by taking C0 sufficiently large, depending on N . Then, it will not be restrictive to assume in the following N > 8. We first compute − [H, xα ∂ β ] = [D2 , xα Dβ ] − [x2 , xα Dβ ] = α(α − 1)xα−2 Dβ + 2αxα−1 Dβ+1 + β(β − 1)xα Dβ−2 + 2βxα+1 Dβ−1 . (6.2.29) The terms with negative exponents do not appear, if α ≤ 1 or β ≤ 1. We insert (6.2.29) in (6.2.26) and we estimate separately the four resulting terms. Consider initially α δ β −1 A1N = ◦ xα−2 Dβ us . (6.2.30) ν μ α(α − 1) H α! β! α+β≤N

Fix first attention on the terms of the sum with α ≥ 4. In this case we may write H −1 ◦ xα−2 Dβ = H −1 ◦ x2 xα−4 Dβ and applying Proposition 6.2.3 we have −1 H ◦ xα−2 Dβ u ≤ C xα−4 Dβ u . s s Hence for < 1, α−4 β −1 α−4 δ β α δ β ◦ xα−2 Dβ us ≤ C0 D us ν μ x ν μ α(α − 1) H α! β! (α − 4)! β!

(6.2.31)

where we choose C0 so that C

αν (α

α(α − 1) ≤ C0 . − 1)ν (α − 2)ν (α − 3)ν

(6.2.32)

Note that the left-hand side of (6.2.32) is bounded with respect to α since ν ≥ 12 . The sum for α + β ≤ N of the terms in the right-hand side of (6.2.31) is estimated ,δ ,δ by C0 SN −4 [u] ≤ C0 SN −1 [u]. It remains to consider the terms in (6.2.30) with α = 2 and α = 3.

248

Chapter 6. Exponential Decay and Holomorphic Extension of Solutions If α = 2, we may assume β ≥ 2 and write H −1 ◦ Dβ = H −1 ◦ D2 Dβ−2 .

Applying Proposition 6.2.3 we have, for < 1, δ < 1, β−2 β−2 2 δ β −1 ◦ Dβ u ≤ C0 δ us μ D μ H ν s 2 β! (β − 2)!

(6.2.33)

for a suitably large C0 . Since 2 + β ≤ N , the right-hand side of (6.2.33) is again a ,δ term of C0 SN −4 [u]. Similarly we argue for α = 3. Summing up, we have proved ,δ that A1N in (6.2.30) can be estimated by C0 SN −1 [u], if C0 is large enough. Consider now A2N =

α+β≤N

−1 α δ β ◦ xα−1 Dβ+1 us . ν μ 2α H α! β!

(6.2.34)

Fixing first attention on the terms of the sum with α ≥ 4, we write xα−1 Dβ+1 = xD ◦ xα−2 Dβ − (α − 2)xα−2 Dβ

(6.2.35)

and estimate consequently the right-hand side (6.2.34). By Proposition 6.2.3 we have, for < 1, α−2 β −1 α−2 δ β α δ β x ◦ xD ◦ xα−2 Dβ us ≤ C0 D us , ν μ 2α H ν μ α! β! (α − 2)! β!

(6.2.36)

where now we choose C0 such that 2C

α ≤ C0 . αν (α − 1)ν

We recognize in the right-hand side of (6.2.36) one of the terms of the sum ,δ C0 SN −2 [u]. On the other hand we have from the second term in the right-hand side of (6.2.35), −1 α δ β ◦ xα−2 Dβ us ν μ 2α(α − 2) H α! β! which we can treat exactly as (6.2.30). Discussing as before the cases α = 1, α = 2, α = 3, we conclude ,δ A2N ≤ C0 SN −1 [u]. Consider then A3N =

α+β≤N

−1 α δ β ◦ xα Dβ−2 us . ν μ β(β − 1) H α! β!

(6.2.37)


249

Assume α ≥ 4, β ≥ 4. It is convenient to write xα Dβ−2 = D2 ◦ xα Dβ−4 − 2αD ◦ xα−1 Dβ−4 + α(α − 1)xα−2 Dβ−4 .

(6.2.38)

Inserting in (6.2.37) we get three terms which we estimate as follows: α δ β β(β − 1) H −1 ◦ D2 ◦ xα Dβ−4 us α!ν β!μ α β−4 α δ β−4 x D ≤ δC0 ν us α! (β − 4)!μ

(6.2.39)

where we may choose C0 so that 2C

β(β − 1) ≤ C0 . β μ (β − 1)μ (β − 2)μ (β − 3)μ

We use here the assumption μ ≥ 12 . As before, C is the constant in Proposition 6.2.3 and δ < 1. Moreover: −1 α δ β ◦ D ◦ xα−1 Dβ−4 us ν μ 2αβ(β − 1) H α! β! α−2 β−4 α−2 δ β−4 x ≤ δC0 D us ν μ (α − 2)! (β − 4)!

(6.2.40)

where now we require 2C

αν (α

−

αβ(β − 1) ≤ C0 . − 1)μ (β − 2)μ (β − 3)μ

1)ν β μ (β

Similarly −1 α δ β ◦ xα−2 Dβ−4 us ν μ α(α − 1)β(β − 1) H α! β! α−4 β−4 α−4 δ β−4 x ≤ δC0 D us . ν μ (α − 4)! (β − 4)!

(6.2.41)

In the right-hand side of (6.2.39), (6.2.40), (6.2.41) we recognize terms from ,δ δC0 SN −4 [u]. We leave to the reader the discussion of the case when α < 4 or β < 4 in (6.2.37), as well as the proof of the estimate A4N =

α+β≤N

−1 α δ β ◦ xα+1 Dβ−1 us ν μ 2β H α! β!

,δ ≤ δC0 SN −1 [u].

We then conclude ,δ 1 2 3 4 A,δ N [u] ≤ AN + AN + AN + AN ≤ C0 ( + δ)SN −1 [u],

for a new constant C0 . Lemma 6.2.4 is proved.

250


Proof of Lemma 6.2.5. Proposition 6.2.3 yields, for some C1 > 0, −1 (α,β) μ,ν ],δ ) ≤ C1 ν [u(α−1,β) ]μ,ν λH ([u ,δ α s s

(6.2.42)

if α ≥ 1 and otherwise δ −1 (0,β) μ,ν ],δ ) ≤ C1 μ [u(0,β−1) ]μ,ν λH ([u ,δ β s s

(6.2.43)

if β ≥ 1. The sum of the terms in the right-hand side of (6.2.42), (6.2.43) is estimated by ,δ C1 ( + δ)SN −1 [u].

Taking into account the case α = 0, β = 0 we obtain Lemma 6.2.5.

In fact, Proof of Lemma 6.2.6. Here we use the assumption u ∈ H (R ), s > s d s d we know from Schauder’s estimates that for u ∈ H (R ), v ∈ H (R ), s > d2 , we have (6.2.44)

uv s ≤ σ u s v s s

d 2.

d

for a constant σ ≥ 1 depending on s. Hence uk ∈ H s (Rd ) in (6.2.13). Therefore, ,δ [u] in (6.2.28) makes in view of (6.2.14) and Proposition 6.2.3, the definition of CN sense for N = 0. Let us assume N ≥ 1 and prove Lemma 6.2.6. As before, we limit now attention to the one-dimensional case. We begin by some inequalities for the nonlinear terms. Define: δ EN [u] =

δγ

Dγ u s . γ!μ

(6.2.45)

γ≤N

,δ δ [u] ≤ SN [u]. We have for μ ≥ 1, Obviously EN δ δ EN [uk ] ≤ σ k−1 (EN [u])k ,

(6.2.46)

where σ is the constant in (6.2.44). In fact, by Leibniz’ rule and Schauder estimates: γ k D u ≤ σk−1 s

γ1 +...+γk =γ

γ!

Dγ1 u s · · · Dγk u s . γ1 ! · · · γk !

(6.2.47)

Hence we may write δγ Dγ uk ≤ σk−1 s γ!μ γ

1 +...+γk =γ

λ(γ) γ1 ,...,γk

δ γ1

∂ γ1 u s γ1 !μ

...

δ γk

∂ γk u s γk !μ

,

(6.2.48)

6.2. Γ-Operators and Semilinear Harmonic Oscillators where

λ(γ) γ1 ,...,γk

=

γ! γ1 ! · · · γk !

251

1−μ ≤1

(6.2.49)

provided μ ≥ 1. From (6.2.48), (6.2.49) we obtain immediately (6.2.46). The same arguments give ,δ k ,δ [u ] ≤ σ k−1 (SN [u])k . (6.2.50) SN In fact, it will be sufficient to write α β k x D u ≤ σ k−1 s

β1 +...+βk =β

α β β β! x D 1 u D 2 u · · · Dβk u s s s β1 ! · · · βk !

and proceed as before, to obtain ,δ k ,δ δ [u ] ≤ σ k−1 SN [u](EN [u])k−1 , SN

which gives in particular (6.2.50). Note that we do not need to strengthen the assumption ν ≥ 12 , for the validity of (6.2.50). ,δ We can now estimate CN [u] in (6.2.28). Let us first consider the case when G[u] = uk , k ≥ 2. We have δβ −1 k (α,β) H −1 Dβ uk H [u ](,δ) = H −1 uk s + μ s β! s β≤N

α+β≤N

β=0

+

α+β≤N α=0

δ H −1 (xα Dβ uk ) . s α!ν β!μ α β

(6.2.51)

We estimate separately the three terms in the right-hand side. From Proposition 6.2.3 and (6.2.44) we have −1 k H u ≤ C uk ≤ Cσ k−1 u k . (6.2.52) s s s On the other hand from Proposition 6.2.3 and (6.2.46) δ β−1 δβ H −1 Dβ uk ≤ C δ Dβ−1 uk μ s μ μ s β! β (β − 1)! β≤N β≤N β=0

β=0

δ k k−1 δ k ≤ CδEN δ(EN −1 [u ] ≤ Cσ −1 [u]) .

(6.2.53)

Similarly from (6.2.50) α+β≤N α=0

α−1 β k α δ β α−1 δ β H −1 (xα Dβ uk ) ≤ C x D u s ν μ ν ν μ s α! β! α (α − 1)! β! α+β≤N α=0

≤

,δ k C SN −1 [u ]

,δ k ≤ Cσ k−1 (SN −1 [u]) . (6.2.54)

252


Applying (6.2.52), (6.2.53), (6.2.54) into (6.2.51), we conclude, for a suitable constant C2 > 0, −1 k (α,β) k ,δ (6.2.55) H [u ](,δ) ≤ C2 ( u s + ( + δ)(SN −1 [u])k ). s

α+β≤N

Let us now consider the case G[u] = xuk . It will be not restrictive to assume N > 2. Writing Dβ (xuk ) = xDβ uk + βDβ−1 uk , we have −1 k (α,β) H [xu ](,δ) ≤ H −1 (xuk )s + α+β≤N

s

0 2. (iii) λ > 0, that is 0 < u0 < u ˜0 . In this case, since √ 0 < λ < λ + 2k − 2 Erfc

?

k−1 x 2

256

Chapter 6. Exponential Decay and Holomorphic Extension of Solutions in view of (6.2.66), the solution is well defined analytic in R and 1

0 < u(x) < λ 1−k e−

x2 2

.

Similar estimates are valid for u (x), u (x). Hence we have u ∈ H 2 (R), therefore Theorem 6.2.2 applies and gives the more precise information u ∈ S 11 (R). Then, the extension u(z) to the complex domain is analytic in a strip 2 {z ∈ C : |Im z| < T }, T > 0, but it is not an entire function, as evident from (6.2.68). In fact, u(z) presents a singularity at z0 ∈ C when

? √ k−1 (6.2.69) z0 = 0, λ > 0. λ + 2k − 2 Erfc 2 The explicit discussion of (6.2.69) is not easy, but we can anyhow appeal to the great Picard theorem in the complex domain, which grants the existence of a solution z0 of (6.2.69) for all λ ∈ C, but for a possible exceptional value. Proposition 6.2.7 is therefore proved.

6.3 G-Pseudo-Differential Operators on Sνμ (Rd) In this section and in the sequel of the chapter we shall fix attention on G-elliptic equations. Basic examples are linear partial differential operators with polynomial coefficients in Rd of the form cαβ xβ Dxα (6.3.1) P = |α|≤m,|β|≤n

with m > 0, n ≥ 0, satisfying the G-ellipticity condition cαβ xβ ξ α ≥ Cξm xn for |x| + |ξ| ≥ R

(6.3.2)

|α|≤m,|β|≤n

for constants C > 0, R > 0. We know from the results of Chapter 3 that all the solutions u ∈ S (Rd ) of the homogeneous equation P u = 0 belong to S(Rd ). To obtain a more precise result in terms of Sνμ (Rd ), in this section we begin to present a class of analytic-type G-pseudo-differential operators. For P in (6.3.1), (6.3.2), the corresponding calculus will give that the solutions u ∈ S (Rd ) of P u = 0 belong to Sθθ (Rd ) for any θ > 1. Let μ, ν be real numbers such that μ ≥ 1, ν ≥ 1, and let m ∈ R, n ∈ R. Let θ ≥ max{μ, ν}. d Definition 6.3.1. For every C > 0, we denote by AGm,n μν (R ; C) the Banach space of all functions p(x, ξ) ∈ C ∞ (R2d ) such that sup C −|α|−|β| (α!)−μ (β!)−ν ξ−m+|α| x−n+|β| Dξα Dxβ p(x, ξ) < ∞, sup α,β∈Nd (x,ξ)∈R2d

(6.3.3)

6.3. G-Pseudo-Differential Operators on Sνμ (Rd )

257

endowed with the norm · C given by the left-hand side of (6.3.3). We set d d AGm,n AGm,n μν (R ) = lim μν (R ; C) −→ C→+∞

with the topology of inductive limit of an increasing sequence of Banach spaces. d Given a symbol p ∈ AGm,n μν (R ), we can consider the associated pseudodifferential operator defined with the standard left quantization by u(ξ) dξ, ¯ u ∈ Sθθ (Rd ) (6.3.4) P u(x) = p(x, D)u(x) = eixξ p(x, ξ) d where dξ ¯ = (2π)− 2 dξ. We denote by OPAGm,n μν (R ) the space of all operators of m,n the form (6.3.4) defined by a symbol p ∈ AGμν (Rd ). We set d

OPAGμν (Rd ) =

d OPAGm,n μν (R ).

(m,n)∈R2

This is a subclass of the G-pseudo-differential operators in Chapter 3 on S(Rd ), S (Rd ). Taking advantage of the estimates (6.3.3) we are able to prove continuity on Sθθ (Rd ). d Theorem 6.3.2. Let p ∈ AGm,n μν (R ) and let θ be a real number such that θ ≥ max{μ, ν}. Then, the operator P defined by (6.3.4) is a linear continuous operator from Sθθ (Rd ) to Sθθ (Rd ) and, when θ > 1, it extends to a linear continuous map from Sθθ (Rd ) to Sθθ (Rd ).

Proof. Let u ∈ Sθθ (Rd ). Since F(Sθθ (Rd )) = Sθθ (Rd ), we may start with u in a bounded subset F of the Banach space defined by the norm 1/θ

sup sup A−|β| (β!)θ ea|ξ| β

ξ∈Rd

|∂ β u (ξ)|

(6.3.5)

for some A > 0, a > 0, cf. (6.1.43). It is sufficient to show that there exist positive constants A1 , B1 , C1 such that, for every α, β ∈ Nd , |α| |β| sup xα Dxβ P u(x) ≤ C1 A1 B1 (α!β!)θ

(6.3.6)

x∈Rd

for all u ∈ F, with A1 , B1 , C1 independent of u ∈ F. We have, for every N ∈ N, β α β α eixξ ξ β Dxβ−β p(x, ξ) x Dx P u(x) = x u(ξ) dξ ¯ β β ≤β β α −2N ixξ N β β−β e ξ (1 − Δ ) D p(x, ξ) u (ξ) dξ. ¯ = x x ξ x β β ≤β

258


By (6.3.3), (6.3.5), we easily obtain the estimate: α β x D P u(x) ≤ C0 B |β|+2N (2N !)θ x|α|+n−2N x 0 β 1 θ ν m −a|ξ| θ ξ × !) (β − β !) e dξ ¯ (β β β ≤β

for some B0 , C0 independent of u ∈ F. Choosing N = min{r ∈ N : 2r ≥ |α| + n}, we obtain that there exist A1 , B1 , C1 > 0 such that (6.3.6) holds for all u ∈ F. This concludes the first part of the proof. To prove the second part, we observe that, for u, v ∈ Sθθ (Rd ), P u(x)v(x)dx = u (ξ)pv (ξ) dξ

where pv (x, ξ) =

eixξ p(x, ξ)v(x) dx. ¯

By the same argument of the first part of the proof, the map v → pv is linear and continuous from Sθθ (Rd ) to itself. Then, we can define, for u ∈ Sθθ (Rd ), P u(v) = u (pv ),

v ∈ Sθθ (Rd ).

This map is linear and continuous from Sθθ (Rd ) to itself and it extends P.

From now on we shall assume μ > 1, ν > 1, hence θ > 1. By Theorems 6.1.13 and 6.3.2, we can associate to P a kernel KP ∈ Sθθ (R2d ) given as standard by (6.3.7) KP (x, y) = (2π)−d ei(x−y)ξ p(x, ξ) dξ, where (6.3.7) is understood in the sense of the Fourier transform of distributions, cf. (1.2.2). We can prove the following result of regularity for the kernel (6.3.7). d Theorem 6.3.3. Let p ∈ AGm,n μν (R ). For k > 0, define

Ωk = {(x, y) ∈ R2d , |x − y| > kx}. Then the kernel KP defined by (6.3.7) is in C ∞ (Ωk ) and there exist positive constants C, a depending on k such that β γ Dx Dy KP (x, y) ≤ C |β|+|γ|+1 (β!γ!)θ exp −a(|x| θ1 + |y| θ1 ) (6.3.8) for every (x, y) ∈ Ωk and for every β, γ ∈ Nd .


259

Lemma 6.3.4. For any given R > 1, we may find a sequence ψN (ξ) ∈ C0∞ (Rd ), ∞ ψN = 1 in Rd , N = 0, 1, 2, . . . , such that N =0

supp ψ0 ⊂ {ξ ∈ Rd : ξ ≤ 3R}, supp ψN ⊂ {ξ ∈ Rd : 2RN θ ≤ ξ ≤ 3R(N + 1)θ }, and

N = 1, 2, . . . ,

α , Dξ ψN (ξ) ≤ C |α|+1 (α!)θ R sup{N θ , 1} −|α|

for every α ∈ Nd and for every ξ ∈ Rd . Proof. Let φ ∈ C0∞ (Rd ) such that φ(ξ) = 1 if ξ ≤ 2, φ(ξ) = 0 if ξ ≥ 3. Assume further φ ∈ Gθ0 (Rd ), i.e., according to (6.1.46): α Dξ φ(ξ) ≤ C |α|+1 (α!)θ for all α ∈ Nd and for all ξ ∈ Rd . We may then define ξ , ψ0 (ξ) = φ R ψN (ξ) = φ

ξ R(N + 1)θ

−φ

ξ RN θ

,

N ≥ 1.

Proof of Theorem 6.3.3. We can assume without loss of generality that p is in d AG0,0 μν (R ). Let us consider a sequence {ψN }N ≥0 as in Lemma 6.3.4. We have, for θ u, v ∈ Sθ (Rd ), ∞ KN , v ⊗ u KP , v ⊗ u = N =0

with KN (x, y) = (2π)−d

so we may decompose KP =

ei(x−y)ξ p(x, ξ)ψN (ξ) dξ ∞

KN .

N =0

Let k > 0 and (x, y) ∈ Ωk . Let h ∈ {1, . . . , d} such that |xh − yh | ≥ for every α, γ ∈ Nd , Dxα Dyγ KN (x, y) = (−1)|γ| (2π)−d

α β≤α

β

k x. d

Then,

ei(x−y)ξ ξ β+γ ψN (ξ)Dxα−β p(x, ξ) dξ

260

Chapter 6. Exponential Decay and Holomorphic Extension of Solutions = (−1)

|γ|+N

(2π)

α

−d

β

β≤α

×

(xh − yh )−N

, ei(x−y)ξ DξNh ξ β+γ ψN (ξ)Dxα−β p(x, ξ) dξ.

Now, given ζ > 0, we consider the operator ∞

ζj 1 (1 − Δξ )j , m2θ,ζ (x − y) j=0 (j!)2θ

L= where

m2θ,ζ (x − y) =

∞

ζj x − y2j . (j!)2θ

j=0

In view of the fact that Lei(x−y)ξ = ei(x−y)ξ , we can integrate by parts obtaining that (xh − yh )−N m2θ,ζ (x − y)

∞ α ζj ei(x−y)ξ λhjN αβγ (x, ξ)dξ × β j=0 (j!)2θ

Dxα Dyγ KN (x, y) =(−1)|γ|+N (2π)−d

β≤α

with

, λhjN αβγ (x, ξ) = (1 − Δξ )j DξNh ξ β+γ ψN (ξ)Dxα−β p(x, ξ) .

(6.3.9)

Let eh be the h-th vector of the canonical basis of Rd and βh = β, eh , γh = γ, eh . Developing in the right-hand side of (6.3.9) we obtain that

λhjN αβγ (x, ξ) =

(−i)N1

N1 +N2 +N3 =N N1 ≤βh +γh

(βh + γh )! N! · N1 !N2 !N3 ! (βh + γh − N1 )!

× (1 − Δξ )j ξ β+γ−N1 eh DξNh2 ψN (ξ)DξNh3 Dxα−β p(x, ξ) . Hence |λhjN αβγ (x, ξ)| ≤

N1 +N2 +N3 =N N1 ≤βh +γh

N! (βh + γh )! |α−β|+N2 +N3 +1 · C N1 !N2 !N3 ! (βh + γh − N1 )! 1 ν

× (N2 !) (N3 !) [(α − β)!] θ

μ

C2j (j!)2θ

1 RN θ

N2

ξ|β|+|γ|−N1 −N3 .

We observe that, on the support of ψN , 2RN θ ≤ ξ ≤ 3R(N + 1)θ . Thus, from standard factorial inequalities, since θ ≥ max{μ, ν}, it follows that N C3 |α|+|γ|+1 (α!γ!)θ C2j (j!)2θ |λhjN αβγ (x, ξ)| ≤ C1 R


261

with C3 independent of R. Observe now that for every c > 1 there exist positive constants , c such that, for t > 0, exp [c t] ≤

∞ j c t j=0

j!

.

(6.3.10) 1

1

Admitting (6.3.10) for a moment, and setting there c = 2θ, t = ζ 2θ x − y θ , we have that 1 1 |m2θ,ζ (x − y)| ≥ exp[c ζ 2θ |x − y| θ ]. From these estimates, choosing ζ < C2−1 , we deduce that α γ Dx Dy KN (x, y) ≤ C |α|+|γ|+1 (α!γ!)θ 1

C4 R

N

1

1

exp[−c ζ 2θ |x − y| θ ]

with C4 = C4 (k) independent of R. Choosing R sufficiently large and observing that |x − y| ≥ c (x + y) on Ωk , we obtain the estimates (6.3.8). Let us now return to the proof of (6.3.10). It will be sufficient to argue for large t, say t > 1. Writing s = 21−c t, and denoting by k = k(t) the smallest integer k > 0 such that (2t)k ≤ k!, we split sj sj + . j! j!

es =

0≤j 1, Qt = {(x, ξ) ∈ R2d : x < t, ξ < t}, Qet = R2d \ Qt .

d Definition 6.3.6. We denote by F AGm,n μν (R ) the space of all the formal sums ∞ 2d pj (x, ξ) such that pj (x, ξ) ∈ C (R ) for all j ≥ 0 and there exist B, C > 0 j≥0

such that sup sup

C −|α|−|β|−2j (α!)−μ (β!)−ν (j!)−μ−ν+1

sup

j≥0 α,β∈Nd (x,ξ)∈Qe

Bj μ+ν−1

× ξ

−m+|α|+j

x−n+|β|+j Dξα Dxβ pj (x, ξ) < ∞.

(6.3.12)

d We observe that every symbol p ∈ AGm,n μν (R ) can be identified with an m,n d element of F AGμν (R ), by setting p0 = p and pj = 0 for all j ≥ 1. d pj , pj in F AGm,n Definition 6.3.7. We say that two sums μν (R ) are equivaj≥0

lent, and we write

pj ∼

j≥0

j≥0

pj ,

j≥0

if there exist constants B, C > 0 such that sup sup

C −|α|−|β|−2N (α!)−μ (β!)−ν (N !)−μ−ν+1

sup

N ∈N α,β∈Nd (x,ξ)∈Qe

BN μ+ν−1

× ξ−m+|α|+N x−n+|β|+N Dξα Dxβ (pj − pj ) < ∞. j 0 :

ϕ0 (x, ξ) ≡ 1 on R2d ,

x ξ , , ϕj (x, ξ) = ϕ Rj μ+ν−1 Rj μ+ν−1

j ≥ 1.


263

We want to prove that if R is sufficiently large, then p(x, ξ) = ϕj (x, ξ)pj (x, ξ) j≥0 d is in AGm,n μν (R ) and p ∼

d pj in F AGm,n μν (R ).

j≥0

Consider Dξα Dxβ p(x, ξ)

=

α β j≥0

δ

γ

γ≤α δ≤β

Dξα−γ Dxβ−δ pj (x, ξ)Dξγ Dxδ ϕj (x, ξ).

If R ≥ B, where B is the same constant of Definition 6.3.6, we can apply the estimates (6.3.12) and obtain α β Dξ Dx p(x, ξ) ≤ C |α|+|β|+1 α!β!ξm−|α| xn−|β| Hjαβ (x, ξ), j≥0

where Hjαβ (x, ξ) =

(α − γ)!μ−1 (β − δ)!ν−1 C 2j−|γ|−|δ| (j!)μ+ν−1 γ!δ! γ≤α δ≤β

×ξ|γ|−j x|δ|−j Dξγ Dxδ ϕj (x, ξ) .

The condition (6.3.13) implies that Hjαβ (x, ξ) ≤ C |α|+|β|+1 (α!)μ−1 (β!)ν−1

C1 R

j ,

large, we obtain that p is in with C1 independent of R. Choosing R sufficiently d d (R ). It remains to prove that p ∼ p in F AGm,n AGm,n j μν μν (R ). Let N be a j≥0

positive integer. We observe that, for (x, ξ) ∈ Qe3RN μ+ν−1 , pj (x, ξ) = pj (x, ξ)ϕj (x, ξ) p(x, ξ) − j 1, ν > 1 and let (m, n, l) ∈ R3 . Definition 6.3.13. For C > 0, we shall denote by Πm,n,l (Rd ; C) the Banach space μν ∞ 3d of all functions a(x, y, ξ) ∈ C (R ) such that sup

sup

α,β,γ∈Nd (x,y,ξ)∈R3d

C −|α|−|β|−|γ| (α!)−μ (β!γ!)−ν × ξ−m+|α| x−n+|β| y−l+|γ| Dξα Dxβ Dyγ a(x, y, ξ) < ∞.

We set Πm,n,l (Rd ) = lim Πm,n,l (Rd ; C). μν μν −→ C→+∞

It is immediate to verify the following relations: i) if a(x, y, ξ) ∈ Πm,n,l (Rd ) for some (m, n, l) ∈ R3d , then the function (x, ξ) → μν a(x, x, ξ) belongs to AGm,n+l (Rd ); μν d 2 m,n,0 ii) if p ∈ AGm,n (Rd ) and μν (R ) for some (m, n) ∈ R , then p(x, ξ) ∈ Πμν m,0,n d p(y, ξ) ∈ Πμν (R ).

266


Given a ∈ Πm,n,l (Rd ), we can associate to a the pseudo-differential operator deμν fined by Au(x) = ei(x−y)ξ a(x, y, ξ)u(y) dy ¯ dξ, ¯ u ∈ Sθθ (Rd ), (6.3.19) with the standard meaning of iterated integrals. Theorem 6.3.2 and Theorem 6.3.3 can be easily rephrased for operators (6.3.19), the kernel of A being now KA (x, y) = (2π)−d ei(x−y)ξ a(x, y, ξ) dξ. In order to prove Proposition 6.3.11 and Theorem 6.3.12, we give first the following result. (Rd ), Theorem 6.3.14. Let A be an operator defined by an amplitude a ∈ Πm,n,l μν 3 (m, n, l) ∈ R . Then we may write A = P +R, where R is a θ−regularizing operator (Rd ) with p ∼ pj , where for θ ≥ μ + ν − 1 and P = p(x, D) ∈ OPAGm,n+l μν j≥0

pj (x, ξ) =

(α!)−1 ∂ξα Dyα a(x, y, ξ)|y=x .

(6.3.20)

|α|=j

Proof. Let χ ∈ C ∞ (R2d ) such that χ(x, y) = and

1 if |x − y| ≤ 14 x, 0 if |x − y| ≥ 12 x

β γ D D χ(x, y) ≤ C |β|+|γ|+1 (β!γ!)ν x

y

for all β, γ ∈ Nd and (x, y) ∈ R2d . We may decompose a as the sum of two elements (Rd ) writing of Πm,n,l μν a(x, y, ξ) = χ(x, y)a(x, y, ξ) + (1 − χ(x, y))a(x, y, ξ). Furthermore, it follows from Theorem 6.3.3 that (1 − χ(x, y))a(x, y, ξ) defines a θ−regularizing operator. Hence, possibly perturbing A with a θ−regularizing operator, we can assume that a(x, y, ξ) is supported on R2d \ Ω 12 × Rd , where Ω 12 is defined as in Theorem 6.3.3. pj defined by (6.3.20) belongs to F AGm,n+l (R2d ). It is trivial to verify that μν j≥0

As in the proof of Theorem 6.3.8 we can find a sequence ϕj ∈ C ∞ (R2d ) depending on a parameter R such that p(x, ξ) = ϕj (x, ξ)pj (x, ξ) j≥0

defines an element of AGm,n+l (Rd ) for R large and p ∼ μν

pj in F AGm,n+l (Rd ), μν

j≥0

cf. (6.3.13) and subsequent formulas. Let P = p(x, D). To prove the theorem it is sufficient to show that the kernel K(x, y) of A − P is in Sθθ (R2d ).


267

We can write a(x, y, ξ) − p(x, ξ) = (1 − ϕ0 (x, ξ))a(x, y, ξ) ∞ + (ϕN − ϕN +1 )(x, ξ) a(x, y, ξ) − pj (x, ξ) . N =0

j≤N

Consequently, we have K(x, y) = K(x, y) +

∞

KN (x, y),

N =0

where −d

ei(x−y)ξ (1 − ϕ0 (x, ξ))a(x, y, ξ) dξ, K(x, y) = (2π) −d pj (x, ξ) dξ. KN (x, y) = (2π) ei(x−y)ξ (ϕN − ϕN +1 )(x, ξ) a(x, y, ξ) − j≤N

A power expansion in the second argument gives, for N = 1, 2, . . ., (α!)−1 (y − x)α ∂yα a(x, x, ξ) + (α!)−1 (y − x)α wα (x, y, ξ) a(x, y, ξ) = |α|≤N

|α|=N +1

with

wα (x, y, ξ) = (N + 1)

1

∂yα a(x, x + t(y − x), ξ)(1 − t)N dt.

0

In view of our definition of the pj (x, ξ)’s, integrating by parts we obtain that

KN (x, y) =WN (x, y) + (2π)−d

1≤|α|≤N 0=β≤α

×

1 β!(α − β)!

ei(x−y)ξ Dξβ (ϕN − ϕN +1 )(x, ξ)(Dξα−β ∂yα a)(x, x, ξ)dξ,

where WN (x, y) = (2π)−d ×

|α|=N +1 β≤α

1 β!(α − β)!

ei(x−y)ξ Dξβ (ϕN − ϕN +1 )(x, ξ)Dξα−β wα (x, y, ξ)dξ

for all N = 1, 2, . . . . Using an absolute convergence argument, we may re-arrange the sums under the integral sign. We also observe that β Dξ (ϕN − ϕN +1 )(x, ξ) = Dξβ ϕ|α| (x, ξ). N ≥|α|

268


Then we have H =K+

Iα +

∞

WN ,

N =0

α=0

where

Iα (x, y) = (2π)−d

0=β≤α

1 β!(α − β)!

ei(x−y)ξ Dξβ ϕ|α| (x, ξ)Dξα−β ∂yα a(x, x, ξ)dξ

and we may write W0 (x, y) for K0 (x, y). To conclude the proof, we want to show ∞ Iα , WN ∈ Sθθ (R2d ). First of all, we have to estimate the derivatives that K, α=0

N =0

of K for (x, ξ) ∈ supp(1 − ϕ0 (x, ξ)), i.e., for x ≤ R, ξ ≤ R. We have k h δ γ x y Dx Dy K(x, y) = (2π)−d xk y h × (−1)|γ1 |

γ1 +γ2 =γ δ1 +δ2 +δ3 =δ

γ1 +γ2 =γ δ1 +δ2 +δ3 =δ

γ!δ! C |γ2 |+|δ2 |+|δ3 | (γ2 !δ2 !δ3 !)ν x − y|γ2 +δ2 | γ1 !γ2 !δ1 !δ2 !δ3 !

× xn yl

γ!δ! γ1 !γ2 !δ1 !δ2 !δ3 !

ei(x−y)ξ ξ γ1 +δ1 Dxδ2 Dyγ2 a(x, y, ξ)Dxδ3 (1 − ϕ0 (x, ξ))dξ

≤ |x||k| |y||h|

ξ ≤R

ξ|γ1 +δ1 | ξm dξ.

Now, a(x, y, ξ) is supported on R2d \ Ω 12 × Rd and in this region |y| ≤ 32 x, so there exist constants C1 , C2 > 0 depending on R such that sup (x,y)∈R2d

k h δ γ x y Dx Dy K(x, y) ≤ C1 R|k|+|h| C |γ|+|δ| (γ!δ!)θ , 2

so K ∈ Sθθ (R2d ). Consider now xk y h Dxδ Dyγ Iα (x, y) = (2π)−d

0=β≤α

×

1 β!(α − β)!

δ1 +δ2 +δ3 =δ

δ! (−1)|γ| xk y h δ1 !δ2 !δ3 !

ei(x−y)ξ ξ γ+δ1 Dxδ2 Dξβ ϕ|α| (x, ξ)Dξα−β Dxδ3 ∂yα a(x, x, ξ) dξ

1 δ! (−1)|γ| (−i)h xk β!(α − β)! δ1 !δ2 !δ3 ! 0=β≤α δ1 +δ2 +δ3 =δ × e−iyξ ∂ξh eixξ ξ γ+δ1 Dxδ2 Dξβ ϕ|α| (x, ξ)Dξα−β Dxδ3 ∂yα a(x, x, ξ) dξ.

= (2π)−d


269

We need the estimates for (x, ξ) ∈ supp Dξβ ϕ|α| (x, ξ) ⊂ Q2R|α|μ+ν−1 \ QR|α|μ+ν−1 . Then, there exist C1 , C2 , C3 > 0 such that |h|+|k|+1

|xk y h Dxδ Dyγ Iα (x, y)| ≤ C1 ×

|α|

|γ|+|δ|

C2 C3

μ−1

(β!)μ−1 [(α − β)!]

0=β≤α

(k!h!γ!δ!)θ (α!)ν x−|α|

|β|

1 R|α|μ+ν−1

ξ ≤2R|α|μ+ν−1

ξ−|α−β| dξ

with C2 independent of R. Now, if (x, ξ) ∈ Q2R|α|μ+ν−1 \ QR|α|μ+ν−1 , we have that |α|

C2 (α!)ν x−|α|

(β!)μ−1 [(α − β)!]μ−1

0=β≤α

1

|β|

R|α|μ+ν−1

×

ξ ≤2R|α|μ+ν−1

ξ

−|α−β|

dξ ≤

C4 R

|α|

with C4 independent of R. Finally, we conclude that sup (x,y)∈R2d

k h δ γ x y D D Iα (x, y) ≤ C |h|+|k|+1 C |γ|+|δ| (k!h!γ!δ!)θ x

2

y

Choosing R > C4 , we obtain that

α=0

C4 R

|α| .

Iα ∈ Sθθ (R2d ).

Arguing as for Iα , we can prove that also sup (x,y)∈R2d

k h δ γ x y D D WN (x, y) ≤ C |h|+|k|+1 C |γ|+|δ| (h!k!γ!δ!)θ x

1

y

2

with C independent of R, which gives, for R sufficiently large, that Sθθ (R2d ). This concludes the proof.

C R

N

∞ N =0

WN is in

Proof of Proposition 6.3.11. By (6.3.18), t P is defined by t P u(x) = ei(x−y)ξ p(y, −ξ)u(y) dy ¯ dξ, ¯ u ∈ Sθθ (Rd ). Thus, t P is an operator of the form (6.3.19) with amplitude p(y, −ξ). By Theorem d 6.3.14, t P = Q + R where R is θ−regularizing and Q = q(x, D) ∈ OPAGm,n μν (R ), with (α!)−1 ∂ξα Dxα p(x, −ξ). q(x, ξ) ∼ j≥0 |α|=j

Proof of Theorem 6.3.12. We can write Q =t (t Q). Then, by Proposition 6.3.11, Q = Q1 + R1 , where R1 is θ−regularizing and Q1 u(x) = ei(x−y)ξ q1 (y, ξ)u(y) dy ¯ dξ, ¯ (6.3.21)

270


,n with q1 (y, ξ) ∈ AGm (R2d ), q1 (y, ξ) ∼ μν

follows that Q 1 u(ξ) =

(α!)−1 ∂ξα Dyα q(y, −ξ). From (6.3.21) it α

e−iyξ q1 (y, ξ)u(y) dy, ¯

u ∈ Sθθ (Rd )

from which we deduce that ¯ dξ ¯ + P R1 u(x). P Qu(x) = ei(x−y)ξ p(x, ξ)q1 (y, ξ)u(y) dy

,n,n We observe that p(x, ξ)q1 (y, ξ) ∈ Πm+m (Rd ); then we may apply Theorem μν 6.3.14 and obtain that P Qu(x) = T u(x) + Ru(x)

,n+n where R is θ−regularizing and T = t(x, D) ∈ OPAGm+m (Rd ) with μν

t(x, ξ) ∼

(α!)−1 ∂ξα p(x, ξ)Dxα q(x, ξ)

j≥0 |α|=j

m+m ,n+n (Rd ). in F Sμν

In Theorem 6.3.12, if p ∼

d pj in F AGm,n μν (R ) and q ∼

j≥0

,n qj in F AGm μν

j≥0

(Rd ), then t(x, ξ) ∼

(α!)−1 ∂ξα ph (x, ξ)Dxα qk (x, ξ)

m+m ,n+n in F Sμν (Rd ).

j≥0 |α|+h+k=j d To be definite, we restate the notion of ellipticity for elements of OPAGm,n μν (R ). It coincides with the definition of G-ellipticity in Chapter 3. d Definition 6.3.15. A symbol p ∈ AGm,n μν (R ) is said to be G-elliptic if there exist B, C > 0 such that

|p(x, ξ)| ≥ Cξm xn

for all (x, ξ) ∈ QeB .

d Theorem 6.3.16. If p ∈ AGm,n μν (R ) is G-elliptic and P = p(x, D), then there exists E ∈ OPAG−m,−n (Rd ) such that EP = I + R1 , P E = I + R2 , where R1 , R2 are μν θ−regularizing operators, for θ ≥ μ + ν − 1.

Proof. Let e0 (x, ξ) be fixed such that e0 (x, ξ) = p(x, ξ)−1

for (x, ξ) ∈ QeB ,

and define by induction, for j ≥ 1, ej (x, ξ) = −e0 (x, ξ) (α!)−1 ∂ξα ej−|α| (x, ξ)Dxα p(x, ξ). 0 1. To conclude this section, we apply Corollary 6.3.17 to the operators P with polynomial coefficients in (6.3.1). Their symbols can be regarded as elements d of AGm,n 1,1 (R ), with m, n as in (6.3.1). The condition (6.3.2) corresponds to Gellipticity in Definition 6.3.15. Therefore, if P u = 0 with u ∈ S (Rd ), or even u ∈ Sθθ (Rd ), then we may conclude u ∈ Sθθ (Rd ) for any θ > 1, in particular |u(x)| ≤ Ce−|x|

1/θ

,

θ>1

(6.3.22)

for suitable positive constants C and . A simple example of an operator of this type is given by P = (1 + |x|2k )(−Δ + 1) + Q1 (x, D),

k ≥ 1,

(6.3.23)

where Q1 (x, D) is any first-order operator with polynomial coefficients of degree 2k − 1. In turn, the operator L in (6.0.10) is a particular case of (6.3.23); this suggests that the estimates (6.3.22) should be valid for θ = 1 as well. In fact, in Section 6.5 we shall improve Corollary 6.3.17, by obtaining u ∈ S11 (Rd ) for d G-elliptic symbols in AGm,n 1,1 (R ).

6.4 A Short Survey on Travelling Waves In the next Section 6.5 we shall discuss regularity in S11 (Rd ) for semilinear equations, having as linear part G-elliptic pseudo-differential operators. The main motivation for such study comes from the theory of travelling (solitary) waves. In fact, a large part of these equations are of the above-mentioned type, with linear part given by G-elliptic partial differential equations with constant coefficients; non-local equations appear as well, whose linear parts are pseudo-differential opd erators with symbol in the classes AGm,n 1,1 (R ) of the preceding Section 6.3. For

272


travelling waves, both exponential decay and extension to the complex domain have relevance in Physics and they are exactly described by regularity in S11 (Rd ), from the Mathematical point of view. Addressing non-experts, we give in the following a short discourse on travelling waves, adding some references. The first documentation of the existence of shallow water waves appeared in 1834 when J. Scott Russell wrote one of the most cited papers about what later became known as soliton theory. Russell observed propagation of a solitary wave in the Glasgow-Edinburgh canal. In 1895 Korteweg and De Vries derived an equation describing shallow water waves, and gave the following interpretation of the solitary wave of Scott Russell. Ignoring some relevant physical aspects and simplifying parameters, we may write for short the KdV equation as vt + 2vvx + vxxx = 0,

(6.4.1)

where t is the time variable, x the point in the canal, v(x, t) the height of the water (let us refer to Bona and Li [22], Porubov [166], Whitham [195] for a much more detailed presentation). Looking for a solitary wave solution, travelling forward with velocity V > 0, we impose v(t, x) = u(x − V t) in (6.4.1) and we obtain d (−V u + u2 + u ) = 0, dx

x ∈ R,

hence u(x) satisfies u − V u + u2 = const. Assuming further const = 0, we are reduced to solving P u = u − V u + u2 = 0, (6.4.2) sometimes called Newton’s equation. Equation (6.4.2) possesses explicit solutions in terms of special functions. If we impose u(x) → 0 for x → ±∞, we obtain simply translations of the function u(x) =

Ch2

where

3 V 2 √

V 2

, x

(6.4.3)

et + e−t . 2 We emphasize two properties of u(x) in (6.4.3): first, it can be extended as an analytic function in a strip of the form {z ∈ C : |Im z| < a} in the complex plane. The second property is the exponential decay for x → ±∞. More precisely, these explicit solutions belong to the space S11 (R). After the discovery of the KdV equation, several related models were proposed. In particular recently, the theory of solitary waves had impressive developments, both concerning applicative aspects and mathematical analysis. Let us mention applications to internal water waves, nerve pulse dynamics, ion-acoustic waves in plasma, population dynamics, etc. In this order of ideas, we observe in Cht =

6.4. A Short Survey on Travelling Waves

273

particular that during the years 1990-2000, several papers were devoted to 5-th order and 7-th order generalizations of KdV, see for example Porubov [164], Chapter 1. The corresponding extension of equation (6.4.2) is of the type N

aj u(j) + Q(u) = 0,

(6.4.4)

j=0

M bj uj and a0 = −V = 0. Because of where Q is a polynomial, Q(u) = Nj=2 j physical assumptions, the equation j=0 aj λ = 0 has no purely imaginary roots, and then all the solutions of the corresponding linear equation have exponential decay/growth. This condition can be read as G-ellipticity of the symbol of the N linear part in (6.4.4): p(ξ) = j=0 aj (iξ)j = 0 for ξ ∈ R, in particular −ξ 2 − V = 0 in (6.4.2), hence |p(ξ)| ≥ CξN for some C > 0. Non-trivial solutions u of (6.4.4) with u(x) → 0 for x → ±∞ may exist or not, according to the coefficients aj , bj , and when they exist, in general they do not have an explicit analytic expression. Exponential decay and holomorphic extensions are granted anyhow, see the next Section 6.5. Let us emphasize that, to reach the exponential decay, the boundedness of u(x) is not sufficient as an initial assumption. We shall express in Section 6.5 a precise threshold in terms of Sobolev estimates; as counter-example, consider here the celebrated Burgers’ equation (1948): vt + vxx + 2vvx = 0.

(6.4.5)

Imposing v(t, x) = u(x−V t) and arguing as before we obtain the Verhulst equation u − V u + u2 = 0

(6.4.6)

which can be regarded as a particular case of (6.4.4). It admits the bounded solution V u(x) = . (6.4.7) 1 + e−V x Assuming V > 0, we have exponential decay only for x → −∞, whereas u(x) → V = 0 as x → +∞. As d-dimensional generalization of (6.4.2), setting for simplicity V = 1, we may consider −Δu + u = up (6.4.8) for an integer p ≥ 2. Such equations in Rd have been widely studied. From the point of view of Mathematical Physics, they appear for example when considering nonlinear Schrödinger equations used in Plasma Physics and Nonlinear Optics. Travelling waves, in this case, have to be understood as stationary wave solutions, defined by time-modulation. From the point of view of Mathematical Analysis, we refer to the recent book of Ambrosetti and Malchiodi [4] for a collection of results of existence and uniqueness, or multiplicity, of the Sobolev solutions of (6.4.8) via

274


variational methods. Concerning exponential decay, we quote the following precise d+2 result in Berestycki and Lions [12]. Assume d ≥ 3. If 1 < p < d−2 , then (6.4.8) 1 d has a positive radial solution u(x) = U (|x|) ∈ H (R ). Such a solution satisfies, for some > 0, U (r) e−r , r = |x|, as r → +∞. We refer to the next section for exponential decay and holomorphic extension of general solutions of (6.4.8) and other semilinear G-elliptic partial differential equations. Non-local equations, i.e., nonlinear partial differential equations involving integral operators, have been proposed as models for different solitary wave phenomena. Let us fix here attention on the so-called intermediate-long-wave equation, see Joseph [124] and recent contributions by J. L. Bona, J. Albert and others: vt + 2vvx − (N v)x + vx = 0,

(6.4.9)

where N is the Fourier multiplier operator defined by (N v)(ξ) = ξ Ctghξ v(ξ).

(6.4.10)

We emphasize the analyticity of the function ξCtgh ξ. Looking for solutions v(t, x) = u(x − V t) and arguing as before, we obtain the non-local equation N u + γu = u2

(6.4.11)

where γ = V − 1. Under the assumption V > 0 a non-trivial solution u ∈ S11 (R) can be computed explicitly. This also will be included in our results in the next section, since the operator N in (6.4.10) can be seen as an example in the class of Fourier multipliers P u(x) =

eixξ p(ξ) u(ξ) dξ, ¯

(6.4.12)

d where p(ξ) belongs to AGm,0 1,1 (R ), m > 0:

|∂ξα p(ξ)| ≤ C |α|+1 α!ξm−|α| .

(6.4.13)

The assumption of G-ellipticity is formally as in Definition 6.3.15: |p(ξ)| ≥ cξm ,

ξ ∈ Rd ,

(6.4.14)

for some c > 0. In particular the symbol of the linear part in (6.4.11) p(ξ) = ξCtgh ξ + γ,

γ > −1,

(6.4.15)

satisfies (6.4.13), (6.4.14) with m = 1. As a final example of a soliton equation, we mention the Benjamin-Ono equation, see for example Amick and Toland [5]: vt + 2vvx − (|D|v)x = 0

(6.4.16)

6.5. Semilinear G-Equations

275

where |D| is the Fourier multiplier operator defined by (|D|u)(ξ) = |ξ| u(ξ). The corresponding equation for the wave profile u(x), with v(t, x) = u(x − V t), V > 0, is (6.4.17) |D|u + V u = u2 , with solution given by u(x) =

2V . 1 + V 2 x2

(6.4.18)

This is the unique non-trivial solution in L2 (R) of (6.4.17), modulo translations. The lack of exponential decay can be related to the singularity at the origin of the symbol |ξ|. Regretfully, the Benjamin-Ono equation will remain outside our arguments in the sequel. In fact, the organization of a pseudo-differential calculus predicting precisely the algebraic decay of (6.4.18) represents a challenging open problem.

6.5 Semilinear G-Equations In this section we shall consider the semilinear equation P u = f + F [u],

(6.5.1)

d where P is a pseudo-differential operator with symbol p(x, ξ) ∈ AGm,n μν (R ), μ ≥ 1, ν ≥ 1, m ≥ 1, n ≥ 0, cf. Section 6.3. Let n be integer. We suppose that P is G-elliptic, according to Definition 6.3.15. We assume f ∈ Sνμ (Rd ) and

F [u] =

Fhlγr xh ul (Dγ u)r ,

(6.5.2)

h,l,γ,r

where h ∈ Nd with 0 ≤ |h| ≤ max{n − 1, 0}, γ ∈ Nd with 0 ≤ |γ| ≤ m − 1 and l, r ∈ N, l + r ≥ 2. We shall prove the following result. Theorem 6.5.1. Under the preceding hypotheses, assume that u ∈ H s (Rd ), s > d2 + m − 1, is a solution of (6.5.1). In the case n = 0 assume further x0 u ∈ H s (Rd ), s > d2 + m − 1, for some 0 > 0. Then u ∈ Sνμ (Rd ). In particular, if μ = ν = 1, we conclude u ∈ S11 (Rd ). For linear equations, i.e., F [u] = 0, this improves Corollary 6.3.17, where the threshold θ = μ = ν = 1 was not reached. For the travelling waves in Section 6.4, see (6.4.4), (6.4.8), (6.4.11), we get the expected result of S11 (Rd ) regularity. To prove Theorem 6.5.1 we shall use Proposition 6.1.11, (b). Namely, to prove u ∈ Sνμ (Rd ) it will be sufficient to obtain separately u s,ν; < ∞ and u {s,μ;T } < ∞ for some s > d2 + m − 1, and sufficiently small > 0, T > 0,

276


cf. (6.1.37), (6.1.38). Equivalently, we shall prove the boundedness of the partial sums: |k| s,ν; HN [u] =

xk u s , N = 0, 1, . . . , (6.5.3) k!ν |k|≤N

s,μ;T [u] = EN

T |j|

∂ j u s , j!μ x

N = 0, 1, . . . .

(6.5.4)

|j|≤N

In the proof of Theorem 6.5.1 we shall not use the full pseudo-differential calculus of Section 6.3, but rely on the following technical proposition. Denote by E the pseudo-differential operator with symbol e0 (x, ξ) = p(x, ξ)−1 for large (x, ξ), cf. the proof of Theorem 6.3.16. Since the cases μ = 1, ν = 1 were excluded in the symbolic calculus of Section 6.3, we have to refer in the following to μ , ν satisfying μ ≥ μ, ν ≥ ν and simultaneously μ > 1, ν > 1. We then have (Rd ), and from Theorem 6.3.12 we obtain EP = I + R where e0 (x, ξ) ∈ AG−m,−n μ ν −1,−1 R ∈ OPAGμ ν (Rd ). For such rough parametrix E of P = p(x, D) we shall use the following proposition. d Proposition 6.5.2. Let p ∈ AGm,n μν (R ), μ ≥ 1, ν ≥ 1, be G-elliptic and let E ∈ (Rd ) be the operator defined as before. For every α, β ∈ Nd denote by OPAG−m,−n μ ν rαβ the symbol of the operator E(Dξα Dxβ p)(x, D). Then, for every σ, θ ∈ Nd , there exists a positive constant C = C(σ, θ) independent of α, β such that θ σ ∂ξ ∂x rαβ (x, ξ) ≤ C |α|+|β|+1 α!μ β!ν ξ−|α|−|θ| x−|β|−|σ| (6.5.5)

for all (x, ξ) ∈ R2d . Proof. By Theorem 6.3.12 we have that rαβ ∼

j≥0

some μ > 1, ν > 1, with rαβj (x, ξ) =

−|α|,−|β|

rαβj in F AGμ ν

(Rd ) for

(γ!)−1 ∂ξγ e(x, ξ)Dξα Dxβ+γ p(x, ξ).

|γ|=j

Leibniz’ rule gives θ σ ∂ξ ∂x rαβj (x, ξ) ≤ C |α|+|β|+|θ|+|σ|+2j+1 α!μ β!ν θ!ν σ!μ j!μ +ν −1 × ξ−|α|−|θ|−j x−|β|−|σ|−j . (6.5.6) Thus we can apply Theorem 6.3.8, taking cut-off functions ϕj (x, ξ) independent of α, β, and we obtain (6.5.5). We start by deriving decay estimates. We first give a result for the linear case. With respect to Corollary 6.3.17 we now allow μ = 1, ν = 1.


277

d μ d d Theorem 6.5.3. Let p ∈ AGm,n μν (R ) be G-elliptic and f ∈ Sν (R ). Let u ∈ S (R ) be a solution of the equation P u = f. Then, for every s ≥ 0 there exists > 0 such that u s,ν; < ∞.

To prove the Theorem we need a preliminary result. d Lemma 6.5.4. Let p ∈ AGm,n μν (R ) be G-elliptic and let E be the operator in Proposition 6.5.2. Then, for every s ≥ 0 there exists a positive constant As such that

E[P, xk ]u s ≤ k!ν

0=β≤k

|β|

As

xk−β u s (k − β)!ν

(6.5.7)

for every k ∈ Nd , k = 0. Proof. Fixing k ∈ Nd , k = 0, and integrating by parts, we have: k x P u(x) = eixξ xk p(x, ξ) u(ξ) dξ ¯ u(ξ)) dξ ¯ = (−1)|k| eixξ Dξk (p(x, ξ) k (−1)|β| = (Dξβ p)(x, D)(xk−β u), β β≤k

from which it follows that E[P, xk ]u =

(−1)|β|+1

0=β≤k

k E(Dξβ p)(x, D)(xk−β u). β

(6.5.8)

We can now apply Proposition 6.5.2 and obtain that E(Dξβ p)(x, D) = rβ (x, D) with rβ (x, ξ) satisfying the estimate θ σ ∂ξ ∂x rβ (x, ξ) ≤ C |β| β!ν ξ−|β|−|θ| x−|σ| for some positive constant C independent of β = 0. Hence, we have, for every s ∈ R: ν (6.5.9)

rβ (x, D) B(H s ) ≤ A|β| s β! ,

cf. Remark 1.5.6. Then, (6.5.8) and (6.5.9) give (6.5.7).

Proof of Theorem 6.5.3. In the linear case we know already that u ∈ S(R ). Moreover, with fixed k ∈ Nd , > 0, we can write d

|k| k |k| k x P u = x f, k!ν k!ν from which we get |k| |k| |k| k k k P (x u) = [P, x ]u + x f. k!ν k!ν k!ν

(6.5.10)

278


Applying to both members of (6.5.10) the operator E in Proposition 6.5.2 we obtain |k| k |k| |k| |k| x u = ν R(xk u) + ν E[P, xk ]u + ν E(xk f ), ν k! k! k! k! d where R ∈ OPAG−1,−1 μ ν (R ), for some μ > 1, ν > 1. Taking now Sobolev norms and summing up for |k| ≤ N , we have s,ν; HN [u] ≤ u s +

|k|

R(xk u) s + k!ν

0 0 we can assume without loss of generality that F [u] = xh u (∂xγ u)r for some , r ∈ N, γ, h ∈ Nd with + r ≥ 2, 0 ≤ |h| ≤ n − 1, 0 ≤ |γ| ≤ m − 1. d Lemma 6.5.6. Let p ∈ AGm,n μν (R ) satisfy the assumptions of Theorem 6.5.5 and let u ∈ H s (Rd ), s > d/2 + |γ| for some γ ∈ Nd with 0 ≤ |γ| ≤ m − 1. Then, for every pair of non-negative integers , r with + r ≥ 2, 0 ≤ |h| ≤ n − 1, the following estimate holds:

0 0. Then, since E ◦ xh+ejk is an operator of orders −|γ|, 0, we can write

E(xk+h u (∂xγ u)r ) s ≤ Cs xk−ejk u (∂xγ u)r s−|γ| . Now, if ≥ 1, applying Schauder’s estimates (0.2.7), we get γ r

xk−ejk u (∂xγ u)r s−|γ| ≤ Cs xk−ejk u s−|γ| · u −1 s−|γ| · ∂x u s−|γ|

≤ Cs xk−ejk u s · u +r−1 . s Hence we have 0 1, ν > 1 and

rγ (x, D) B(H s ) ≤ Bs|γ| γ!μ . Hence, since μ ≥ 1, we have j γ!μ Bs|γ| ∂xj−γ u s

E[P, ∂ ]u s ≤ γ j

0=γ≤j

≤ j!μ

0=γ≤j

The lemma is then proved.

|γ|

Bs

∂ j−γ u s . (j − γ)!μ x

d μ d Theorem 6.5.11. Let p ∈ AGm,n μν (R ) be G-elliptic and let f ∈ Sν (R ). Let u ∈ d S (R ) be a solution of the equation P u = f . Then there exists T > 0 such that u {s,μ;T } < ∞.


283

Proof. As in Theorem 6.5.3, we have indeed u ∈ S(Rd ) from the results of Chapters 1 and 3. Moreover, for any T > 0 and j ∈ Nd we have T |j| j T |j| j ∂ P u = ∂ f, j!μ x j!μ x from which we get T |j| T |j| T |j| j j j P (∂ u) = [P, ∂ ]u + ∂ f. x x j!μ j!μ j!μ x

(6.5.20)

Hence, arguing as in the proof of Theorem 6.5.3, we obtain T |j| j T |j| T |j| T |j| j j ∂ u = R(∂ u) + E[P, ∂ ]u + E(∂xj f ) x x j!μ x j!μ j!μ j!μ for some operator R of orders −1, −1. Taking now Sobolev norms and summing up for |j| ≤ N , we have s,μ;T EN [u] ≤ u s +

0

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