Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends

Operator Theory: Advances and Applications Volume 219 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Black...

Author: B. Malcolm Brown | Jan Lang | Ian G. Wood

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Operator Theory: Advances and Applications Volume 219 Founded in 1979 by Israel Gohberg

Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA)

Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)

B. Malcolm Brown Jan Lang Ian G. Wood Editors

Spectral Theory, Function Spaces and Inequalities New Techniques and Recent Trends

Editors B. Malcolm Brown School of Computer Science and Informatics Cardiff University Cardiff, CF24 3XF UK

Jan Lang Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, OH 43210 USA

Ian G. Wood School of Mathematics, Statistics and Actuarial Science University of Kent Cornwallis Building Canterbury, Kent CT2 7NF UK

ISBN 978-3-0348-0262-8 e-ISBN 978-3-0348-0263-5 DOI 10.1007/978-3-0348-0263-5 Springer Basel Dordrecht Heidelberg London New York Library of Congress Control Number: 2011941087 Mathematics Subject Classification (2010): Primary: 11R09, 26A12, 26D10, 34A40, 34B24, 35J60, 35Q40, 35P99, 42B20, 46E30, 46E35, 47B25, 47F05, 47G30; Secondary: 26D07, 26D15, 34L05, 34L40, 35J70, 35P15, 35S05, 47B10, 47B35, 81Q35  Springer Basel AG 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Printed on acid-free paper

Springer Basel AG is part of Springer Science + Business Media (www.birkhauser-science.com)

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii B.M. Brown, J. Lang and I.G. Wood David Edmunds’ Mathematical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix B.M. Brown, J. Lang and I.G. Wood Desmond Evans’ Mathematical Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi B.M. Brown and M.S.P. Eastham Generalised Meissner Equations with an Eigenvalue-inducing Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B.M. Brown and K.M. Schmidt On the HELP Inequality for Hill Operators on Trees . . . . . . . . . . . . . . . 21 F. Cobos, L.M. Fern´ andez-Cabrera and A. Mart´ınez Measure of Non-compactness of Operators Interpolated by Limiting Real Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 R.L. Frank and E.H. Lieb A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . . 55 D.D. Haroske Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 I. Knowles and M.A. LaRussa Lavrentiev’s Theorem and Error Estimation in Elliptic Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 V. Kokilashvili and A. Meskhi Two-weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 J. Lang and O. Méndez Modular Eigenvalues of the Dirichlet p(·)-Laplacian and Their Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 R.T. Lewis Spectral Properties of Some Degenerate Elliptic Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

vi

Contents

B. Opic Continuous and Compact Embeddings of Bessel-Potential-Type Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Y. Sait¯ o and T. Umeda A Sequence of Zero Modes of Weyl–Dirac Operators and an Associated Sequence of Solvable Polynomials . . . . . . . . . . . . . . . . . . . . . 197 A.V. Sobolev A Szeg˝ o Limit Theorem for Operators with Discontinuous Symbols in Higher Dimensions: Widom’s Conjecture . . . . . . . . . . . . . . 211 V.D. Stepanov On a Supremum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 H. Triebel Entropy Numbers of Quadratic Forms and Their Applications to Spectral Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Preface This is a a collection of contributed papers by David and Des’s friends and colleagues and is issued to mark their respective 80th and 70th birthday. For the past forty years they have made fundamental contributons in the area of differential equations, operator theory and function space theory, and it is fitting that these contributions reflect that. Our thanks must also go to Dr Thomas Hempfling, Executive Editor, Mathematics Birkhäuser and Ms Sylvia Lotrovsky for the help and assistance given to us during the preparation of this volume. Finally we thank all the authors who contributed papers to this special edition to mark David and Des’s birthdays. B.M. Brown J. Lang I.G. Wood

Operator Theory: Advances and Applications, Vol. 219, ix-x c 2012 Springer Basel AG

David Edmunds’ Mathematical Work B.M. Brown, J. Lang and I.G. Wood

David Edmunds has influenced and made major contributions to numerous branches of mathematics. These include spectral theory, functional analysis, approximation theory, the theory of function spaces, operator theory, ordinary and partial differential equations. The breadth of his impact is demonstrated by his publication record, which consists of 5 books and more than 190 research papers, and by his winning the LMS P´ olya prize in 1996 and the Bolzano Medal of the Czech Academy of Sciences, in 1998. He was awarded the Ph.D. degree by the University of Wales in 1955, having been supervised by R.M. Morris. After some years working for EMI Electronics on guided missiles, he held positions of Lecturer and then Senior Lecturer at the University of Wales, Cardiff, leaving in 1966 to take up a Readership at the University of Sussex. He was awarded a Personal Chair there in 1970 and is still affiliated to Sussex as well as additionally being appointed Honorary Professor, School of Mathematics, Cardiff University, 2004. In his early works he focused on problems of fluid dynamics, including moving aerofoils, magneto-hydrodynamics and the nature of solutions of the Navier-Stokes equations (studying questions of stability, backward uniqueness, asymptotic behaviour and removable singularities). Then his interests shifted towards the study of more general non-linear problems, elliptic equations and inequalities, and functional analysis. His first joint paper with W.D. Evans appeared in 1973. In this, by deriving new weighted embeddings on unbounded domains in Lp spaces, results were obtained for the Dirichlet problem concerning elliptic equations. This work began their long and fruitful collaboration and established their common interest in the properties of function spaces, embedding theorems, integral operators and spectral theory. These matters were also the main topics of their two joint books, the well-known ‘Spectral theory and differential operators’ (OUP) and the more recent ‘Hardy operators, function spaces and embeddings’ (Springer). Edmunds’ work on the properties of Besov and Lizorkin spaces has often been motivated by his interest in the nature of eigenvalues and eigenvectors of operators acting on non-Hilbert spaces. Many of his papers on this topic are

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concerned with the qualitative and quantitative properties of embeddings of function spaces, such as the behaviour of their entropy and s-numbers. These results, including those obtained with Hans Triebel, formed the basis of their joint book, ‘Function spaces, entropy numbers, differential operators’ (CUP). Interest in functional analysis led him to study interpolation theory. His recent results with Yuri Netrusov settled a long-standing conjecture concerning the behaviour of entropy numbers under real interpolation. In the theory of integral operators he has concentrated on maps of Hardy or Volterra type, acting on function spaces with and without weights; many of his results in this area are presented in his book ‘Bounded and compact integral operators’, with V. Kokilashvili and A. Meskhi. His very recent book ‘Eigenvalues, embeddings and generalised trigonometric functions’, with J. Lang, has as its basis their work on the properties of s-numbers of Hardy-type operators, which involves the study of eigenfunctions of integral and differential operators together with certain generalisations of the trigonometric functions that are of importance in non-linear analysis. Another of his interests is the currently popular theory of Lp(x) spaces, the so-called variable exponent spaces, fundamental work on which was carried out by J. Rákosn´ık, and together with whom results were obtained that have proved stimulating for many analysts. His standing attracted a number of mathematicians, such as D. Vassiliev and A. Sobolev, who started their professional careers in the UK at Sussex. He was also active in making contacts with colleagues from other countries. He has supervised 18 Ph.D. students, including J.M. Ball (Oxford), J.R.L. Webb (Glasgow) and V. Mustonen (Oulu); according to the Mathematics Genealogy project he has 73 descendants. David Edmunds’ contribution to mathematics is not only long, but wide and deep. His superb professional work and his warm personality have deeply influenced a large part of the mathematical community. B.M. Brown School of Computer Science, Cardiff University, Cardiff CF24 3XF UK e-mail: [email protected] J. Lang Department of Mathematics, The Ohio State University, 231 West 18th Avenue Columbus, OH 43210-1174 USA e-mail: [email protected] I.G. Wood School of Mathematics, Statistics and Actuarial Science, Cornwallis Building University of Kent, Canterbury, Kent CT2 7NF UK e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, xi-xii c 2012 Springer Basel AG

Desmond Evans’ Mathematical Work B. M. Brown, J. Lang and I. G. Wood

Desmond Evans has made major influential contributions to numerous branches of mathematical analysis which include the spectral theory of both ordinary and partial differential equations, mathematical physics and functional analysis. The breadth and depth of his achievements are recorded in his 142 published papers and 3 books. Following a B.Sc (Wales) from University College, Swansea in 1961 he went up to Jesus College, Oxford to work for a D.Phil. under the guidance of E. C. Titchmarsh, one of the leading analysts of the day. After the sudden death of Titchmarsh in 1963, Des completed his studies under the direction of another leading analyst, J.B.McLeod. The degree was awarded in 1965. In 1964 he was appointed to a lectureship in Pure Mathematics at the then University College of South Wales and Monmouthshire, which later became Cardiff University, and has remained at Cardiff all his working life, progressing through the grades of senior lecturer and reader before being awarded a Personal Chair by the University of Wales in 1977. During this time at Cardiff he has supervised 13 Ph.D. students. Following some early work on the Dirac system he worked on the limitpoint, limit-circle classification problem for ordinary differential equations, inequalities related to differential and difference equations (in particular the HELP inequality and its later variants), and on spectral problems associated with non-selfadjoint differential systems. His work on partial differential equations has often been motivated by physical questions, a significant portion having been concerned with problems arising in the study of non-relativistic quantum mechanics. This research contains work on the spectrum of relativistic one-electron atoms and on the zero modes of Pauli and Weyl-Dirac operators. His many papers in this and related areas cover an impressive range of topics. These include the spectral analysis of N-body operators for atoms and molecules; quantum graphs; Hardy and Rellich inequalities with magnetic potentials; Schrödinger operators and biharmonic operators with magnetic fields. He has been active also in functional analysis and operator theory, especially in areas concerning the properties of Hardy-type operators acting

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between function spaces, estimates and asymptotic results for their approximation numbers and related inequalities. Much of this work was motivated by his study of the properties of embedding maps between Sobolev spaces defined on irregular (including fractal) domains, in which he and Desmond Harris introduced the notion of a generalised ridged domain and developed techniques for reducing problems to analogous ones on associated trees. His two research monographs with David Edmunds, “Spectral Theory and Differential Equations” and “Hardy Operators, Function Spaces and Embeddings” have become standard texts in his main areas of activity. Recently, with Alex Balinsky, he has published “Spectral analysis of relativistic operators”, which includes, in particular, an account of their numerous contributions to problems concerning the stability of matter. As well as his mathematical contributions, Des has been active in various administrative roles both within and outside Cardiff University. In particular, he has been Head of School at Cardiff on several occasions, and served the London Mathematical Society over many years as Editorial Advisor, Editor of the Proceedings and Council member. Recently he has played an important part in establishing the Wales Institute of Mathematical and Computational Sciences. In recognition of these achievements he was elected a fellow of the Learned Society of Wales in 2011. (This is an expanded version of the Laudatum in JCAM V208 1 November 2007) B. M. Brown School of Computer Science Cardiff University, Cardiff, CF24 3XF, U.K. e-mail: [email protected] J. Lang Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 43210-1174 USA e-mail: [email protected] I. G. Wood School of Mathematics, Statistics and Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 1–20 c 2012 Springer Basel AG

Generalised Meissner Equations with an Eigenvalue-inducing Interface B.M. Brown and M.S.P. Eastham To David and Des

Abstract. An interface situation is considered, where a periodic differential equation is given on one side x > 0 of the interface and a general Sturm–Liouville equation is given on a finite interval (−X, 0) on the other side of the interface. A boundary condition is imposed at −X. The emphasis is on a periodic discontinuous weight function, which has the effect of widening the spectral gaps (instability intervals). It is shown that the interface can induce eigenvalues in all the gaps beyond some point. The dependence on X of the number of eigenvalues in each gap is noted. The general theory is supported by step-function examples. Mathematics Subject Classification (2010). Primary 34B24; Secondary 34L05. Keywords. Periodic, interface, Meissner.

1. Introduction A basic property of the periodic differential equation y (x) + {λw(x) − q(x)}y(x) = 0

(1.1)

on an unbounded x-interval I is the existence of stability and instability λintervals on the real λ-axis. Here the weight function w(x) and the potential function q(x) are real-valued and Lloc (I) with w(x) > 0, and they have a common period a; I can be either (−∞, ∞) or a semi-infinite interval, let us say [0, ∞). The stability intervals are specified in terms of the eigenvalues λn and μn (n ≥ 0) of (1.1) arising respectively from the periodic boundary conditions y(0) = y(a), y (0) = y (a) (1.2) and the semi-periodic boundary conditions y(0) = −y(a), y (0) = −y (a).

(1.3)

2

B.M. Brown and M.S.P. Eastham

These eigenvalues have the ordering λ0 < μ0 ≤ μ1 < λ1 ≤ λ2 < μ2 ≤ μ3 < λ3 · · · .

(1.4)

Then the stability intervals are the open intervals (λ2m , μ2m ), (μ2m+1 , λ2m+1 ) (m ≥ 0)

(1.5)

which, together possibly with their end-points, comprise the values of λ for which all solutions of (1.1) are bounded in I. In terms of the spectral theory of (1.1) in the Hilbert space L2 (I), the intervals (1.5) exhibit the band structure of the associated spectrum in the sense that the essential spectrum is formed by the closures of (1.5). We refer to the standard books [4, chapters 8–9], [6], [17, chapter 13] and [22, chapter 21] for all this basic theory of (1.1). We also refer to [8] for (1.4) in a wider context of more general boundary conditions. Moving on to the particular topic considered in this paper, we note first that, when I is (−∞, ∞), (1.1) has no eigenvalues in the spectral gaps between the intervals (1.5) but, when I is [0, ∞) and the usual type of boundary condition is imposed at x = 0, any given λ in a gap can be an eigenvalue arising from an associated boundary condition [4, p. 257], [9, Theorem 2.5.3]. If however the periodicity of (1.1) is compromised by some perturbation, there is the possibility that eigenvalues can appear in the gaps as a result of the perturbation, that is, new spectral points may arise. Historically, the first type of perturbation to be considered in this context is the addition to q(x) of a non-periodic function p(x) which is small in some sense when |x| is large. On the one hand, we have the result that if p(x) = O(|x|)−2 ) (|x| → ∞), then only a finite number of eigenvalues can appear in sufficiently distant spectral gaps [19, Corollary 3] (see also [18]). On the other hand, if a fixed spectral gap is considered and p(x) contains a coupling parameter c, the number of eigenvalues in the gap can become large for large c, and the asymptotic distribution is investigated in [3], [20], [21] (see also [1]). Another type of perturbation is based on the idea of introducing an interface where (1.1) holds on one side of the interface (say for x > 0) and a different periodic equation holds on the other side x < 0. In [13] a dislocation situation is considered in which w(x) = 1 and the potential for x < 0 is q(x + t) where t ∈ R is the dislocation parameter. It is shown that a spectral point (an eigenvalue or a resonance) λ(t) is produced in each spectral gap, and its behaviour in terms of t is discussed [13, pp. 474, 480]. A biperiodic situation is introduced in [14] where q has a different period for x < 0, and here it is shown that up to two spectral points appear in each spectral gap. Recently, similar interface problems have been considered in [5] again with w(x) = 1 and using the method of C 1 gluing across the interface. In [5, section 3] explicit conditions on q are derived which guarantee the appearance of up to two eigenvalues in the first two spectral gaps (−∞, λ0 ) and (μ0 , μ1 ). In this paper we consider a different interface situation where a new feature is that arbitrarily many eigenvalues can occur in the spectral gaps (μ2m , μ2m+1 ) and (λ2m+1 , λ2m+2 ) (m ≥ 0). Our main focus is on the weight

Generalised Meissner Equations

3

function w(x) and, in (1.1), we take w(x) to be non-constant for x > 0 and, in particular, w(x) has discontinuities. This has the effect of widening the gaps [6, section 4.5], [7], [16]. On the other side (x < 0) of the interface the differential equation is y (x) + {λ − q1 (x)}y(x) = 0

(1.6)

on a finite interval [−X, 0) with an arbitrary q1 (x) and a non-trivial boundary condition (1.7) c1 y(−X) + c2 y (−X) = 0. Our spectral setting is therefore on the x-interval [−X, ∞) with (1.1) for x > 0, (1.6) for x < 0 and the boundary condition (1.7). A simple relative compactness argument shows that the essential spectrum of our interface problem retains the band structure noted above. In section 2, we recall the basic Floquet theory from (for example) [6] which we require, and formulate the eigenvalue equation arising from (1.7). Then in section 3 we formulate and prove a general theorem (Theorem 3.1) on the existence of interfaceinduced eigenvalues in sufficiently distant spectral gaps, subject to a condition concerning the length of the gaps. The dependence of the number of these eigenvalues on the value of X is noted. In sections 4 and 5, we consider the case where w(x) has two discontinuities in its period, a two-valued stepfunction being an example, and we lead up to situations where the length condition in Theorem 3.1 is satisfied. We mention here that the eigenvalues λn and μn for step-function examples have been discussed in [6, section 2.2], [10, section 50], [12], and one contribution of our paper in sections 4 and 5 is to develop properties of these eigenvalues which are not confined to the stepfunction case. In section 6, we discuss briefly the case where w(x) has just one discontinuity in its period, that is, w(0) = w(a). Finally in section 7 we discuss in more detail some step-function examples where induced eigenvalues appear in all the spectral gaps (except (−∞, λ0 )), not just the distant ones. We conclude this introduction by mentioning that the adjective Meissner is applied to any periodic equation (1.1) in which w and q are step functions (cf. [2], [10], [12]). This follows the original equation of this kind formulated by Meissner [15] (concerning locomotive coupling rods). In our paper we are not confined to step-functions, but the discontinuities in w(x) are essential.

2. Formulation of the eigenvalue problem We begin with the solutions φ1 (x) and φ2 (x) of (1.1) which have the initial values 1, 0 and 0, 1 (2.1) respectively at x = 0, with the dependence on λ not indicated until necessary. Since we are dealing with λ in a spectral gap, the basic theory of [6, chapters 1 and 2] shows that there are also solutions ψk (x) (k = 1, 2) of (1.1) such that (2.2) ψk (x + a) = ρk ψk (x) (x ≥ 0)

4


where the ρk are the two distinct and real solutions of the quadratic ρ2 − Dρ + 1 = 0 with D = φ1 (a) + φ2 (a).

(2.3)

Further [6, section 1.1], ψk (x) can be written as either ψk (x) = φ2 (a)φ1 (x) − {φ1 (a) − ρk }φ2 (x).

(2.4)

or, in case (2.4) is a trivial linear combination of φ1 (x) and φ2 (x), ψk (x) = {φ2 (a) − ρk }φ1 (x) − φ1 (a)φ2 (x).

(2.5)

We shall generally keep to (2.4) and comment on the change to (2.5) as necessary. Since ρ1 ρ2 = 1, we take it that |ρ1 | < 1 and |ρ2 | > 1.

(2.6)

We are looking for a solution of (1.1) which is L2 (0, ∞), and it follows from (2.2) and (2.6) that this solution must be y(x) = ψ1 (x)

(x ≥ 0)

(2.7)

to within a constant multiple. We have now to continue this solution into x < 0 and substitute the result into the boundary condition (1.7) to complete the formulation of the eigenvalue problem. We continue to denote by φ1 (x) and φ2 (x) the solutions now of (1.6) in [−X, 0) but still satisfying (2.1). We also note that, by (2.1) and (2.4), ψ1 (x) has the initial values φ2 (a), −{φ1 (a) − ρ1 } at x = 0. Hence, as a linear combination of φ1 (x) and φ2 (x), (2.7) is continued into x < 0 as y(x) = φ2 (a)φ1 (x) − {φ1 (a) − ρ1 }φ2 (x) (x < 0).

(2.8)

Then (1.7) gives the equation to determine the eigenvalues in the spectral gaps as φ2 (a){c1 φ1 (−X) + c2 φ1 (−X)} − {φ1 (a) − ρ1 }{c1 φ2 (−X) + c2 φ2 (−X)} = 0.

(2.9)

We shall examine (2.9) firstly for sufficiently distant gaps and then, in an example, for all gaps. To prepare for the former in the next section, we require a slightly more precise version of a familiar result for the SturmLiouville equation y (x) + {λ − Q(x)}y(x) = 0

(0 ≤ x ≤ A) √ with any Q in Lloc [0, A]. We consider λ > 0 and write ν = λ. Lemma 2.1. Let y(x) satisfy (2.10) and let A |Q(t)|dt. ν≥ 0

(2.10)

(2.11)


5

Then 1 1 E(x)} cos νx + {y (0) + F (x)} sin νx, ν ν 1 y (x) = −ν{y(0) + E(x)} sin νx + {y (0) + F (x)} cos νx, ν y(x) = {y(0) +

where |E(x)|, |F (x)| ≤ (|y(0)| +

1 |y (0)|)(e − 1) ν

(2.12) (2.13)

A

0

|Q(t)|dt.

(2.14)

Proof. The integral formulation of (2.10) is 1 1 x y(x) = y(0) cos νx + y (0) sin νx + sin{ν(x − t)}Q(t)y(t)dt. (2.15) ν ν 0 Hence 1 x 1 |Q(t)||y(t)|dt, |y(x)| ≤ |y(0)| + |y (0)| + ν ν 0 and this Gronwall inequality gives x 1 1 |y(x)| ≤ (|y(0)| + |y (0)|) exp |Q(t)|dt . (2.16) ν ν 0 We now write (2.15) in the form (2.12) with x E(x) = − (sin νt)Q(t)y(t)dt 0

and similarly for F (x) with cos νt in place of − sin νt. Then, by (2.16), x 1 t 1 |Q(t)| exp |Q(u)|du dt |E(x)|, |F (x)| ≤ (|y(0)| + |y (0)|) ν ν 0 0 x 1 1 = (|y(0)| + |y (0)|)ν{exp |Q(t)|dt − 1} ν ν 0 yielding (2.14) when (2.11) holds. This proves (2.12), and (2.13) follows similarly from differentiation of (2.15). We note that the lemma also holds for A < 0 if the integration range in (2.11) and (2.14) is replaced by (A, 0).

3. Dirichlet and Neumann boundary conditions The Dirichlet condition is the case c2 = 0 of (1.7), and then (2.9) becomes φ2 (a, λ)φ1 (−X, λ) − {φ1 (a, λ) − ρ1 (λ)}φ2 (−X, λ) = 0,

(3.1)

where we are now indicating the dependence on λ. The Neumann condition is the case c1 = 0 of (1.7) and, as usual, it is typical of the situation when c2 = 0. When c1 = 0, (2.9) becomes φ2 (a, λ)φ1 (−X, λ) − {φ1 (a, λ) − ρ1 (λ)}φ2 (−X, λ) = 0.

(3.2)

Before proceeding further with (3.1) and (3.2), it is convenient at this point to refer to the familiar Dirichlet and Neumann problems for (1.1) over

6


the basic periodicity interval (0, a) [6, section 3.1]. In any instability interval, each of these problems has a unique eigenvalue ΛD and ΛN for which φ1 (a, ΛN ) = 0

φ2 (a, ΛD ) = 0,

[6, Theorem 3.1]. As we indicated in section 2, there is therefore the possibility that (2.4) is a trivial linear combination when λ = ΛD . But both cannot occur together should ΛD = ΛN . In case (2.4) is trivial, (2.9), (3.1) and (3.2) can be expressed instead interms of (2.5). Then (3.2) for example becomes {φ2 (a, λ) − ρ1 (λ)}φ1 (−X, λ) − φ1 (a, λ)φ2 (−X, λ) = 0.

(3.3)

In what follows, we avoid this slight complication by simply excluding the value ΛD from our considerations. Thus we work with (3.2) (λ = ΛD ) rather than (3.3). We can now state and prove a theorem which, in general terms, is the main result of the paper. It contains a general condition (3.4) which will be analysed in subsequent sections. The theorem concerns λ-solutions of (3.2) lying in a spectral gap (λ , λ ) of (1.1), and we recall that these solutions are the eigenvalues induced in the gap by the interface represented by (1.6) and (1.7) with a√variable X being √ allowed for. A similar result holds for (3.1). We write ν = λ and ν = λ . Theorem 3.1. Suppose that there exist a fixed number K (K > 0) such that ν − ν ≥ K

(3.4)

for a sequence of spectral gaps receding to infinity. Then there is a number ν0 (X) such that (3.2) has λ-solutions in (λ , λ ) when ν ≥ ν0 (X), and the number of such solutions exceeds 3KX/4π − 4.

(3.5)

Further, if q1 is defined in (−∞, 0) and satisfies q1 ∈ L(−∞, 0),

(3.6)

then ν0 (X) can be taken to be independent of X. Proof. We begin by applying Lemma 2.1 to (1.6) on [−X, 0). By (2.11) we are considering values of ν such that 0 ν≥ |q1 (t)|dt. (3.7) −X

Then, recalling the initial values (2.1), we obtain from (2.13) and (2.14) φ1 (−X, λ) = ν(1 +

1 E1 ) sin νX + F1 cos νX ν

φ2 (−X, λ) = E2 sin νX + (1 + F2 ) cos νX, where E1 = E1 (−X) etc, and |E1 |, |F1 | ≤ (e − 1)

0

−X

|q1 (t)|dt := I(X),

(3.8)


7

say, and |E2 |, |F2 | ≤

1 I(X). ν

(3.9)

On substituting into (3.2), we obtain (1 + ν1 E1 ) sin νX + ν1 F1 cos νX 1 = H(ν), E2 sin νX + (1 + F2 ) cos νX ν

(3.10)

H(ν) = {φ1 (a, λ) − ρ1 (λ)}/φ2 (a, λ) (λ = ΛD ).

(3.11)

where Let us now denote the left-hand side of (3.10) by T (ν) (ν < ν < ν ). We wish to show that T (ν) behaves sufficiently like tan νX, the relevant property of the latter being that it increases from −∞ to +∞ in any ν-interval 1 1 ((k − )π/X, (k + )π/X) 2 2

(3.12)

with k an integer. Then, in any such interval which lies within (ν , ν ), the graph of tan νX crosses that of ν1 H(ν), producing a solution of the equation tan νX = ν1 H(ν) (excepting possibly only an interval (3.12) which contains the point ΛD ). Since, however, we have T (ν) rather than tan νX, we begin by considering instead of (3.12) intervals I(k) of the form 1 1 ({(k − )π + η}/X, {(k + )π − η}/X), 2 2

(3.13)

sin η = I(X)/{ν − I(X)}.

(3.14)

where Also, further to (3.7), we assume that ν > 3I(X),

(3.15)

where I(X) is as in (3.8). Then (3.14) and (3.15) imply that 0 < η < π/6.

(3.16)

Denoting the denominator in T (ν) by C(ν), we show first that C(ν) > 0 or C(ν) < 0 in I(k) according to k being even or odd. Taking k even (odd is similar), it follows from (3.9) that, in I(k), C(ν) := E2 sin νX + (1 + F2 ) cos νX > −

1 1 I(X) + {1 − I(X)} sin η = 0 ν ν

by (3.14), as required. Next, considering also I(k + 1) in (3.13), it follows from what we have just proved that, in the interval 1 1 L(k) = [{(k + )π − η}/X, {(k + )π + η}/X], 2 2

(3.17)

8


C(ν) has a least zero νl (k) and a greatest zero νg (k). Also, in L(k), the numerator S(ν) in (3.10) is positive (k even) or negative (k odd). This is easily seen because (if k is even for example) 1 1 E1 ) sin νX + F1 cos νX ν ν 1 1 ≥ {1 − I(X)} cos η − I(X) sin η ν ν 1 > (2 cos η − sin η) > 0. 3

S(ν) := (1 +

Altogether, then, we have shown that T (ν) → +∞ as

ν → νl (k) − 0

and T (ν) → −∞ as

ν → νg (k − 1) + 0

with T (ν) continuous in (νg (k − 1), νl (k)). Further, both νg (k − 1) and νl (k) lie in the interval L(k − 1) ∪ I(k) ∪ L(k),

(3.18)

which is an interval of total length (π + 2η)/X. It follows that (3.10) has at least one solution ν in each interval (3.18), with the possible exception of at most two intervals which contain ΛD . The number of complete intervals (3.18) which lie in [ν , ν ] is at least KX/(π + 2η) − 2, by (3.4). Then, discounting at most two intervals which contain ΛD and using (3.16), we arrive finally at (3.5) with 0 |q1 (t)|dt (3.19) ν0 (X) = 3(e − 1) −X

by (3.8) and (3.15). Also, the concluding statement of the theorem, where (3.6) holds, is now clear There is one further observation to be made concerning the condition (3.6). If we consider X → ∞ in (1.7), our eigenvalue problem approximates to the problem with two singular end-points where the differential equation is (1.6) in (−∞, 0) and (1.1) in (0, ∞).When (3.6) holds, this latter problem has [λ0 , ∞) as an interval of continuous spectrum [23, sections 3.1, 3.8, 5.6 and 5.14]. Our Theorem 3.1 is therefore in accord with this property in that, at least for ν > ν0 (∞) (see(3.19)), (3.13), (3.17) and (3.18) show that the interface eigenvalues become everywhere dense in (ν , ν ) as X → ∞, thus filling up the spectral gaps of (1.1). Finally, we note that Theorem 3.1 continues to apply when (3.4) holds, not for a fixed K, but for a sequence Kn → 0 as n → ∞. Then for (3.5) to guarantee at least one eigenvalue, we require Kn ≥ 20π/3X. Thus only a finite number of the Kn is allowed for a fixed X.


9

4. Two discontinuities In this section and the next we suppose that the weight function w(x) in (1.1) is given by w1 (x) (0 ≤ x < a1 ) w(x) = (4.1) w2 (x) (a1 ≤ x < a), where w1 and w2 have continuous second derivatives in [0, a1 ] and [a1 , a] respectively, but w1 (a1 − 0) = w2 (a1 ) and w2 (a − 0) = w1 (0). (To be brief, we omit “−0” in the sequel.) We write σ1 = (w2 /w1 )1/4 (a1 ),

σ2 = (w1 /w2 )1/4 (a).

(4.2)

At this point we note that, in the step-function case where w1 and w2 are different constants, σ1 and σ2 are connected by the relation σ1 σ2 = 1. We shall refer later to this relation, but our analysis is not dependent on it. In order to examine the spectral gaps in the case (4.1), and in particular to verify (3.4), we need to determine the eigenvalues λn and μn associated with (1.2) and (1.3). These eigenvalues are the solutions of the equations D(λ) = ±2 with D as in (2.3) [6, chapter√2]. In the following lemma, we obtain the form of D(λ), at least for ν(= λ) large enough. In the lemma and its proof we use the general notation M (ν) to denote any expression satisfying an inequality |M (ν)| ≤ C (4.3) where C is independent of ν and is expressible explicitly in terms of w1 , w2 and q. Lemma 4.1. With the notation (4.1)–(4.3), 1 ) cos νA1 cos νA2 φ1 (a, λ) + φ2 (a, λ) = (σ1 σ2 + σ1 σ2 σ1 σ2 1 −( + ) sin νA1 sin νA2 + M (ν), σ2 σ1 ν where a a 1

A1 =

0

and ν≥

a

0

1/2

w1 (x)dx,

1/2

A2 =

(4.4)

w2 (x)dx

(4.5)

|(qw−1/2 − w−1/4 (w−1/4 ) )(x)|dx.

(4.6)

a1

Proof. We make the Liouville transformation of (1.1) in each of the two intervals indicated in (4.1): x 1/2 −1/4 z1 , t1 = 0 w1 (u)du (0 ≤ x < a1 ) y = w1 x 1/2 (4.7) −1/4 y = w2 z2 , t2 = a1 w2 (u)du (a1 ≤ x < a) This gives where

d2 zj /dt2j + {λ − Qj (tj )}zj = 0 −3/4

Qj (tj ) = (q/wj − wj

−1/4

(wj

) )(x)

(4.8) (4.9)

10


and 0 ≤ tj < Aj (j = 1, 2) [6, section 4.1]. The solutions φ1 and φ2 of (1.1) are transformed into solutions Φ1,j and Φ2,j of (4.8), and we shall obtain (4.4) by applying Lemma 2.1 to (4.8) We first note the connection between the values of z1 and z2 and their derivatives which follows from the continuity of y and y at x = a1 . Thus, from (4.7), we have 1/4

z2 (0) = [w2 y](a1 ) = σ1 z1 (A1 ) 1 dz1 dz2 (0) = W (a1 )z1 (A1 ) + (A1 ) dt2 σ1 dt1 where σ1 is as in (4.2) and −1/2

W = w2

1/4

−1/4

(w2 w1

).

(4.10) (4.11)

(4.12)

Let us now consider φ1 and its transforms Φ1,1 and Φ1,2 . By (2.1) and (4.7), Φ1,1 (t1 , λ) has the initial values 1/4

−1/2

w1 (0),

[w1

1/4

(w1 ) ](0)

at t1 = 0. Then Lemma 2.1 gives 1 −1/2 1/4 1 1/4 Φ1,1 (A1 , λ) = w1 (0) cos νA1 + [w1 (w1 ) ](0) sin νA1 + E1 (ν) (4.13) ν ν dΦ1,1 1/4 −1/2 1/4 (A1 , λ) = −νw1 (0) sin νA1 + [w1 (w1 ) ](0) cos νA1 + F1 (ν) dt1 (4.14) where A1 1 −1/2 1/4 1/4 (w1 ) ](0)|}(e − 1) |Q1 (t1 )|dt1 |E1 (ν)|, |F1 (ν)| ≤ {w1 (0) + |[w1 ν 0 (4.15) and ν≥

A1

0

|Q1 (t1 )|dt1

(4.16)

by (2.11). Using | sin νA1 | ≤ 1 and | cos νA1 | ≤ 1, we can write (4.13) and (4.14) as 1 1/4 Φ1,1 (A1 , λ) = w1 (0) cos νA1 + M (ν) ν dΦ1,1 1/4 (A1 , λ) = −νw1 (0) sin νA1 + M (ν) dt1 where M (ν) has the form indicated in (4.3). It then follows from (4.10) and (4.11) that the initial values of Φ1,2 (t2 ) at t2 = 0 are 1 1/4 σ1 w1 (0) cos νA1 + M (ν) ν and ν 1/4 (4.17) − w1 (0) sin νA1 + M (ν). σ1


11

Turning to Lemma 2.1 again, applied now to Φ1,2 (t2 , λ) over the interval 0 ≤ t2 ≤ A2 , and using the initial values (4.17), we have 1 1 sin νA1 sin νA2 } + M (ν) σ1 ν (4.18) dΦ1,2 1 1/4 (A2 , λ) = −w1 (0)ν{σ1 cos νA1 sin νA2 + sin νA1 cos νA2 } + M (ν) dt2 σ1 (4.19) where, by (2.11), A2 |Q2 (t2 )|dt2 . (4.20) ν≥ 1/4

Φ1,2 (A2 , λ) = w1 (0){σ1 cos νA1 cos νA2 −

0

Next we consider similarly φ2 and its transforms Φ2,1 and Φ2,2 . By (2.1) and (4.7), Φ2,1 (t1 , λ) has the initial values 0,

−1/4

w1

(0)

at t1 = 0. Then Lemma 2.1 gives 1 −1/4 1 w (0) sin νA1 + 2 M (ν) ν 1 ν 1 dΦ2,1 −1/4 (A1 , λ) = w1 (0) cos νA1 + M (ν), dt1 ν Φ2,1 (A1 , λ) =

where (4.16) holds. It then follows from (4.10) and (4.11) that the initial values of Φ2,2 (t2 , λ) at t2 = 0 are 1 σ1 −1/4 w (0) sin νA1 + 2 M (ν) ν 1 ν and 1 −1/4 1 w1 (0) cos νA1 + M (ν). σ1 ν

(4.21)

Turning to Lemma 2.1 once again, applied now to Φ2,2 (t2 , λ) over the interval 0 ≤ t ≤ A2 , and using the initial values (4.21), we have 1 −1/4 1 1 w (0){σ1 sin νA1 cos νA2 + cos νA1 sin νA2 } + 2 M (ν) ν 1 σ1 ν (4.22) dΦ2,2 1 1 −1/4 (A2 , λ) = w1 (0){−σ1 sin νA1 sin νA2 + cos νA1 cos νA2 }+ M (ν) dt2 σ1 ν (4.23) where (4.20) holds. To complete the proof of (4.4), we use (4.7), (4.18), (4.22) and (4.23) to revert to the values of φ1 and φ2 at x = a. Thus, with σ2 as in (4.2),

Φ2,2 (A2 , λ) =

−1/4

φ1 (a, λ) = w2

(a)Φ1,2 (A2 , λ)

= σ1 σ2 cos νA1 cos νA2 −

σ2 1 sin νA1 sin νA2 + M (ν) σ1 ν

(4.24)

12


and dΦ2,2 (A2 , λ) dt2 1 σ1 1 = cos νA1 cos νA2 − sin νA1 sin νA2 + M (ν). σ1 σ2 σ2 ν −1/4

φ2 (a, λ) = (w2

1/4

) (a)Φ2,2 (A2 , λ) + w2 (a)

(4.25)

Now (4.4) follows since (4.6) accommodates both (4.16) and (4.20), and the lemma is proved. Let us now write (4.4) as D(λ) = where

1 1 1 1 1 1 1 (σ1 + )(σ2 + ) cos νI + (σ1 − )(σ2 − ) cos νJ + M (ν), 2 σ1 σ2 2 σ1 σ2 ν (4.26) a I = A1 + A2 = w1/2 (x)dx, J = |A1 − A2 |. (4.27) 0

Then the equations D(λ) = ±2 for the periodic and semi-periodic (respectively) eigenvalues become cos νI = f± (ν) +

1 M (ν), ν

(4.28)

where f± (ν) = (σ1 +

1 −1 1 1 1 ) (σ2 + )−1 {−(σ1 − )(σ2 − ) cos νJ ± 4}. (4.29) σ1 σ2 σ1 σ2

In the next section, we discuss the solutions of (4.28), and we are interested in identifying situations where the main condition (3.4) in Theorem 3.1 is satisfied, and the ν1 term in (4.28) does not materially affect the analysis for large ν. We note in passing that a similar type of equation to (4.28) and (4.29) (with σ1 σ2 = 1) appears in [11, (1.5)] for a different step-function example, and the difficulty of analysing the equation is commented upon. Nevertheless, even without the restriction to the step-function relation σ1 σ2 = 1, we shall extract sufficient information for our purposes from (4.28) and (4.29).

5. Lower bounds concerning the spectral gaps We first note that the solutions of (4.28)–(4.29) are known from other sources, but not

sufficiently

accurately for our purposes. Thus [7, Theorem 1], [16], for ν = λ2m+1 or λ2m+2 , we have |ν − 2(m + 1)πI −1 | ≤ I −1 (ω1 + ω2 ) + o(1) (m → ∞), where

(5.1)

1 1 1 (5.2) ωj = tan−1 ( |σj − |) (0 < ωj < π) 2 σj 2 √ √ and I is as in (4.27). For ν = μ2m or μ2m+1 , we replace 2(m + 1) by 2m + 1 in (5.1). However, (5.1) only requires w1 and w2 to be differentiable


13

once. Under our assumptions of twice differentiability, we can go further in the following proposition which identifies a situation where (3.4) is satisfied. Proposition 5.1. Suppose that σ1 = σ2 and that σ1 σ2 = 1. Then

λ − λ2m+1 } ≥ 2α/I + O(m−1 ) √ √ 2m+2 μ2m+1 − μ2m

(5.3)

as m → ∞, where cos α = (σ1 +

1 −1 1 1 1 ) (σ2 + )−1 {|(σ1 − )(σ2 − )| + 4} σ1 σ2 σ1 σ2

(5.4)

and 0 < α < π/2. Proof. We first check that α is well defined by (5.4). Thus we need to check that 1 1 1 1 |(σ1 − )(σ2 − )| + 4 < (σ1 + )(σ2 + ). (5.5) σ1 σ2 σ1 σ2 There are two cases to consider: (i) σ1 > 1 and σ2 < 1 (or vice versa) (ii) σ1 > 1 and σ2 > 1 (or σ1 < 1 and σ2 < 1). In case (i), (5.5) simplifies to σ1 σ2 +1/σ1 σ2 > 2, which is true when σ1 σ2 = 1. In case (ii), (5.5) simplifies to σ1 /σ2 + σ2 /σ1 > 2, which is true when σ1 = σ2 . To obtain the

upper inequality in (5.3) we first note from (5.1) and (5.2)

that λ2m+1 and λ2m+2 lie in the open interval ((2m+1)π/I, (2m+3)π/I). We therefore consider ν to lie in this interval and let ν1 and ν2 be the solutions of 1 cos νI = cos α + M (ν), ν where M (ν) is as in (4.28). Thus ν1 , ν2 = 2(m + 1)π/I ± α/I + O(m−1 ).

(5.6)

In (ν1 , ν2 ) we have 1 1 M (ν) ≥ f+ (ν) + M (ν) ν ν

by (4.29) and (5.4), and hence D(λ) > 2 in (ν1 , ν2 ). Hence λ2m+1 ≤ ν1 and

√ √ λ2m+2 ≥ ν2 , and (5.3) follows from (5.6). The proof for μ2m+1 − μ2m is similar , completing the proof of the proposition. cos νI > cos α +

There are two, more specialised situations where (5.3) can be improved to an asymptotic formula. The cases are described by I and J in (4.27), and we present them in the following subsections.

14


5.1. The case I = 2J (= 0) Proposition 5.2. Let I = 2J. Then there are numbers ψ± and ω± with ψ− ∈ (π/2, π], ω− ∈ [0, π/2), ψ+ ∈ (π/2, π), ω+ ∈ (0, π/2) such that, as m → ∞,

λ4m−1

λ4m

λ4m+1

λ4m+2 √ μ4m √ μ4m+1 √ μ4m+2 √ μ4m+3

= 2(2mπ − ω− )/I + O(m−1 ) = 2(2mπ + ω− )/I + O(m−1 ) = 2(2mπ + ψ− )/I + O(m−1 ) = 2{2(m + 1)π − ψ− }/I + O(m−1 ) = 2(2mπ + ω+ )/I + O(m−1 ) = 2(2mπ + ψ+ )/I + O(m−1 ) = 2{2(m + 1)π − ψ+ }/I + O(m−1 ) = 2{2(m + 1)π − ω+ }/I + O(m−1 ).

Further, ψ− < π if σ1 = σ2 and ω− > 0 if σ1 σ2 = 1. Proof. Defining θ = νJ and using I = 2J, we write (4.28)-(4.29) as 1 1 1 1 1 (σ1 + )(σ2 + )(2 cos2 θ−1)+(σ1 − )(σ2 − ) cos θ∓4 = M (ν). (5.7) σ1 σ2 σ1 σ2 ν The left-hand side here is a quadratic p− (c) (or p+ (c)) in cos θ (c = cos θ). It is easy to check that p± (0) < 0 and √ 1 p± (1) = 2( σ1 σ2 ± √ )2 ≥ 0 σ1 σ2

p± (−1) = 2( σ1 /σ2 ± σ2 /σ1 )2 ≥ 0, where the inequalities are strict for p+ while, for p− , the first inequality is strict if σ1 σ2 = 1 and the second if σ1 = σ2 . Thus each p± (c) has two distinct zeros c± and d± with −1 < c+ < 0,

0 < d+ < 1

−1 ≤ c− < 0, 0 < d− ≤ 1 and c− > −1 if σ1 = σ2 and d− < 1 if σ1 σ2 = 1.

Now define ψ± and ω± by cos ψ± = c± cos ω± = d±

(π/2 < ψ+ < π, π/2 < ψ− ≤ π) (0 < ω+ < π/2 0 ≤ ω− < π/2)

Then, for any integer m > 0, consider θ in the ranges 2mπ − ω− < θ < 2mπ + ω− 2mπ + ψ− < θ < 2(m + 1)π − ψ−

.

(5.8)

2mπ + ω+ < θ < 2mπ + ψ+ 2(m + 1)π − ψ+ < θ < 2(m + 1)π − ω+

(5.9) .

(5.10)


15

The ranges (5.9) are maximal ones in which p− (c) > 0, and (5.10) are maximal ones in which p+ (c) < 0. By (4.26) and (5.7), these ranges (5.9) and (5.10) correspond to the spectral gaps in which D(λ) > 2 and D(λ) < −2 respectively, with an error O(m−1 ) when m is large. The spectral gaps are the intervals (λ2n+1 , λ2n+2 ) and (μ2n , μ2n+1 ) and, since θ = νJ = 12 νI, the values stated in the proposition follow from (5.9) and (5.10),√where the choices n = 2m − 1, n = 2m, n = 2m + 1 tally with (5.1) (for λ) and with the √ corresponding formula for μ. By (4.27), the criterion I = 2J means that A1 = 3A2 (or A2 = 3A1 ), and a step-function of this type was examined in [6, section 2.2] (see also [10, section 50]). Proposition 5.2 generalises these findings in [6]. It is also possible to consider in the same way (N − 1)A1 = (N + 1)A2 , N being a positive integer, and the criterion is then I = N J. As a corollary, we state the conclusions of Proposition 5.2 in a form which shows that (3.4) is satisfied. Corollary 5.3. Let I = 2J. Then

4ω− /I + O(m−1 ) (n = 2m − 1) λ2n+2 − λ2n+1 = 4(π − ψ− )/I + O(m−1 ) (n = 2m), √ √ μ2n+1 − μ2n = 2(ψ+ − ω+ )/I + O(m−1 ), where ω− > 0 if σ1 σ2 = 1 and π − ψ− > 0 if σ1 = σ2 5.2. The case J = 0 When J = 0 (i.e., A1 = A2 ), (4.28) simplifies to 1 −1 1 −1 1 1 1 ) (σ2 + ) {−(σ1 − )(σ2 − ) ± 4} + M (ν). σ1 σ2 σ1 σ2 ν In the same way as in the proof of Proposition 5.2 we write this equation as cos νI = (σ1 +

cos νI = cos α± +

1 M (ν) ν

(5.11)

where (as is easily verified) 0 ≤ α+ < π/2,

π/2 < α− ≤ π.

Further, α+ > 0 if σ1 σ2 = 1 and α− < π if σ1 = σ2 . Then (5.11) gives

λ2m+2 , λ2m+1 = 2(m + 1)π/I ± α+ /I + O(m−1 ) √ √ μ2m+1 , μ2m = (2m + 1)π/I ± (π − α− )/I + O(m−1 ). Thus, corresponding to Corollary 5.3, we have the simpler formulae

λ2m+2 − λ2m+1 = 2α+ /I + O(m−1 ) √ √ μ2m+1 − μ2m = 2(π − α− )/I + O(m−1 ), giving another situation where (3.4) is satisfied.

(5.12)

16


6. One discontinuity Here we mention briefly the situation where w(x) has just one discontinuity, which we can take as w(0) = w(a), and w(x) is otherwise twice differentiable as before. The periodic and semi-periodic eigenvalues have been determined asymptotically in [7, section 3] (see also [16]) using the Liouville transformation as in (4.7). In place of (4.28), we have the simpler equation cos νI = ±2σ/(σ 2 + 1) +

1 M (ν), ν

(6.1)

where σ = {w(0)/w(a)}1/4 . Formally, this tallies with (4.28)–(4.29) if the discontinuity at a1 is made to disappear by taking σ1 = 1. If we now define α by cos α = 2σ/(σ 2 + 1) (0 < α < π/2), we have the same situation in (6.1) as in (5.11) but now with α+ = α and α− = π − α. Then, referring again to (3.4), (5.12) shows that ν − ν = 2α/I + O(m−1 ) for all spectral gaps as m → ∞. Although Theorem 3.1 is applicable, there is no simple illustrative example when w(x) has just the one discontinuity. A step-function is excluded, and the next obvious example is w(x) = x + 1 and q(x) = 0 for which (1.1) has the Bessel function solutions (x + 1)1/2 J 13 { 23 λ1/2 (x + 1)3/2 } and (x + 1)1/2 Y 13 { 32 λ1/2 (x + 1)3/2 } [23, section 4.13].

7. Step-function examples In this section we take q = 0, q1 = 0 and w to be a step-function. Then (1.1) and (1.6) can be solved explicitly in terms of trigonometric or hyperbolic functions. The E and F terms in (3.8)-(3.10), and the M (ν) terms in section 4, are now all zero, and there is no need for a restriction on ν such as (3.15). Since w is a step function, (4.2) gives σ1 = 1/σ2

(= σ, say),

(7.1)

and the formulae (4.24)-(4.26) becomes D(λ) =

1 1 1 1 (σ + )2 cos νI − (σ − )2 cos νJ 2 σ 2 σ

1 1 1 σ{(σ + ) cos νI + (σ − ) cos νJ} 2 σ σ 1 1 −1 1 φ2 (a, λ) = σ {(σ + ) cos νI − (σ − ) cos νJ}. 2 σ σ We also require the similar formula for φ2 (a, λ) which, by (4.22) and is now 1 1 1 φ2 (a, λ) = ν −1 {w1 (0)w2 (a)}−1/4 {(σ + ) sin νI + (σ − ) sin νJ}. 2 σ σ φ1 (a, λ) =

(7.2) (7.3) (7.4) (4.7), (7.5)


17

In section 1, we have indicated that there are no induced eigenvalues in the first spectral gap, which is now (−∞, 0), and we deal with this point first. Then we move on to the other spectral gaps. 7.1. The first gap (−∞, 0) Proposition 7.1. Let q = q1 = 0 and let w be a step-function. Then (3.2) has no solution in (−∞, 0). Proof. In √ the first gap, D(λ) > 2 [6, section 2.3] and hence (2.6) gives ρ1 = 1 {D − D2 − 4}. Then, by (2.3), (3.10) becomes 2

(7.6) tan νX = {φ1 (a, λ) − φ2 (a, λ) + D2 (λ) − 4}/{2νφ2 (a, λ)}. √ Since we are considering λ < 0, we write ν = λ = iμ with real μ > 0. By (7.2)-(7.4), the numerator in (7.6) is real, and we claim now that it is positive, that is, D2 (λ) − 4 > {φ1 (a, λ) − φ2 (a, λ)}2 . By (2.3), this reduces to φ1 (a, λ)φ2 (a, λ)} > 1 which, by (7.3) and (7.4), in turn becomes 1 1 (σ + )2 cosh2 μI − (σ − )2 cosh2 μJ > 4, σ σ i.e., 1 (σ 2 + 2 )(cosh2 μI − cosh2 μJ) + 2(sinh2 μI + sinh2 μJ) > 0. σ This inequality is clearly true since I > J by (4.27) . Thus, by (7.5), (7.6) can be written 1 1 tanh μX = −{positive}{w1 (0)w2 (a)}1/4 /{(σ + ) sinh μI + (σ − ) sinh μJ}. σ σ Since the denominator here is also positive, there are no solutions for μ > 0 as required. 7.2. Induced eigenvalues in all other gaps We now show that it is possible for eigenvalues to appear in all the spectral gaps (except (−∞, 0)) if X is chosen suitably. The example which we highlight here is where 1 a = 1, a1 = , w1 (x) = 9, w2 (x) = 1 2 and, consequently, σ(= σ1 ) = √13 [6, pp. 25-26]. Also, (4.27) gives I = 2 and J = 1. These values are substituted into (7.2)–(7.5). Then, for λ > 0 and ν real, (3.10) becomes tan νX = H± (ν) (7.7) where, as in (7.6), H± (ν) =

1 {4(cos 2ν − cos ν) ± 3 D2 (λ) − 4}/(2 sin 2ν − sin ν) 2

(7.8)

18


and 2 (4 cos 2ν − cos ν). 3 Here H+ refers to a λ-gap in which D(λ) > 2, and H− to a μ-gap in which D(λ) < −2. The periodic eigenvalues are given by

λ4m−1 = λ4m = 2mπ,

λ4m+1 = 2mπ + α, λ4m+2 = 2(m + 1)π − α, (7.9) D(λ) =

and the semi-periodic eigenvalues by √ √ μ4m = 2mπ + β, μ4m+1 = 2mπ + γ, √ √ μ4m+2 = 2(m + 1)π − γ, μ4m+3 = 2(m + 1)π − β

(7.10)

where α = cos−1 (−7/8)

(π/2 < α < π), √ √ β = cos−1 {(1 + 33)/16}, γ = cos−1 {(1 − 33)/16}

(0 < β < γ < π)

[6, p. 26]. These values are in line with the case ω− = 0, ψ− < π of Proposition 5.2. We begin our consideration of (7.7) by examining the first λ-gap (λ1 , λ2 ). The corresponding ν-interval as given by (7.9) is (α, 2π − α) and, in this interval, H+ (ν) has a vertical asymptote at ν = π (corresponding to the Dirichlet eigenvalue ΛD = π 2 ). The graph of H+ (ν) can be exhibited using Mathematica. Consequently, √ we can say that, as ν increases from α to π, H+ (ν) decreases from − 13 15 to −∞ and, as ν increases from π to 2π − α, √ H+ (ν) decreases from +∞ to 13 15. The solutions of (7.7) in (α, 2π − α) can be obtained computationally for a mesh of values of X. This interval is (2.64, 3.65) to 2d.p. and there are no solutions for X = 1 and X = 2. For other integer values of X there are solutions 2.73 and 3.55 (X = 3), 2.81 and 3.47 (X = 4), 2.86 and 3.42 (X = 5)), and there are four solutions when X = 10. The number of solutions generally increases with X in line with the method of proving Theorem 3.1. The situation for a general λ-gap (λ4m+1 , λ4m+2 ) is similar, and it is particularly simple when X is an integer because then, by (7.9), the problem in terms of ν is just translated by 2mπ, and (7.7) is invariant under this translation. The situation is also similar for the μ-gaps, and we only mention the first one (μ0 , μ1 ). The corresponding ν-interval as given by (7.10) is (β, γ), which is (1.14, 1.87). In this interval, H− (ν) has a vertical asymptote at ν = cos−1 (1/4) = 1.32 (corresponding to a Dirichlet eigenvalue). As with H+ (ν), H− (ν) decreases from −3.43 to −∞ and from +∞ to 0.51 in the two parts of (β, γ). The ν-solutions of (7.7) are 1.68 (X = 1), none (X = 2), 1.52 (X = 3), 1.18 and 1.81 (X = 4), 1.53 (X = 5), 1.41 and 1.69 (X = 10). Again, these values can be translated by 2mπ for other μ-gaps.


19

7.3. Repeated eigenvalues The step-function example in section 7.2 is described in [10, section 50] as exhibiting a remarkable distribution of single and double eigenvalues for periodic boundary conditions (1.2) (i.e., infinitely many of both simple and double periodic eigenvalues). Here, as an aside to the main topic of the paper, we take the opportunity to develop this observation from [10] by considering I = N J in (7.2), where N is an integer. Thus (7.2) is now 1 1 1 D(λ) = (σ + )2 cos(νN J) − (σ − )2 cos νJ 2 σ σ and we consider the following choices for ν with m as a positive integer. 1. ν = 2mπ/J (= 2mN π/I). Here D(λ) = 2 and D (λ) = 0. Hence (2N mπ/I)2 are double periodic eigenvalues, as in section 7.2 where N = 2. 2. ν = (2m + 1)π/J (= (2m + 1)N π/I), N odd. This time D(λ) = −2 and D (λ) = 0. Hence {(2m + 1)N π/I}2 are double semi-periodic eigenvalues. Thus we can add to the observation in [10] by noting that, when N is odd, there are infinitely many double eigenvalue in both the periodic and semiperiodic problems. For further information on double eigenvalues, we refer to [12].

References [1] L. Aceto, P. Ghelardoni, and M. Marletta. Numerical computation of eigenvalues in spectral gaps of Sturm-Liouville operators. J. Comput. Appl. Math., 189(1-2):453–470, 2006. [2] J. M. Almira and P. J. Torres. Invariance of the stability of Meissner’s equation under a permutation of its intervals. Ann. Mat. Pura Appl. (4), 180(2):245–253, 2001. [3] B. M. Brown, M. S. P. Eastham, A. M. Hinz, and K. M. Schmidt. Distribution of eigenvalues in gaps of the essential spectrum of Sturm-Liouville operators—a numerical approach. J. Comput. Anal. Appl., 6(1):85–95, 2004. [4] E. A. Coddington and N. Levinson. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [5] T. Dohnal , M. Plum, and W. Reichel. Localized modes of the linear periodic Schr¨ odinger operator with a nonlocal perturbation. SIAM J. Math. Anal., 41(5):1967-1993, 2009. [6] M. S. P. Eastham. The spectral theory of periodic differential equations. Scottish Academic Press, Edinburgh, 1973. [7] M. S. P. Eastham. Results and problems in the spectral theory of periodic differential equations. In Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad J¨ orgens), pages 126–135. Lecture Notes in Math., Vol. 448. Springer, Berlin, 1975. [8] M. S. P. Eastham, Q. Kong, H. Wu, and A. Zettl. Inequalities among eigenvalues of Sturm-Liouville problems. J. Inequal. Appl., 3(1):25–43, 1999.

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[9] M. S. P. Eastham and H. Kalf. Schr¨ odinger-type operators with continuous spectra, volume 65 of Research Notes in Mathematics. Pitman (Advanced Publishing Program), Boston, Mass., 1982. [10] W. N. Everitt. A catalogue of Sturm-Liouville differential equations. In SturmLiouville theory, pages 271–331. Birkh¨ auser, Basel, 2005. [11] S. Gan and M. Zhang. Resonance pockets of Hill’s equations with two-step potentials. SIAM J. Math. Anal., 32(3):651–664 (electronic), 2000. [12] H. Hochstadt. A special Hill’s equation with discontinuous coefficients. Amer. Math. Monthly, 70:18–26, 1963. [13] E. Korotyaev. Lattice dislocations in a 1-dimensional model. Comm. Math. Phys., 213(2):471–489, 2000. [14] E. Korotyaev. Schr¨ odinger operator with a junction of two 1-dimensional periodic potentials. Asymptot. Anal., 45(1-2):73–97, 2005. ¨ [15] E. Meissner. Uber Sch¨ uttelswingungen in Systemen mit periodisch ver¨ anderlicher Elastizit¨ at. Schweizer. Bauzeitung, 72:95–98, 1918. [16] A. A. Ntinos. Lengths of instability intervals of second order periodic differential equations. Quart. J. Math. Oxford (2), 27(107):387–394, 1976. [17] M. Reed and B. Simon. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York, 1978. [18] F. S. Rofe-Beketov. A finiteness test for the number of discrete levels which can be introduced into the gaps of the continuous spectrum by perturbations of a periodic potential. Soviet Math., 5 689-692, 1964. [19] F. S. Rofe-Beketov. Kneser constants and effective masses for band potentials. Soviet Physics, 29:391–393, 1984. [20] K. M. Schmidt. Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators. Comm. Math. Phys., 211(2):465– 485, 2000. [21] A. V. Sobolev. Weyl asymptotics for the discrete spectrum of the perturbed Hill operator. In Estimates and asymptotics for discrete spectra of integral and differential equations (Leningrad, 1989–90), volume 7 of Adv. Soviet Math., pages 159–178. Amer. Math. Soc., Providence, RI, 1991. [22] E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Part 2. Clarendon Press, Oxford, 1958. [23] E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Part 1. Second Edition. Clarendon Press, Oxford, 1962. B.M. Brown School of Computer Science Cardiff University, Cardiff CF24 3XF UK e-mail: [email protected] M.S.P. Eastham School of Computer Science Cardiff University, Cardiff CF24 3XF UK


On the HELP Inequality for Hill Operators on Trees B.M. Brown and K.M. Schmidt To David and Des

Abstract. The validity of a generalised HELP inequality for a Schr¨ odinger operator with periodic potential on a rooted homogeneous tree is related to the quasi-stability or quasi-instability of the associated differential equation. A numerical approach to the determination of the optimal constant in the HELP inequality is presented. Moreover, we give an example to illustrate that the generalised Weyl–Titchmarsh m function for the tree operator fails to capture all of its spectral properties. Mathematics Subject Classification (2010). Primary 34A40; Secondary 34B24, 34L40, 81Q35. Keywords. Differential inequality, regular quantum trees, Hill operator, Weyl–Titchmarsh function.

1. Introduction The classical HELP inequality for a Sturm–Liouville operator τf =

1 (−(pf ) + qf ) w

with locally integrable real-valued coefficients w, p1 , q; w, p > 0 on an interval (a, b) is 2 b b b 2 2 (p |f | + q |f | ) ≤K |f |2 w |τ f |2 w a

a

a

[6]. Typically one considers a situation where a is a regular, b a singular end-point in the limit-point case; the crucial point about the inequality is that, when valid, it holds for all functions f for which the derivatives exist in a weak sense and the right-hand side is finite, irrespective of the boundary values at a.

22

B.M. Brown and K.M. Schmidt

HELP is a generalisation of an inequality of Hardy and Littlewood [8] which covers the case w = p = 1, q = 0 and is valid with the optimal constant K = 4. The validity of the more general HELP inequality with some constant K can be shown to be equivalent to a certain property of the Weyl–Titchmarsh m function for the Sturm–Liouville operator in a cone-like neighbourhood of the imaginary axis [7]. The recent work [3] has shown that a similar criterion using a generalised m function can be obtained for a HELP inequality on trees of infinite length; in fact this is a particular instance of the abstract HELP inequality established by [1] and [10]. In the present study we focus on a situation which is analogous to Hill’s equation (the periodic Sturm–Liouville equation) on a half-line; we consider a tree composed of infinitely many identical intervals, each carrying the same potential q, and such that at each end-point with a single exception (the tree root) a fixed number of intervals are joined together. The spectrum of the graph Laplacian on such trees has been studied by [11]; it consists of bands of purely absolutely continuous spectrum with an additional eigenvalue in each gap and thus is analogous to that of a Hill operator on a half-line, except that the eigenvalues of the tree Laplacian have infinite multiplicity. [11] then went on to consider the effect of adding a decaying potential which is symmetric in the sense of only depending on the distance from the tree root. The generalised Hill operator on a perfectly homogeneous, rootless tree has been analysed in detail by [5] under the assumption that the potential, equal on each tree edge, is an even function on the interval. Our paper is organised as follows. In Section 2 we show, based on its strong limit-point property, that the maximal tree-Hill operator has deficiency indices (1, 1) and that, as a consequence, the Weyl solutions for nonreal spectral parameter are symmetric on the tree branches and their evolution is governed by a period-transfer matrix. This leads to an analogue to Floquet theory which we pursue in Section 3. In Section 4 we state the HELP inequality and study the associated generalised m function to relate the validity of the inequality to the quasi-stability or quasi-instability, in the sense of the generalised Floquet theory, of the tree-Hill equation at spectral parameter 0. Section 5 reports on a numerical approach to calculate the optimal HELP constant and shows results in the example of a shifted piecewise linear (sawtooth-type) potential in dependence on the constant offset. Finally, an appendix illustrates the observation that, due to the discrepancy between the finite deficiency indices of the operator and the a priori unboundedness of the multiplicity of its eigenvalues, the generalised m function does not carry complete spectral information, unlike the classical Sturm–Liouville case.

2. Hill operators on trees Consider a regular tree Γ with constant branching number b ∈ N+1 and fixed edge length l. Thus, b copies of the interval [0, l] (1st generation edges) are attached to the right-hand end-point of the interval [0, l] (the 0th generation),

Hill Operators on Trees

23

b copies (2nd generation edges) are attached to each of the b right-hand endpoints of 1st generation edges, and so on ad infinitum. The maximal domain for the tree Laplacian on Γ in the Hilbert space L2 (Γ) is D := {f : Γ → C | f, f a.c. on edges, f continuous, Kirchhoff conditions, f, f ∈ L2 (Γ)} (Kirchhoff conditions meaning, as usual, that at each junction the outgoing derivatives of f add up to 0). As shown in [3] Thm 4.1, the tree Laplacian on Γ, generally defined as −f for functions f satisfying the regularity, continuity and Kirchhoff conditions of D, has the Strong Limit-Point Property that for all f, g ∈ D,

lim f (x)g (x) = 0. r→∞

|x|=r

(here | · | denotes the metric distance from the tree root, 0). In fact, this property holds for general trees of infinite length, i.e., those for which any forward path can be extended indefinitely. In the following, we consider the tree-Hill operator on Γ, i.e., the operator −f + qf

(f ∈ D),

where q : Γ → R is an l-periodic, bounded function of the distance from the root only: in other words, each edge carries a (directed) copy of the same potential function. Thus the tree-Hill operator is a generalisation of the classical Hill, or one-dimensional Schrödinger, operator on a half-line. Let θ φ (·, λ) Φ(·, λ) = θ φ be the canonical fundamental system of the one-dimensional Schrödinger equation −u + qu = λu on the interval [0, l], λ ∈ C; then Φ(x, ·) is an entire function for each x ∈ [0, l]. For non-real spectral parameter, the strong limit-point property of the tree Laplacian yields the following characterisation of square-integrable solutions of the eigenvalue equation for the tree Laplacian. Theorem 2.1. Let λ ∈ C \ R and ψ ∈ D a solution of −f + qf = λf

(1)

on Γ. Then a) ψ can be taken to be non-trivial, and any other solution of (1) in D is linearly dependent on ψ; ˜ b) there is a function ψ˜ : [0, ∞) → C such that ψ(x) = ψ(|x|) (x ∈ Γ);

24 c)


˜ ˜ ψ(nl+) ψ((n − 1)l+) = A(λ) ˜ ψ˜ (nl+) ψ ((n − 1)l+) for all n ∈ N, with the transfer matrix θ(l, λ) φ(l, λ) A(λ) = 1 . 1 b θ (l, λ) b φ (l, λ)

Remark 2.2. Part a) shows that the minimal tree-Hill operator, defined on the subspace of D of functions of compact support in the interior of Γ, has deficiency indices (1, 1). Self-adjoint realisations are obtained by restricting the maximal operator, defined on D, by means of a boundary condition, e.g. of Dirichlet or Neumann type, at the tree root. Part b) shows that the generalised Weyl solutions, i.e., solutions in D for non-real spectral parameter, are all what we shall call symmetric functions in the following; these are functions which are symmetric under arbitrary permutations of the tree branches which leave the tree structure intact. We shall call Dsym := {f ∈ D | f symmetric} the symmetric subspace. Proof. a) Assume we have two linearly independent solutions ψ, ξ ∈ D of (1); let us assume for the moment that ψ(0), ξ(0) = 0 (the following argument ψ(0) ∈ C and will also show that this must always be the case). Then α := − ξ(0) f := ψ + αξ ∈ D is also a solution of (1), with f (0) = 0. Integrating by parts, we find

2 |f | = − f f + f (x) f (x) − f (0) f (0), Γr

Γr

|x|=r

where Γr := {x ∈ Γ | |x| ≤ r}; so passing to the limit and using the strong limit-point property, the boundary value at the root and the eigenvalue equation, Γ

|f |2 =

Γ

(λ − q) |f |2 .

Taking the imaginary parts on either side gives 0 = (Im λ)

Γ

|f |2 . In particu-

lar, if λ ∈ / R, then ψ and ξ are linearly dependent. Clearly, if either ψ(0) = 0 or ξ(0) = 0, the same argument gives ψ = 0 or ξ = 0, respectively. The existence of a non-trivial, symmetric solution of (1) can be inferred from the fact that, on the symmetric subspace, the tree-Hill operator is equivalent to a Sturm–Liouville operator with a singular right-hand end-point in the limit-point case; see [3] Section 3 for details. b) By part a), the function arising from ψ by any rearrangement of tree branches which leaves the distance to the root invariant must be a solution of (1) linearly dependent on ψ; as it coincides with ψ on the first edge, the two must be identical. Hence ψ itself is a symmetric function. c) In view of the symmetry shown in b), the continuity and Kirchhoff condition at the junctions joining the (n − 1)-st to the n-th generation edges

Hill Operators on Trees imply

25

˜ ˜ ψ(nl+) ψ(nl−) = 1 ˜ ψ˜ (nl+) b ψ (nl−)

for all n ∈ N. Also

˜ ˜ ψ((n − 1)l+) ψ(nl−) = Φ(l, λ) ψ˜ ((n − 1)l+) ψ˜ (nl−)

by solving the differential equation along the edges, so we get the transfer 1 0 relation with A(λ) = Φ(l, λ). 0 1b

3. Quasi-Floquet theory The transfer matrix A(λ) plays a similar role to the standard monodromy matrix in the Floquet theory of Hill’s equation. There is, however, the essential difference that it has determinant 1b instead of 1. From Theorem 2.1 c) it is clear that if ψ ∈ D is a non-trivial solution of (1) for λ ∈ C \ R, then ψ(0) will be an eigenvector of A(λ). The corresponding eigenvalue will ψ (0) have modulus < √1b for the following reason. Generally, if for any λ ∈ C, μ is an eigenvalue of A(λ) and u : Γ → C is the corresponding unique symmetric solution of (1) whose start phase vector is an eigenvector, u u (0) = μ (0), A(λ) u u then u will be equal on all tree edges, except for a factor μn on the n-th generation edges (of which there are bn copies). Hence the square integral of u is ∞ l ∞ l

√ 2 n 2n 2 2n |u| = b |μ| |u| = ( b|μ|) |u|2 , Γ

n=0

0

n=0

0

which is finite if and only if |μ| < √1b . Let us now consider a real spectral parameter. As uniqueness of solution of initial-value problems holds only on the symmetric subspace, the quasiFloquet theory will not give a complete overview of the solutions of (1), but only of symmetric solutions; however, we shall be interested in the situation of real λ as limiting values of λ in the complex upper half-plane where all solutions in D are symmetric. The two eigenvalues of A(λ) satisfy μ1 μ2 = 1b , μ1 + μ2 = D(λ), where D(λ) := traceA(λ) = θ(l, λ)+ 1b φ (l, λ) is a quasi-discriminant ; if λ ∈ R, then D(λ) ∈ R. Solving the characteristic equation, we find D(λ) ± D(λ)2 − 4b . μ= 2

26


Thus the eigenvalues are real and of equal sign if D(λ)2 > 4/b (and then both eigenvalues have equal sign), complex conjugates if D(λ)2 < 4/b, and in the limiting case they are ± √1b , with the same sign as D(λ).

In the case of real eigenvalues, the one closer to 0 has modulus |μ| < √1b . Hence the corresponding symmetric (quasi-Floquet) solution u whose start phase vector is an eigenvector for this eigenvalue, will be square-integrable; in the exceptional case where it also satisfies the boundary condition imposed at the tree root, λ will be an eigenvalue of the corresponding self-adjoint tree-Hill operator with symmetric eigenfunction. If the eigenvalues of A(λ) are non-real complex conjugates, they will have modulus |μ| = √1b , so the corresponding quasi-Floquet solutions are not square-integrable. Remark 3.1. We remark that the intervals on the real λ-axis where D(λ)2 > 4/b and where D(λ)2 < 4/b formally correspond to the instability and stability intervals of Hill’s equation, respectively; the terminology refers to the fact that for the periodic Sturm–Liouville equation the trivial solution u = 0 is respectively unstable or stable in the two situations. By analogy, we shall refer to them as quasi-instability and quasi-stability intervals; but note that the trivial solution of the equation on the tree may still be asymptotically stable for λ inside a quasi-instability interval: this will happen when √1b < |μ| < 1 for the eigenvalue of larger modulus. For the purpose of locating the quasi-stability and quasi-instability intervals, we observe that the transition points, i.e., the values of λ ∈ R where the quasi-discriminant is D(λ) = ± √2b and μ = ± √1b correspondingly, are the eigenvalues of an associated quasi-(anti-)periodic boundary value problem on the interval [0, l]. To see this, assume μ is an eigenvalue of A(λ) — for real λ — and v a corresponding eigenvector. Then the solution u of the differential equation −u + qu = λu u u on [0, l] with initial data (0) = v is given as = Φ(·, λ) v, and hence u u 1 0 1 0 u(0) u . (l) = Φ(l, λ) v = A(λ) v = μ v=μ 0 b 0 b bu (0) u At end-points of the quasi-stability intervals, μ = ± √1b , so we find that, depending on the sign, u will be a solution of the quasi-periodic boundary value problem √ 1 u (l) = b u (0), u(l) = √ u(0), b or of the quasi-anti-periodic boundary value problem √ 1 u (l) = − b u (0). u(l) = − √ u(0) b


27

4. Weyl–Titchmarsh m function and HELP inequality For spectral parameter λ in the complex upper half-plane, we can construct a Weyl–Titchmarsh m function in analogy to the standard Sturm–Liouville theory. Indeed, by Theorem 2.1 there is exactly one non-trivial solution ψ ∈ D (up to multiplication with a constant) and it is symmetric under branch permutations of the tree. The solution ψ cannot have ψ (0) = 0, as λ would then be a non-real eigenvalue of the self-adjoint realisation of our operator with Neumann boundary condition at the root; so by multiplication with a suitable constant we can assume that ψ (0) = 1 and hence represent ψ on [0, l] in terms of the canonical fundamental system, ψ = φ − mθ. The coefficient m gives the Weyl–Titchmarsh function; clearly ψ(0) = −m. In fact, m is the standard Weyl–Titchmarsh function for the Sturm–Liouville operator equivalent to the tree-Hill operator restricted to the symmetric subspace. In particular, m has the property that Im m(λ) > 0 (Im λ > 0). As observed of the previous section, the initial phase vector atthe beginning −m ψ(0) = is an eigenvector of the transfer matrix A(λ) with eigen1 ψ (0) value μ(λ) of modulus |μ(λ)| < √1b , which gives the following characterisation of the m as a function of the spectral parameter. (Note that μ(λ) denotes the eigenvalue of A(λ) of smaller modulus.) Lemma 4.1. For λ ∈ C \ R, m(λ) =

φ (l, λ) − bμ(λ) φ(l, λ) = . θ(l, λ) − μ(λ) θ (l, λ)

In particular, although m is the generalised Weyl–Titchmarsh function for the tree-Hill operator, it can be calculated from the solutions on the single interval [0, l] only. We remark that the relationship of this m function to the spectral properties of the tree-Laplace operator turns out to be more tenuous than in the case of Sturm–Liouville operators, where knowing the m function is equivalent to knowing the operator’s spectral function. From the point of view of Sturm–Liouville theory, it may already appear paradoxical that the minimal tree Laplacian has finite deficiency indices, as shown in Theorem 2.1, while it is well known [11] that the self-adjoint realisation of the Laplacian on Γ with Dirichlet boundary condition has eigenvalues of infinite multiplicity. Here it is important to observe that the statement of Theorem 2.1 a) only holds for non-real λ, but will be false for real λ in general, as is obvious from the proof. In particular, eigenfunctions for real λ need not be symmetric functions under branch permutations — each of the eigenvalues given in [11] has one symmetric eigenfunction, whereas the infinitely many additional eigenfunctions are not symmetric.

28


This observation may indicate some severe limitations to the scope of Titchmarsh-Weyl m function theory when applied to trees (or, more generally, graphs). The definition of the m function as characteristic coefficient in the representation of the distinguished square-integrable solution for non-real λ only captures the behaviour of the operator on the symmetric subspace and, in particular, will only flag a simple real eigenvalue where the tree Laplacian has in fact an eigenvalue of infinite multiplicity. The underlying reason for this effect is the lack of uniqueness of the solution of initial-value problems on the tree. Indeed, there are non-trivial solutions of (1) which vanish identically near the root. This peculiarity may seem inconsequential in the present case of a fully regular tree with constant branching number and edge lengths, where the spectrum of the Laplacian just consists of infinitely many copies of the spectrum on the symmetric subspace; but for more general trees one may expect to lose spectral information in the m function. We illustrate this effect with a simple example in the Appendix below. A similar spectral incompleteness of the generalised m function has been observed for operators of a different type by [2]. The properties of the function m play a decisive role for the validity of a HELP inequality for the tree-Hill operator, as shown in [3]. Theorem 4.2. a) The following statements are equivalent. (i) (HELP Inequality) There is a constant K > 0 such that 2 2 2 2 (|f | + q|f | ) ≤ K |f | | − f + qf |2 (f ∈ D). Γ

Γ

Γ

(ii) There exist ϑ+ , ϑ− ∈ [0, π/2) such that Im(−λ2 m(λ)) ≥ 0

(λ ∈ C \ {0}, arg λ ∈ [ϑ+ , π − ϑ− ]).

(2)

ˆ −2 , The optimal constant for the HELP inequality is given by K = (cos ϑ) ˆ where ϑ := min{ϑ ∈ [0, π/2) | (2) holds for all λ ∈ C \ {0}, arg λ ∈ [ϑ, π − ϑ]}. b) (2) is satisfied if and only if it holds locally both at 0 and at ∞. If m(λ) ∼ cλα (λ → ∞ in an open sector containing the imaginary axis) with c = 0, α ∈ [−1, 1] \ {0}, then (2) holds at infinity. If the non-tangential limit m0 := limλ↓0 m(λ) exists, then (2) is satisfied at 0 if m0 ∈ C \ R and (2) is not satisfied at 0 if m0 ∈ R \ {0}. 1 Moreover, if either m is analytic at 0 and m(0) = 0 or if m is 1 analytic at 0 and m (0) = 0, then (2) is satisfied at 0. For the proof of these statements, we refer to the cited references, with the exception of the last sentence in the theorem, for which we give a proof now. If m is analytic at 0 and m(0) = 0, then m(λ) = aλ + o(|λ|) with a > 0 (since Im m(λ) > 0 in the complex upper half-plane). Hence we find for all λ = reiϑ with sufficiently small r > 0 and ϑ ∈ ( π3 , 2π 3 ) that Im(−λ2 m(λ)) = −ar3 Im e3iϑ + o(r3 ) ≥ 0.


29

1 1 1 Similarly, if m is analytic at 0 and m (0) = 0, then m (λ) = aλ + o(λ) with iϑ a < 0, and therefore for all λ = re with sufficiently small r > 0 and ϑ ∈ (0, π)

Im(−λ2 m(λ)) = |m(λ)|2 Im(−λ2 (aλ + o(|λ|)) = |m(λ)|2 (−ar3 Im eiϑ + o(r3 )) ≥ 0. Part b) of the preceding theorem shows that, in order to decide whether a HELP inequality holds, we only need to study the limiting behaviour of the m function at ∞ and at 0. In the following we shall prove that for the tree-Hill operator, the condition at ∞ is always satisfied; the behaviour of m at 0 depends on whether we have quasi-stability or quasi-instability at λ = 0. The behaviour of m at ∞ We have the following observation, valid under general hypotheses. Lemma 4.3. Let q : [0, l] → ∞ be integrable and Z := {z ∈ C \ {0} | arg z ∈ / (−α, α) mod 2π}, with any α > 0. Then m(λ) ∼ √iλ (λ → ∞, λ ∈ Z). Proof. We begin with obtaining the asymptotics of φ(l, λ) and θ(l, λ) by a variation of constants estimate (cf., e.g., [4]). Let λ ∈ Z. Then the free Schrödinger equation on [0, l], −y = λy, has the canonical fundamental system √ √cos( λx) √ − λ sin( λx)

Φ0 (x) =

√ √ sin( λx)/ λ √ cos( λx)

(x ∈ [0, l]),

√ using the convention Im λ > 0; so solving the initial-value problem −u + q(x)u = λu with given u(0), u (0) by variation of constants, we find

u(x) u (x)

l 0 u0 (x) −1 Φ0 (x)Φ0 (t) = + q(t) u0 (x) 0

0 0

u(t) u (t)

dt

(x ∈ [0, l])

and hence u(x) − u0 (x) =

0

x

√ sin( λ(x − t)) √ u(t) dt q(t) λ

(x ∈ [0, l]),

where u0 is the solution of −y = λy with the same initial data as u. Thus, writing √ √ sin( λt) √ u (0), u(t) = (u(t) − u0 (t)) + cos( λt) u(0) + λ

30


we can estimate √

|ei

λx

| |u(x) − u0 (x)| √ x 1 |e2i λ(x−t) − 1| i√λt |e ≤ √ |q(t)| | |u(t) − u0 (t)| 2 |i λ| 0 √

≤

x

0

√

|e2i λt − 1| √ |u (0)| dt |u(0)| + + 2 2|i λ| x √ |u (0)| |q(t)| i λt |q(t)| √ |e √ dt (|u(0)| + √ ). | |u(t) − u0 (t)| dt + | λ| | λ| 0 | λ| |e2i

λt

+ 1|

This is an integral inequality of Gronwall type, x x f (x) ≤ c g+ fg 0

0

with g ≥ 0; it follows that x x x x g+ f g e− 0 g f (x) e− 0 g ≤ c 0 0 y x y y c + (f (y) − c = g− f g) g(y) e− 0 g dy 0 0 0 x y x f (y) e− 0 g dy = c(1 − e− 0 g ) ≤c 0

(for the second line, we rewrite the previous expression as the integral of its derivative). Thus we obtain l √ |u (0)| √1 |q| |u(l) − u0 (l)| ≤ (|u(0)| + √ ) |e−i λl | (e λ 0 − 1). | λ|

In particular, √

√ e−i θ0 (l, λ) = cos( λl) ∼ 2

λl

√ (Im λ → ∞)

and |θ(l, λ) − θ0 (l, λ)| ≤ |e so θ(l, λ) ∼

and

√ −i λl

| (e

√1 | λ|

l 0

|q|

√ (Im λ → ∞); similarly, √ √ sin( λl) i e−i λl √ √ φ0 (l, λ) = ∼ λ 2 λ √

e−i 2

− 1) ∼ |e

√ −i λl

1 | √ | λ|

l

0

|q|,

λl

√

(Im

√ λ → ∞)

√ |e−i λl | |√1λ| 0l |q| 1 (e − 1) ∼ |e−i λl | |φ(l, λ) − φ0 (l, λ)| ≤ √ |λ| | λ| √ √ −i λl so φ(l, λ) ∼ i e2√λ (Im λ → ∞).

l 0

|q|,


31 √

The assertion now follows, bearing in mind that |μ(λ)| ≤ √1b = o(|e−i √ and that Im λ → ∞ whenever λ → ∞ in Z.

λl

|)

The behaviour of m at 0 The type of limiting value of m at 0 essentially depends on whether the tree-Hill equation with spectral parameter 0 has quasi-stability or quasiinstability. Theorem 4.4. a) If λ = 0 is a point of quasi-stability, then limλ↓0 m(λ) ∈ C \ R. Consequently, a HELP inequality holds. b) If λ = 0 is a point of quasi-instability, then either • limλ↓0 m(λ) ∈ R \ {0} and there is no valid HELP inequality, or 1 (0) = 0, 0 is a • m(0) = 0, 0 is a Dirichlet eigenvalue, or m Neumann eigenvalue, and in either case, a HELP inequality holds. Proof. First we observe that if φ(l, 0) = 0 or θ (l, 0) = 0, then 0 is not a point of quasi-stability. Indeed, then either the top-right or the bottom-left entries 1 , and of A(0) vanish, so in view of the determinant of A(0), φ (l, 0) = θ(l,0) the quasi-discriminant satisfies 2 1 2 1 √ 1 4 = bθ(l, 0) + √ ≥ . D(0)2 = θ(l, 0) + θ(l, 0)b b b bθ(l, 0) a) If 0 is a point of quasi-stability, then observing that all entries of A are entire functions of λ and hence approach a finite real limit as λ → 0, while μ(λ) ∈ C \ R in the limit, the formulae given in Lemma 4.1 show that m has a non-real limit, and the assertion follows. 1 b) If 0 is a point of quasi-instability and m(0) = 0 = m (0), then m will have a finite limit as the ratio of non-vanishing analytic functions given in Lemma 4.1 (note μ is analytic except at the quasi-(anti-)symmetric eigenvalues; cf. [9] p. 64); moreover, as all entries of A and μ are real in the limit, so will m be, and by Theorem 4.2 b) there will be no valid HELP inequality. If m(0) = 0, then the formulae of Lemma 4.1 show that φ(l, 0) = 0 and φ (l, 0) = bμ(0), so the transfer matrix must have the form 1 0 μ(0)b A(0) = θ (l,0) . μ(0) b 0 This matrix clearly has eigenvector with eigenvalue μ(0) of modulus 1 |μ(0)| < √1b ; in other words, the extension of φ as a symmetric function on the tree is a Dirichlet eigenfunction. As the two eigenvalues of A(0) are distinct, θ(l, 0) − μ(0) = 0, so m is analytic at 0. Theorem 4.2 b) shows that a HELP inequality holds. 1 (0) = 0 is analogous; in this case, The situation m μ(0) φ(l, 0) A(0) = , 1 0 μ(0)b

32


Now the extension of θ as a symmetric function on the tree is a Neumann 1 eigenfunction, φ (l, 0) = bμ(0), and m is analytic at 0. Again by Theorem 4.2 b), we conclude that a HELP inequality holds. Remark 4.5. If 0 is a point of transition between quasi-stability and quasiinstability, i.e., a quasi-(anti-)periodic eigenvalue, then the analysis is very much like part b) of the preceding theorem, with the difference that in the 1 exceptional cases m(0) = 0 or m (0) = 0, the two eigenvalues of A(0) are equal and φ or θ, resp., will not be Dirichlet or Neumann eigenfunctions. It would seem that in this exceptional situation the local validity of (2) will need to be verified separately for the particular operator under study.

5. Calculating the optimal constant Theorem 4.4 only provides a criterion for the existence or otherwise of a HELP inequality with a finite constant; however, the actual determination of that constant will require more detailed knowledge of the m function, beyond the bare asymptotics at 0 and at ∞. To illustrate the process, we studied the tree-Hill operator with edge length l = 1, branching number b = 3 and the potential qτ (x) := x + τ (x ∈ [0, 1]) on each edge. The offset τ is introduced with a view to investigating the effect of closeness of λ = 0 to an end-point of the region of quasi-stability. More precisely, the first two quasistability intervals for potential q0 are approximately [0.68772, 7.15072] and [14.0335, 33.5907]; we therefore considered the cases τ = −3, τ = −0.7 and τ = −1 in order to study situations where 0 is near the middle, close to the end, or at an intermediate position in the quasi-stability interval, respectively. For the calculation, we use (2) directly on semicircular arcs in the complex upper half-plane, i.e., for λ = Reiϑ , ϑ ∈ (0, π); r > 0. The Weyl– Titchmarsh function m is computed, using Lemma 4.1, with Mathematica software from numerical solutions of −y + qτ y = λy on [0, 1]. For fixed r > 0, 1 Im(λ2 m(λ)) = Im(e2iϑ m(reiϑ )) |λ|2 then is a function taking a non-negative value at ϑ = 0, a negative value at ϑ = π2 and a non-negative value at ϑ = π (see Fig. 1). To determine the maximal symmetric interval of negativity around ϑ = π2 for this function, we computed its nearest zero ϑ0 (r) by the bisection method, taking ϑ0 (r) ∈ [0, π2 ) w.l.o.g. (otherwise subtract from π). The supremum ϑˆ of ϑ0 (r), taken over all r > 0, will then determine the optimal constant ˆ −2 . K = (cos ϑ) The numerical results show three different situations.


33

1.0

0.5

0.5

1.0

1.5

2.0

2.5

3.0

0.5

Figure 1. Fig 1. Plot of Im(e2iϑ m(eiϑ )). In the first case, we have τ = −3, and the profile curve of the zero is given in Fig. 2 a). The curve approaches its supremum asymptotically as r → ∞, and ϑˆ = π3 gives the optimal constant K = 4. Since the asymptotic value at infinity will always be the same, as shown in Lemma 4.3, K = 4 is clearly the minimum value for an optimal HELP constant. In the second case, where τ = −0.7, the behaviour of the curve near ∞ is the same as before, but the supremum is now achieved near 0, giving a HELP constant of about 60, cf. Fig. 2 b). This behaviour seems to be typical of a situation where 0 is close to the end of an interval of quasi-stability; it is plausible that the HELP constant should tend to infinity as 0 approaches a region of quasi-instability, in the interior of which the inequality was shown to fail. Finally, the intermediate choice τ = −1 gives rise to a situation where the optimal constant is determined by a local maximum, cf. Fig. 2c), illustrating the point that knowledge of the asymptotics is not sufficient in general.

6. Appendix: m function and spectra We give a simple example to show that the m function for the Schrödinger operator on a tree does not provide full information on its spectral properties. Consider the tree-Hill operator with potential q = 0, i.e., the tree Laplacian on our regular tree Γ, with a Dirichlet boundary condition at 0. As shown in [11], the spectrum of this operator consists of absolutely continuous bands coinciding with the closure of the quasi-stability intervals and of eigenvalues of infinite multiplicity, situated in quasi-instability intervals. The high multiplicity arises from the fact that any Dirichlet eigenfunction can be shifted down, multiplied with the b-th roots of unity, into the first-generation

34


1.0

0.9

0.8

0.7

10

8

6

4

2

2

4

2

4

2

4

1.4

1.3

1.2

1.1

1.0

10

8

6

4

2

1.05

1.00

0.95

10

8

6

4

2

Figure 2. Fig. 2. Zero ϑ0 as a function of log r for τ = −3 (a), τ = −0.7 (b), and τ = −1 (c).

subtrees and extended by 0 on the 0-th generation edge, to obtain another Dirichlet solution in D. Thus, starting from a symmetric eigenfunction, an


35

unlimited number of linearly independent eigenfunctions can be created by repeated application of this mechanism. As apparent from the considerations in the proof of Theorem 4.4 b) above, the Dirichlet eigenvalue will, by virtue of its symmetric eigenfunction, be accompanied with a zero of the m function. While the m function thus flags up all eigenvalues for the operator with full tree-Hill symmetry, although not their multiplicities, the disparity becomes more obvious when this symmetry of periodic type is broken. Let us therefore consider a modified operator which differs from the above by the addition of a potential q˜ of compact support in (0, l) on the 0th generation edge only. The essential spectrum of this perturbed operator will be the same as before, and the infinitely many non-symmetric Dirichlet eigenfunctions constructed above will still be Dirichlet eigenfunctions, as they vanish on the 0-th generation edge and hence know nothing about the potential. However, the Dirichlet eigenvalues on the symmetric subspace will feel the perturbation and hence change in general, as will the m function. Regarding the latter, we can calculate it for the perturbed operator as follows. Let m be the original m function, calculated as previously for the tree-Hill operator from the canonical fundamental system Φ, which in our case is equal to the Φ0 of the proof of Lemma 4.3 above. Now let ˜ ˜ ˜= θ φ Φ θ˜ φ˜ be the canonical fundamental system on [0, l] of −u + q˜ u = λu. For λ ∈ C \ R, the distinguished solution ψ ∈ D will be the same, except on the 0-th generation edge, for the unperturbed and perturbed operators, so ˜ we find ψ(l−) ˜ denoting the solution in the perturbed case with ψ, = ψ(l−), ψ˜ (l−) = ψ (l−) and therefore ψ˜ ψ ψ˜ −1 −1 ˜ ˜ (0) = Φ (l) ˜ (l) = Φ (l)Φ(l) (0) ψ ψ ψ˜ ˜ ψ (0) (θ φ˜ − φ θ˜ )(l) ψ(0) + (φ θ˜ − θ φ)(l) . = ˜ ψ (0). (θ φ˜ − φ θ˜ )(l) ψ(0) + (φ θ˜ − θ φ)(l) For the perturbed m function, we thus obtain m ˜ =−

˜ ˜ (θ φ˜ − φ θ˜ )(l) m − (φ θ˜ − θ φ)(l) ψ(0) = . ˜ ˜ ˜ ˜ ˜ (φ θ − θ φ)(l) − (θ φ − φ θ )(l) m ψ (0)

This function m ˜ has a zero at the position of a simple Dirichlet eigenvalue for the perturbed operator with symmetric eigenfunction. However, at the position of the Dirichlet eigenvalue of infinite multiplicity, m has a zero and hence (cf. Lemma 4.1) φ(l) = 0 and, by linear independence, θ(l) = 0. Thus we obtain ˜ θ(l) φ(l) , m ˜ = ˜ − θ (l) φ(l) ˜ φ (l) θ(l)

36


˜ = 0. This will not be the case for most perwhich will be zero only if φ(l) turbations q˜, and hence m ˜ cannot be used to detect the eigenvalue of infinite multiplicity inherited from the unperturbed problem.

References [1] Bennewitz C: A general version of the Hardy-Littlewood-P´ olya-Everitt (HELP) inequality. Proc. Roy. Soc. Edinburgh Sect. A 97 (1984) 9–20 [2] Brown BM, Hinchcliffe J, Marletta M, Naboko S, Wood IG: The abstract Titchmarsh-Weyl M -function for adjoint operator pairs and its relation to the spectrum. Integr. Equ. Oper. Theory 63 (2009) 297–320 [3] Brown BM, Langer M, Schmidt KM: The HELP inequality on trees. Proceedings of Symposia in Pure Mathematics 77 (2008) 337–354 [4] Brown BM, Peacock RA, Weikard R: A local Borg-Marchenko theorem for complex potentials. J. Comput. Appl. Math. 148 (2002) 115-131 [5] Carlson R: Hill’s equation for a homogeneous tree. Electron. J. Differential Equations (1997) No. 23, 30 pp [6] Everitt WN: On an extension to an integro-differential inequality of Hardy, Littlewood and P´ olya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1972) 295–333 [7] Evans WD, Everitt WN: A return to the Hardy-Littlewood integral inequality. Proc. Roy. Soc. London Ser. A 380 (1982) 447–486 [8] Hardy GH, Littlewood JE: Some integral inequalities connected with the calculus of variations. Quart. J. Math. Oxford 3 (1932) 241–252 [9] Kato T: Perturbation Theory for Linear Operators. Springer, Berlin 1980 [10] Langer M: A general HELP inequality connected with symmetric operators. Proc. Roy. Soc. London Ser. A 462 (2006) 587–606 [11] Sobolev AV, Solomyak MS: Schr¨ odinger operators on homogeneous metric trees: spectrum in gaps. Rev. Math. Phys. 14 (2002) 421–467 B.M. Brown School of Computer Science Cardiff University, Cardiff CF24 3XF UK e-mail: [email protected] K.M. Schmidt School of Mathematics Cardiff University, Cardiff CF24 4AG UK e-mail: [email protected]


Measure of Non-compactness of Operators Interpolated by Limiting Real Methods Fernando Cobos, Luz M. Fernández-Cabrera and Antón Mart´ınez Dedicated to Professor David E. Edmunds and Professor W. Desmond Evans on the occasion of their 80th and 70th birthday, respectively.

Abstract. We establish formulae for the measure of non-compactness of operators interpolated by limiting methods that come up by the choice θ = 0 and θ = 1 in the definition of the real method. Mathematics Subject Classification (2010). 46B70, 47B06. Keywords. Limiting interpolation methods, real interpolation, measure of non-compactness, compact operators.

1. Introduction Interpolation theory is a branch of functional analysis with many important applications in partial differential equations, approximation theory, harmonic analysis, operator theory and other areas of mathematics (see, for example, the monographs by Butzer and Berens [4], Bergh and Löfstr¨ om [3], Triebel [24], Bennett and Sharpley [2], Connes [13] or Amrein, Boutet de Monvel and Georgescu [1]). Two main interpolation methods have been developed: The real method (A0 , A1 )θ,q , connected with the Marcinkiewicz interpolation theorem, and the complex method (A0 , A1 )[θ] , associated with the RieszThorin theorem. Here 0 < θ < 1 and 1 ≤ q ≤ ∞. The real method can be described using the Peetre’s K- and J-functionals. If we restrict to ordered couples of Banach spaces, i.e., A0 → A1 , and we make a natural change in the usual definition, then it was shown by Gomez and Milman [20] that the real method in the K-form is meaningful when θ = 1 and produces very interesting spaces (see also [21]). The extreme Jspaces with θ = 0 have been studied more recently by K¨ uhn, Ullrich and two of the present authors in [7] (see also [9] and [11]). The authors have been supported in part by the Spanish Ministerio de Educaci´ on y Ciencia (MTM2010-15814).

38

F. Cobos, L.M. Fernández-Cabrera and A. Mart´ınez

The limiting spaces with θ = 1 are very close to A1 and, if θ = 0, they are very near to A0 . This fact produces a number of important changes in their theory in comparison with the real method. For example, consider the behaviour of compact operators. Cwikel [14] and Cobos, K¨ uhn and Schonbek [12] have shown that if any restriction of the operator is compact, then the interpolated operator by the real method is also compact (see [6] for the background of this result). In contrast to this, compactness of T : A0 −→ B0 is not enough to imply that the interpolated operator by the limiting Kmethod is compact (see [7]). The limiting J-method has a similar negative behaviour but now for the restriction T : A1 −→ B1 . However, if T : Ai −→ Bi is compact, then the interpolated operator by the extreme method with θ = i is compact as well. This result has been proved in [7] as well as some estimates for entropy numbers in degenerated cases. In this paper we continue their research by establishing formulae for the measure of non-compactness of operators interpolated by the limiting methods. In the case of the real method, this problem was settled by Fern´ andezMart´ınez and two of the present authors in [10]. They proved that β(T : (A0 , A1 )θ,q −→ (B0 , B1 )θ,q )

(1.1)

≤ C β(T : A0 −→ B0 )1−θ β(T : A1 −→ B1 )θ . No similar result is known for the complex method. Inequality (1.1) led to the problem if a similar formula holds for entropy numbers. This question has been recently solved in the negative by Edmunds and Netrusov [16]. An extension of (1.1) to the so-called abstract real method was given by Szwedek [23] and the present authors [8]. The techniques we use here are based on the tools developed in [10] and [8]. The organization of the paper is simple. In Section 2 we recall some basic results on the measure of non-compactness and we introduce limiting interpolation methods. Section 3 contains the formula for the limiting Jmethod and, in Section 4, we consider the case of the limiting K-method.

2. Preliminaries Let A, B be Banach spaces and let T ∈ L(A, B) be a bounded linear operator acting from A into B. The (ball) measure of non-compactness β(T ) = β(TA,B ) = β(T : A −→ B) of T is defined to be the infimum of the set of numbers σ > 0 for which there is a finite subset {b1 , . . . , bn } ⊆ B such that T (UA ) ⊆

n

{bj + σUB },

j=1

where UA (respectively, UB ) stands for the closed unit ball in A (respectively, B).

Measure of Non-compactness of Operators Interpolated

39

It is easy to check that β(T ) = 0 if and only if T is compact. Note that β(T ) is the limit of the sequence of entropy numbers of the operator T (see [17]). Clearly, β(TA,B ) ≤ T A,B . Other properties of β can be found in [15, 5, 22]. Let K(A, B) be the set of compact linear operators from A into B. One can show that β(TA,B ) = inf{σ > 0 : there is a Banach space E and V ∈ K(E, B) such that T (UA ) ⊆ σUB + V (UE )}. (2.1) Non-compactness of T ∈ L(A, B) can be also measured by the seminorm γ(T ) = γ(TA,B ) defined to be the infimum of the set of numbers σ > 0 for which there is a Banach space Z and a compact operator V ∈ K(A, Z) such that

T x B ≤ σ x A + V x Z for all x ∈ A. The seminorm γ(T ) coincides with the infimum of all η > 0 such that there is a subspace M of A with finite codimension, such that

T x B ≤ η x A

for all x ∈ M.

Under this form, the seminorm γ(T ) is studied in [22]. It is shown there that β and γ are equivalent: 1 β(T ) ≤ γ(T ) ≤ 2 β(T ) for any T (2.2) 2 (see [22, Thm. 14.36]). Let A¯ = (A0 , A1 ) be a couple of Banach spaces with A0 → A1 . Here the symbol → means continuous inclusion. For t > 0, the Peetre’s K- and J-functionals are defined by K(t, a) = inf{ a0 A0 + t a1 A1 : a = a0 + a1 , ai ∈ Ai } ,

a ∈ A1 ,

and J(t, a) = max{ a A0 , t a A1 } , a ∈ A0 . Let 1 ≤ q ≤ ∞. The limiting J-space A¯0,q;J = (A0 , A1 )0,q;J , realized in discrete way, is formed by all those a ∈ A1 which can be represented as a=

∞

un

(convergence in A1 ), with (un ) ⊆ A0

(2.3)

n=1

and

∞

q1 J(2n , un )q

β(TAi ,Bi ) for i = 0, 1. Our plan is to show first that σ 1 ¯log,q;K ≤ C1 σ0 1 + log β T πP2m : q (Gn ) −→ B , for all m ∈ N, σ0 + (3.2) then β Rm jT πQ2m : q (Gn ) −→ q (n−1 Fn ) −→ 0 as m → ∞, (3.3)

44


and finally that there is N ∈ N such that for any m ≥ N we have σ1 −1 β Sm jT πQ2m : q (Gn ) −→ q (n Fn ) ≤ C2 σ0 1 + log . (3.4) σ0 + Assuming these three facts, the choice of m ∈ N big enough yields that σ1 β(TA¯0,q;J ,B¯log,q;K ) ≤ Cσ0 1 + log . σ0 + Consequently, if β(TA0 ,B0 ) = 0 , letting σ0 → 0, we obtain case (a) of the theorem. If β(TA1 ,B1 ) = 0 , letting σ1 → 0, we derive case (b). Finally, if β(TAi ,Bi ) > 0 for i = 0, 1, take ε > 0 and write σi = (1 + ε)β(TAi ,Bi ). It follows that β(TA1 ,B1 ) . β(TA¯0,q;J ,B¯log,q;K ) ≤ C(1 + ε)β(TA0 ,B0 ) 1 + log β(TA0 ,B0 ) + This gives case (c) by letting ε → 0. We proceed to prove (3.2). By the equivalence (2.2) between the measures β and γ, we can find Banach spaces Zi and compact linear operators Vi ∈ K(Ai , Zi ), i = 0, 1, such that

T x Bi ≤ 2σi x Ai + Vi x Zi ,

x ∈ Ai ,

i = 0, 1.

(3.5)

For any n ∈ N, put Wn = (Z0 ⊕ Z1 ) 1 . Let λ > 0 and r ∈ N ∪ {0} be fixed numbers to be specified later and consider the operators V, Lλ,r : q (Gn ) −→ q (Wn ) defined by V (un ) = (vn ) and Lλ,r (un ) = (wn ) where (V0 un , 2n V1 un ) if 1 ≤ n ≤ 2m, vn = (0 , 0) otherwise, and

⎧ r+1 ⎪ ⎪ , 0) if n = 1, (λV u 0 j ⎨ j=1 n wn = (V0 ur+n , 2 V1 ur+n ) if 2 ≤ n ≤ 2m − r, ⎪ ⎪ ⎩ (0 , 0) otherwise.

It is easy to check that V and Lλ,r are linear and compact. Now we distinguish two cases. If β(TA1 ,B1 ) < β(TA0 ,B0 ), then we can suppose that 1 < σ0 /σ1 . For u = (un ) ∈ q (Gn ), we have

un Ai ≤ 2−in J(2n , un ) ≤ 2−in (σ0 /σ1 )i J(2n , un ),

n ∈ N,

Using (3.5), we derive

T un Bi ≤ 2σi 2−in (σ0 /σ1 )i J(2n , un ) + Vi un Zi .

i = 0, 1.


45

It follows that

T πP2m u 0,q;J ≤ ≤

2m

2m

1/q n

q

J(2 , T un )

n=1

1/q q

(max{2σ0 J(2 , un ) + V0 un Z0 , 2σ0 J(2 , un ) + 2 V1 un Z1 }) n

n

n=1

≤ 2σ0

∞

n

1/q n

q

J(2 , un )

+ V u q (Wn ) .

n=1

Since V is compact, this yields that

β (T πP2m ) ≤ C1 σ0 = C1 σ0

σ1 1 + log . σ0 +

Suppose now β(TA0 ,B0 ) ≤ β(TA1 ,B1 ). This time we may assume that 1 ≤ σ1 /σ0 . Let r = [log2 (σ1 /σ0 )] where the logarithm is taken in the base 2 and [·] is the greatest integer function. Clearly r ≥ 0 and 2r ≤ σ1 /σ0 < 2r+1 . Given any u = (un ) ∈ q (Gn ), let v = (vn ) be the sequence defined by v1 =

r+1

uj , v2 = ur+2 , . . . , v2m−r = u2m and vn = 0 for n > 2m − r.

j=1

Then

v1 A0 ≤

r+1

uj A0 ≤ (r + 1) max

1≤j≤r+1

j=1

Using (3.5), we get

T v1 B0 ≤ 2σ0 (r + 1) max

1≤j≤r+1

2 T v1 B1 ≤ 2C3 T v1 B0 ≤ 4C3 σ0 (r + 1) max

1≤j≤r+1

J(2, T v1 ) ≤ 4(1 + C3 )σ0 (r + 1) max

1≤j≤r+1

J(2j , uj ) .

J(2j , uj ) + V0 v1 Z0 .

Moreover, since B0 → B1 , we obtain

So,

J(2j , uj ) + 2C3 V0 v1 Z0 .

J(2j , uj ) + (1 + 2C3 ) V0 v1 Z0 .

For 2 ≤ n ≤ 2m − r, using again (3.5), we obtain with i = 0, 1 2in T vn Bi ≤ 2in 2σi 2i(−r−n) J(2r+n , ur+n ) + 2in Vi ur+n Zi σ i 0 ≤ 2σi 2 J(2r+n , ur+n ) + 2in Vi ur+n Zi σ1 ≤ 4σ0 J(2r+n , ur+n ) + 2in Vi ur+n Zi . Whence, J(2n , T vn ) ≤ 4σ0 J(2r+n , ur+n ) + max{ V0 ur+n Z0 , 2n V1 ur+n Z1 }.

46

F. Cobos, L.M. Fernández-Cabrera and A. Mart´ınez Let λ = 1 + 2C3 and consider the compact operator Lλ,r . Since

T πP2m u 0,q;J = T πP2m v 0,q;J ≤

2m−r

1/q n

q

J(2 , T vn )

n=1

≤ 4(1 + C3 )σ0 (r + 1) (un ) q (Gn ) + Lλ,r (un ) q (Wn ) and r ≤ C4 log(σ1 /σ0 ), it follows that β (T πP2m ) ≤ C1 σ0

σ1 1 + log . σ0 +

This completes the proof of (3.2). To establish (3.3), we first use (3.1), Lemma 3.1 and the norm estimate (2.5) to obtain β Rm jT πQ2m ≤ Rm jT πQ2m q (Gn ), q (n−1 Fn ) ≤ C5 Rm jT πQ2m ( 1 (Gn ), 1 (2−n Gn ))0,q,J ,( ∞ (Fn ), ∞ (2−n Fn ))0,q,J ≤ C6 Rm jT πQ2m 1 (Gn ), ∞ (Fn )

Rm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log .

Rm jT πQ2m 1 (Gn ), ∞ (Fn ) + Using the factorization Q2m

jT π

1 (Gn ) −−−−−−→ 1 (2−n Gn ) −−−−−−→ ∞ (2−n Fn ) −−−−m−−→ ∞ (Fn ) R

and the estimates for the norms of the projections, we derive

Rm jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ 2−(2m+1) T A1 ,B1 2m = 2−(m+1) T A1,B1 . On the other hand, the factorization Q2m

jT π

1 (2−n Gn ) −−−−−−→ 1 (2−n Gn ) −−−−−−→ ∞ (2−n Fn ) −−−−m−−→ ∞ (2−n Fn ) R

yields that

Rm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) ≤ T A1,B1 for all m ∈ N. Hence, lim Rm jT πQ2m 1 (Gn ), ∞ (Fn )

Rm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log =0

Rm jT πQ2m 1 (Gn ), ∞ (Fn ) +

m→∞

and (3.3) follows.


47

Finally, to prove (3.4), we use again (3.1), Lemma 3.1 and (2.4) to get β Sm jT πQ2m ≤ Sm jT πQ2m q (Gn ), q (n−1 Fn ) ≤ C5 Sm jT πQ2m ( 1 (Gn ), 1 (2−n Gn ))0,q,J ,( ∞ (Fn ), ∞ (2−n Fn ))0,q,J ≤ C6 Sm jT πQ2m 1 (Gn ), ∞ (Fn )

Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log .

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) + Clearly,

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ jT πQ2m 1 (Gn ), ∞ (Fn ) . Moreover,

jT πQ2 1 (Gn ), ∞ (Fn ) ≥ jT πQ4 1 (Gn ), ∞ (Fn ) ≥ · · · ≥ 0. Whence, there is λ ≥ 0 such that

jT πQ2m 1 (Gn ), ∞ (Fn ) −→ λ as m → ∞. We are going to estimate λ from above by using σ0 . For this aim, let wm ∈ U 1 (Gn ) such that

jT πQ2mwm ∞ (Fn ) −→ λ as m → ∞, and find vectors {b1 , . . . , bs } ⊆ B0 such that s {bk + σ0 UB0 } . T π U 1 (Gn ) ⊆ k=1

Since the number of bk is finite, there is some k, say k = 1, and a subsequence (m ) of N such that

T πQ2m wm − b1 B0 ≤ σ0 for all m .

Using that Q2m 1 (Gn ), 1 (2−n Gn ) = 2−(2m +1) , we derive for any n ∈ N K(2n , b1 ) ≤ b1 − T πQ2m wm B0 + 2n T πQ2m wm B1 ≤ σ0 + 2n T A1,B1 Q2m wm 1 (2−n Gn )

≤ σ0 + 2n−(2m +1) T A1 ,B1 −→ σ0

as m → ∞.

Hence,

jb1 ∞ (Fn ) = sup {K(2n , b1 )} ≤ 2σ0 . n∈N

It follows that λ = lim

jT πQ2m wm ∞ (Fn ) m →∞ ≤ sup jT πQ2m wm − jb1 ∞ (Fn ) + jb1 ∞ (Fn ) m

≤ sup { T πQ2m wm − b1 B0 + 2σ0 } ≤ 3σ0 . m

This yields that there is N1 ∈ N such that for all m ≥ N1 we have

jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ 4σ0 .

48


Therefore

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) ≤ 4σ0 for all m ≥ N1 . Now we turn our attention to Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) . We can find a finite set {d1 , . . . , dr } ⊆ U 1 (2−n Gn ) , formed by sequences having only a finite number of non-zero coordinates, such that r jT π U 1 (2−n Gn ) ⊆ jT πdk + 2σ1 U ∞ (2−n Fn ) . k=1

Since Sm ∞ (Fn ), ∞ (2−n Fn ) = 2−(m+1) and {jT πdk : 1 ≤ k ≤ r} ⊆ ∞ (Fn ), there is N2 ∈ N such that

Sm jT πdk ∞ (2−n Fn ) ≤ σ1 for 1 ≤ k ≤ r and m ≥ N2 . We claim that

Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) ≤ 3σ1 for all m ≥ N2 . Indeed, take any u ∈ U 1 (2−n Gn ) . Since Q2m u ∈ U 1 (2−n Gn ) , we can find 1 ≤ k ≤ r such that jT πQ2m u − jT πdk ∞ (2−n Fn ) ≤ 2σ1 . Therefore, for m ≥ N2 , we derive

Sm jT πQ2m v ∞ (2−n Fn ) ≤ Sm jT πQ2m v − Sm jT πdk ∞ (2−n Fn ) + Sm jT πdk ∞ (2−n Fn ) ≤ jT πQ2m v − jT πdk ∞ (2−n Fn ) + σ1 ≤ 3σ1 . Consequently, for m ≥ max{N1 , N2 } we conclude that β Sm jT πQ2m ≤ C6 Sm jT πQ2m 1 (Gn ), ∞ (Fn )

Sm jT πQ2m 1 (2−n Gn ), ∞ (2−n Fn ) × 1 + log

Sm jT πQ2m 1 (Gn ), ∞ (Fn ) + σ1 ≤ C7 σ0 1 + log . σ0 + This establishes (3.4) and completes the proof.

As a direct consequence of Theorem 3.2, we recover the compactness result for the limiting J-method proved in [7]. ¯ = (B0 , B1 ) be pairs of Banach spaces with Corollary 3.3. Let A¯ = (A0 , A1 ), B ¯ B). ¯ If T : A0 −→ B0 A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and let T ∈ L(A, is compact, then T : (A0 , A1 )0,q;J −→ (B0 , B1 )0,q;J is also compact.


49

4. Limiting K-method and measure of non-compactness To deal with the limiting K-method, we shall need the following interpolation formula for vector-valued sequence spaces established in [7, Lemma 7.12]: (4.1) ∞ (Wn ), ∞ (2−n Wn ) 1,q;K → q (2−n Wn ). We shall also require the following result. Lemma 4.1. Let (Wn ) be a sequence of Banach spaces and let 1 ≤ q < ∞. Then 1 + n Wn → 1 (Wn ), 1 (2−n Wn ) 1,q;K . q n 2 Proof. We view (1 (Wn ), 1 (2−n Wn ))1,q;K as a J-space, endowed with the equivalent norm · ρ,q;J (see Section 2). Take any w = (wn ) ∈ q 1+n 2n Wn . m m Let um = (um n )n∈N with un = 0 if n = m and un = wm if n = m. We have J(2m , um ) = max um 1 (Gn ) , 2m um 1 (2−n Gn ) = max wm Gm , 2m 2−m wm Gm = wm Gm . Whence,

w ρ,q;J ≤

∞

1+n

2n

n=1

=

∞

1+n

2n

n=1

q 1q J(2 , u ) n

n

q 1q

wn Gn

= w 1+n

q

2n

Wn

.

For the limiting K-method the result is as follows ¯ = (B0 , B1 ) be pairs of Banach spaces with Theorem 4.2. Let A¯ = (A0 , A1 ), B ¯ B). ¯ Then we have: A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and T ∈ L(A, (a) β(TA¯1,q;K ,B¯1,q;K ) = 0 if β(TA1 ,B1 ) = 0, (b) β(TA¯1,q;K ,B¯1,q;K ) ≤ Cβ(TA1 ,B1 ) if β(TA0 ,B0 ) = 0, β(TA0 ,B0 ) (c) β(TA¯1,q;K ,B¯1,q;K ) ≤ Cβ(TA1 ,B1 ) 1+ log β(TA ,B ) if β(TAi ,Bi ) > 0 1

1

+

for i = 0, 1. Here C is a constant independent of T . Proof. This time the trivial case corresponds to q = ∞ because (A0 , A1 )1,∞;K = A1 and (B0 , B1 )1,∞;K = B1 . Assume then 1 ≤ q < ∞. We shall use the notation introduced in the proof of Theorem 3.2 for vector-valued sequence spaces, maps and projections. The relevant diagram says now π

T

j

1 (Gn ) −−−−→A0 −−−−→ B0 −−−−→ ∞ (Fn ) j

1 (2−n Gn ) −−−−→A1 −−−−→ B1 −−−−→ ∞ (2−n Fn ) π

q

1 + n 2n

T

j π T ¯1,q;K −−− Gn −−−−→ A¯ρ,q;J =A¯1,q;K −−−−→ B −→ q (2−n Fn )

50


For each m ∈ N, we have β TA¯1,q;K ,B¯1,q;K ≤ 2β jT : A¯1,q;K −→ q (2−n Fn ) ≤ 2β R2m jT : A¯1,q;K −→ q (2−n Fn ) 1 + n Gn −→ q (2−n Fn ) + C1 β S2m jT π : q n 2 ≤ 2β R2m jT : A¯1,q;K −→ q (2−n Fn ) 1 + n −n −→ G (2 F ) + C1 β S2m jT πPm : q n q n 2n 1 + n −n −→ G (2 F ) . + C1 β S2m jT πQm : q n q n 2n Take σi > β(TAi ,Bi ) for i = 0, 1. Proceeding as in the proof of (3.3) but using now (4.1), Lemma 4.1, (2.6) and the factorization

jT π

2m −−→ ∞ (2−n Fn ) 1 (2−n Gn ) −−−−m−−→ 1 (Gn ) −−−−−−→ ∞ (Fn ) −−−−

P

S

we derive 1 + n −n −→ G (2 F ) −→ 0 as m → ∞. (4.2) β S2m jT πPm : q n q n 2n Moreover, using a similar argument as in the proof of (3.4), we obtain that there is N ∈ N such that for any m ≥ N 1 + n σ0 −n β S2m jT πQm : q Gn −→ q (2 Fn ) ≤ C2 σ1 1 + log . 2n σ1 + (4.3) For the remaining term, we claim that σ0 −n ¯ β R2m jT : A1,q;K −→ q (2 Fn ) ≤ C3 σ1 1 + log . (4.4) σ1 + Indeed, by (2.1), there are Banach spaces Ei , i = 0, 1, and compact linear operators Vi : Ei −→ Bi such that T (UAi ) ⊆ σi UBi + Vi (UEi ),

i = 0, 1.

(4.5)

For each n ∈ N, let En = (E0 ⊕ E1 ) ∞ . Let r ∈ N ∪ {0} be a fixed number to be determined later, and define operators V, Lr : q (2−n En ) −→ q (2−n Fn ) by V (x0,n , x1,n ) = (vn ), Lr (x0,n , x1,n ) = (wn ) where 2 (V0 x0,n + 2−n V1 x1,n ) if 1 ≤ n ≤ 2m, vn = 0 otherwise, and

4(1 + r) V0 x0,n + (2n σ1 /σ0 )−1 V1 x1,n if 1 ≤ n ≤ 2m, wn = 0 otherwise.

It is not hard to check that V, Lr ∈ K(q (2−n En ), q (2−n Fn )). Suppose first β(TA0 ,B0 ) ≤ β(TA1 ,B1 ). Then we may assume that 0 < σ0 ≤ σ1 . Take any 0 = a ∈ UA¯1,q;K and put dn = dn (a) = 2K(2n, a).


51

Since K(2n , a) < dn , we can decompose a = a0,n + a1,n with ai,n ∈ Ai , n −1

d−1 n a0,n A0 ≤ 1 and 2 dn a1,n A1 ≤ 1. Using (4.5), we can find xi,n ∈ UEi such that

T ai,n − Vi (2−in dn xi,n ) Bi ≤ 2−in dn σi ≤ 2−in dn σ1 .

(4.6)

Write x = ((dn /2)(x0,n , x1,n )), which belongs to the closed unit ball of q (2−n Wn ). Let V x = (vn ). We have vn = dn (V0 x0,n + 2−n V1 x1,n ),

1 ≤ n ≤ 2m.

Using (4.6), we get K(2n , T a − vn ) ≤ 2dn σ1 ,

1 ≤ n ≤ 2m.

Therefore

R2m jT a − V x q (2−n Fn ) ≤ 2σ1 (dn ) q (2−n Fn ) ≤ 4σ1 . This yields that β R2m jT ≤ 4σ1 = 4σ1

σ0 1 + log . σ1 +

Suppose now β(TA1 ,B1 ) < β(TA0 ,B0 ). We may assume this time that 0 < σ1 < σ0 . Let r = [log2 (σ0 /σ1 )]. Then r ≥ 0 and 2r ≤ σ0 /σ1 < 2r+1 . Given any 0 = a ∈ UA¯1,q;K , let again dn = dn (a) = 2K(2n , a), n ∈ N. If r = 0, we put dk = d1 for 1 − r ≤ k ≤ 0. We have K(2n (σ1 /σ0 ), a) ≤ K(2n−r , a) < dn−r , n ∈ N. Hence, we can decompose a = a0,n + a1,n with ai,n ∈ Ai , d−1 n−r a0,n A0 ≤ 1 and (2n σ1 /σ0 )d−1 a

≤ 1. By (4.5), there exist x ∈ UEi such that 1,n A i,n n−r 1

T ai,n − Vi ((2n σ1 /σ0 )−i dn−r xi,n ) Bi ≤ (2n σ1 /σ0 )−i dn−r σi = 2−in σ0 dn−r . (4.7) Let x = 4−1 (1 + r)−1 (dn−r x0,n , dn−r x1,n ). Then x ∈ U q (2−n Wn ) because 1/q r ∞

1 (2−j d1 )q + (2−j dj−r )q

x q (2−n Wn ) = 4(1 + r) j=1 j=r+1 r ∞ 1/q 1/q 1 −j q −r −j q ≤ (2 d1 ) +2 (2 dj ) 4(1 + r) j=1 j=1 ≤

1 (4r + 2−r 2) ≤ 1. 4(1 + r)

Put Lr x = (wn ). So, wn = V0 (dn−r x0,n ) + (2n σ1 /σ0 )−1 V1 (dn−r x1,n ),

1 ≤ n ≤ 2m.

Using (4.7) we get K(2n , T a − wn ) ≤ 2σ0 dn−r ,

1 ≤ n ≤ 2m.

52


Consequently,

R2m jT a − Lr x q (2−n Fn ) ≤ 2σ0 2−r

2m

(2−(j−r) dj−r )q

1/q

j=1

σ0 σ0 ≤ 8σ0 2−r (1 + r) ≤ 16σ1 1 + log2 ≤ C4 σ1 1 + log . σ1 σ1 This establishes (4.4). From (4.2), (4.3) and (4.4), we conclude the proof of the theorem by letting σi −→ β(TAi ,Bi ) , i = 0, 1. As a direct consequence of Theorem 4.2 we can derive the compactness result for the limiting K-method given in [7, Thm. 7.14]. ¯ = (B0 , B1 ) be pairs of Banach spaces with Corollary 4.3. Let A¯ = (A0 , A1 ), B ¯ B). ¯ If T : A1 −→ B1 A0 → A1 , B0 → B1 , let 1 ≤ q ≤ ∞ and let T ∈ L(A, is compact, then T : (A0 , A1 )1,q;K −→ (B0 , B1 )1,q;K is also compact.

References [1] W.O. Amrein, A. Boutet de Monvel and V. Georgescu, “C0 -Groups, Commutator Methods and Spectral Theory of N -Body Hamiltonians”, Progress in Math. 135, Birkh¨ auser, Basel, 1996. [2] C. Bennett and R. Sharpley, “Interpolation of Operators”, Academic Press, Boston, 1988. [3] J. Bergh and J. L¨ ofstr¨ om, “Interpolation Spaces. An Introduction”, Springer, Berlin 1976. [4] P.L. Butzer and H. Berens, “Semi-Groups of Operators and Approximation”, Springer, New York, 1967. [5] B. Carl and I. Stephani, “Entropy, Compactness and the Approximation of Operators”, Cambridge Univ. Press, Cambridge 1990. [6] F. Cobos, Interpolation Theory and Compactness, in “Function Spaces, Inequalities and Interpolation (Paseky 2009), pages 31-75, Ed. J. Lukeˇ s and L. Pick, Matfyzpress, Prague, 2009. [7] F. Cobos, L.M. Fern´ andez-Cabrera, T. K¨ uhn and T. Ullrich, On an extreme class of real interpolation spaces, J. Funct. Anal. 256 (2009) 2321-2366. [8] F. Cobos, L.M. Fern´ andez-Cabrera and A. Mart´ınez, Abstract K and J spaces and measure of non-compactness, Math. Nachr. 280 (2007) 1698-1708. [9] F. Cobos, L.M. Fern´ andez-Cabrera and M. Mastylo, Abstract limit J-spaces, J. London Math. Soc. (2) 82 (2010), 501–525. [10] F. Cobos, P. Fern´ andez-Mart´ınez and A. Mart´ınez, Interpolation of the measure of non-compactness by the real method, Studia Math. 135 (1999) 25-38. [11] F. Cobos and T. K¨ uhn, Equivalence of K- and J-methods for limiting real interpolation spaces, J. Funct. Anal. (to appear).


53

[12] F. Cobos, T. K¨ uhn and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo functors, J. Funct. Anal. 106 (1992) 274-313. [13] A. Connes, “Noncommutative Geometry”, Academic Press, San Diego, 1994. [14] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992) 333-343. [15] D. E. Edmunds and W. D. Evans, “Spectral Theory and Differential Operators”, Clarendon Press, Oxford 1987. [16] D. E. Edmunds and Yu. Netrusov, Entropy numbers and interpolation, Math. Ann., DOI:10.1007/s00208-010-0624-1. [17] D. E. Edmunds and H. Triebel, “Function Spaces, Entropy Numbers, Differential Operators”, Cambridge Univ. Press, Cambridge 1996. [18] W. D. Evans, B. Opic and L. Pick, Real Interpolation with logarithmic functors, J. of Inequal. & Appl. 7 (2002) 187-269. [19] A. Gogatishvili, B. Opic and W. Trebels, Limiting reiteration for real interpolation with slowly varying functions, Math. Nachr. 278 (2005) 86-107. [20] M.E. Gomez and M. Milman, Extrapolation spaces and almost-everywhere convergence of singular integrals, J. London Math. Soc. 34 (1986) 305-316. [21] M. Milman, “Extrapolation and Optimal Decompositions”, Springer, Lect. Notes in Math. 1580, Berlin, 1994. [22] M. Schechter, “Principles of Functional Analysis”, Amer. Math. Soc., Providence 2002. [23] R. Szwedek, Measure of non-compactness of operators interpolated by real method, Studia Math. 175 (2006) 157-174. [24] H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland, Amsterdam 1978.

Fernando Cobos Departamento de An´ alisis Matem´ atico Facultad de Matem´ aticas Universidad Complutense de Madrid Plaza de Ciencias 3 28040 Madrid Spain e-mail: [email protected] Luz M. Fern´ andez-Cabrera Secci´ on Departamental de Matem´ atica Aplicada Escuela de Estad´ıstica Universidad Complutense de Madrid 28040 Madrid Spain e-mail: luz [email protected]

54


Ant´ on Mart´ınez Departamento de Matem´ atica Aplicada E.T.S. Ingenieros Industriales Universidad de Vigo 36200 Vigo Spain e-mail: [email protected]

Operator Theory: Advances and Applications, Vol. 219, 55–67 c 2012 by the authors

A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality Rupert L. Frank and Elliott H. Lieb Dedicated to D. E. Edmunds and W. D. Evans

Abstract. We show that the sharp constant in the Hardy–Littlewood– Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range. Mathematics Subject Classification (2010). Primary 39B62; Secondary 26A33, 26D10, 46E35. Keywords. Sharp constants, Sobolev inequality, Hardy–Littlewood–Sobolev inequality.

1. Introduction In a recent paper [11] we showed how to compute the sharp constants for the analogue of the Hardy–Littlewood–Sobolev (HLS) inequality on the Heisenberg group. Unlike the situation for the usual HLS inequality on RN , there is no known useful symmetric decreasing rearrangement technique for the Heisenberg group analogue. A radically new approach had to be developed and that approach can, of course, be used for the original HLS problem as well, thereby providing a genuinely rearrangement-free proof of HLS on RN . That will be given here. The HLS inequality (more precisely, the diagonal case) on RN is 1−λ/N Γ(N ) f (x) g(y) λ/2 Γ((N − λ)/2) dx dy

f p g p ≤ π λ Γ(N − λ/2) Γ(N/2) RN ×RN |x − y| (1.1) where 0 < λ < N and p = 2N/(2N − λ). The constant in (1.1) is sharp and inequality (1.1) is strict unless f and g are proportional to a common This paper may be reproduced, in its entirety, for non-commercial purposes. Support by U.S. NSF grant PHY 0965859 (E.H.L.) is acknowledged.

56

R.L. Frank and E.H. Lieb

translate or dilate of −(2N −λ)/2 H(x) = 1 + |x|2 .

(1.2)

An equivalent formulation of (1.1), which has been noted before in the special cases s = 1 and s = 1/2 [19, Thms. 8.3 and 8.4], is the sharp fractional Sobolev inequality (−Δ)s/2 u

2

≥

22s π Γ((N + 2s)/2) Γ(s) Γ((N − 2s)/2)

Γ(N/2) Γ(N )

2s/N

u 2q

(1.3)

for 0 < s < N/2 and q = 2N/(N − 2s). This follows from (1.1) by a duality argument (see [19, Thm. 8.3]), using the fact that the Green’s function of (−Δ)s is 2−2s π −N/2 Γ((N − 2s)/2)/(Γ(s)|x|N −2s ) for 0 < s < N/2 [19, Thm. 5.9]. In particular, for s = 1, (1.3) is the familiar Sobolev inequality 2/N 2π (N +1)/2 N (N − 2) 2 |∇u| dx ≥

u 2q (1.4) 4 Γ((N + 1)/2) RN for N ≥ 3 and q = 2N/(N − 2) in the sharp form of [22, 2, 24]. To recall, briefly, the previous proofs of (1.1) we first mention the papers [13, 14, 23], where the inequality was initially derived, but with a nonsharp constant. The sharp version was found in [18] by noting the conformal invariance of the problem and relating it, via stereographic projection, to a conformally equivalent, but more tractable problem on the sphere SN . Riesz’s rearrangement inequality (see [19, Thm. 3.7]) was used in the proof of the existence of a maximizer, and its strong version ([17], see also [19, Thm. 3.9]) was used to prove that the constant function is a maximizer – in the spherical version. There are other, by now standard, ways to prove the existence of a maximizer; that is not the issue. The main point is to prove that (1.2) is a maximizer and that it is, essentially, unique. Then Carlen and Loss [6] cleverly utilized the translational symmetry in RN in competition with the rotational symmetry on the sphere, together with the strong Riesz inequality, to conclude the same thing. Another proof, but only for N − 2 ≤ λ < N , was recently given in [10]. This was done by proving a form of reflection positivity for inversions in spheres in RN , and generalizing a theorem of Li and Zhu [16]. This is the first rearrangement-free proof of HLS, but it is not valid for 0 < λ < N − 2. An elegant, rearrangement-free proof, this time only for λ = N − 2, is in [5]. In this note we show how the new method developed in [11] can be adapted to the HLS problem to yield a proof for all 0 < λ < N . We also apply the method directly to a proof of (1.4) in Section 3.

2. Main result We shall prove

Sharp HLS Inequality

57

Theorem 2.1. Let 0 < λ < N and p := 2N/(2N − λ). Then (1.1) holds for any f, g ∈ Lp (RN ). Equality holds if and only if f (x) = c H(δ(x − a)) ,

g(y) = c H(δ(x − a))

for some c, c ∈ C, δ > 0 and a ∈ RN (unless f ≡ 0 or g ≡ 0). Here H is the function in (1.2). In other words, we prove that the function H in (1.2) is the unique optimizer in inequality (1.1) up to translations, dilations and multiplication by a constant. The stereographic projection (see Appendix A) defines a bijection between RN and the punctured sphere SN \ {(0, . . . , 0, −1)}. We consider the sphere SN as a subset of RN +1 with coordinates (ω1 , . . . , ωN +1 ) satisfying N +1 2 j=1 ωj = 1, and (non-normalized) measure denoted by dω. Via stereographic projection Theorem 2.1 is equivalent to Theorem 2.2. Let 0 < λ < N and p := 2N/(2N − λ). Then for any f, g ∈ Lp (SN ) 1−λ/N Γ(N ) f (ω) g(η) λ/2 Γ((N − λ)/2) dω dη ≤ π

f p g p SN ×SN |ω − η|λ Γ(N − λ/2) Γ(N/2) (2.1) with equality if and only if f (ω) =

c , (1 − ξ · ω)(2N −λ)/2

g(ω) =

c , (1 − ξ · ω)(2N −λ)/2

(2.2)

for some c, c ∈ C and some ξ ∈ RN +1 with |ξ| < 1 (unless f ≡ 0 or g ≡ 0). In particular, with ξ = 0, f = g ≡ 1 are optimizers. We conclude this section by recalling that (2.1) can be differentiated at the endpoints λ = 0 and λ = N , where the inequality turns into an equality. In this way one obtains the logarithmic HLS inequality [7, 4] and a conformally invariant logarithmic Sobolev inequality [3].

3. The sharp Sobolev inequality on the sphere In this section we derive the classical Sobolev inequality (1.4). This case is simpler than the general λ case of the HLS inequality, but it already contains the main elements of our strategy. It is easiest for us to work in the formulation on the sphere SN . N +1 2 We consider SN as a subset of RN +1 , i.e., {(ω1 , . . . , ωN +1 ) : j=1 ωj = 1}. We recall that the conformal Laplacian on SN is defined by L := −Δ +

N (N − 2) , 4

58


where Δ is the Laplace-Beltrami operator on SN , and we denote the associated quadratic form by N (N − 2) 2 |∇u|2 + dω . E[u] := |u| 4 SN The sharp Sobolev inequality on SN is Theorem 3.1. For all u ∈ H 1 (SN ) one has (N −2)/N 2/N N (N − 2) 2π (N +1)/2 |u|2N/(N −2) dω , E[u] ≥ 4 Γ((N + 1)/2) SN (3.1) with equality if and only if u(ω) = c (1 − ξ · ω)−(N −2)/2 for some c ∈ C and some ξ ∈ R

N +1

(3.2)

with |ξ| < 1.

See Appendix A for the equivalence of the RN -version (1.4) and the S -version (3.1) of the Sobolev inequality. In the proof of Theorem 3.1 we shall make use of the following elementary formula. N

Lemma 3.2. For all u ∈ H 1 (SN ) one has N +1

E[ωj u] = E[u] + N

j=1

SN

|u|2 dω .

(3.3)

Proof. We begin by noting that for any smooth, real-valued function ϕ on SN one has |∇(ϕu)|2 = ϕ2 |∇u|2 + |u|2 |∇ϕ|2 + ϕ∇ϕ · ∇(|u|2 ) . Hence an integration by parts leads to 2 ϕ |∇u|2 − ϕ(Δϕ)|u|2 dω . |∇(ϕu)|2 dω = SN

SN

We apply this identity to ϕ(ω) = ωj . Using the fact that −Δωj = N ωj , we find

SN

|∇(ωj u)|2 dω =

SN

ωj2 |∇u|2 + N |u|2 dω .

Summing over j yields (3.3) and completes the proof.

We are now ready to give a short Proof of Theorem 3.1. It is well-known that there is an optimizer U for inequality (3.1). (Using the stereographic projection, one can deduce this for instance from the existence of an optimizer on RN ; see [18].) As a preliminary remark we note that any optimizer is a complex multiple of a non-negative function. Indeed, if u = a + ib with a and b


59

real functions, then E[u] = E[a] + E[b]. We also note that the right side of (3.1) is a2 + b2 q/2 with q = 2N/(N − 2) > 2. By the triangle inequality, a2 + b2 q/2 ≤ a2 q/2 + b2 q/2 . This inequality is strict unless a ≡ 0 or b2 = λ2 a2 for some λ ≥ 0. Therefore, if U = A + iB is an optimizer for (3.1), then either one of A and B is identically equal to zero or else both A and B are optimizers and |B| = λ|A| for some λ > 0. For any real u ∈ H 1 (SN ) its positive and negative parts u± belong to H 1 (SN ) and satisfy ∂u± /∂ωk = ±χ{±u>0} ∂u/∂ωk in the sense of distributions. (This can be proved similarly to [19, Thm. 6.17].) Thus E[u] = E[u+ ] + E[u− ] for real u. Moreover, u 2q ≤ u+ 2q + u− 2q for real u with strict inequality unless u has a definite sign. Therefore, if U = A + iB is an optimizer for (3.1), then both A and B have a definite sign. We conclude that any optimizer is a complex multiple of a non-negative function. Hence we may assume that U ≥ 0. It is important for us to know that we may confine our search for optimizers to functions u satisfying the ‘center of mass condition’ ωj |u(ω)|q dω = 0 , j = 1, . . . , N + 1 . (3.4) SN

It is well known, and used in many papers on this subject (e.g., [15, 21, 8]), that (3.4) can be assumed, and we give a proof of this fact in Appendix B. It uses three facts: one is that inequality (3.1) is invariant under O(N + 1) rotations of SN . The second is that the stereographic projection, that maps RN to SN , leaves the optimization problem invariant. The third is that the RN -version, (1.4), of inequality (3.1) is invariant under dilations F (x) → δ (N −2)/2 F (δx). Our claim in the appendix is that by a suitable choice of δ and a rotation we can achieve (3.4). Therefore we may assume that the optimizer U satisfies (3.4). Imposing this constraint does not change the positivity of U . We shall prove that the only optimizer with this property is the constant function (which leads to the stated expression for the sharp constant). It follows, then, that the only optimizers without condition (3.4) are those functions for which the dilation and rotation, just mentioned, yields a constant. In Appendix B we identify those functions as the functions stated in (3.2). The second variation of the quotient E[u]/ u 2q around u = U shows that q U dω − (q − 1)E[U ] U q−2 |v|2 dω ≥ 0 (3.5) E[v]

SN

SN

for all v with U q−1 v dω = 0. Because U satisfies condition (3.4) we may choose v(ω) = ωj U (ω) in (3.5) and sum over j. We find N +1

j=1

E[ωj U ] ≥ (q − 1) E[U ] .

(3.6)

60


On the other hand, Lemma 3.2 with u = U implies N +1

E[ωj U ] = E[U ] + N U 2 dω , SN

j=1

which, together with (3.6), yields U 2 dω ≥ (q − 2) E[U ] . N SN

Recalling that q − 2 =

4 N −2 ,

we see that this is the same as |∇U |2 dω ≤ 0 . SN

We conclude that U is the constant function, as we intended to prove.

4. The sharp HLS inequality on the sphere Our goal in this section is to compute the sharp constant in inequality (2.1) on the sphere SN . We outline our argument in Subsection 4.1 and reduce everything to the proof of a linear inequality. After some preparations in Subsection 4.2 we shall prove this inequality in Subsection 4.3. 4.1. Strategy of the proof Step 1. The optimization problem corresponding to (2.1) admits an optimizing pair with f = g. The fact that one only needs to consider f = g follows from the positive definiteness of the kernel |x − y|−λ . The existence of an optimizer has been proved in [18] for the inequality (1.1) on RN and follows, as explained in Appendix A, via stereographic projection for the inequality on the sphere; for a rearrangement-free proof, see [20] and also the arguments in [11], which easily carry over to the RN case. We claim that any optimizer for problem (2.1) with f = g is a complex multiple of a non-negative function. Indeed, if we denote the left side of (1.1) with g = f by I[f ] and if f = a + ib for real functions a and b, then I[f ] =

I[a] + I[b].

Moreover, for any numbers α, β, γ, δ ∈ R one has 2 2 γ 2 + δ 2 with strict inequality unless αγ + βδ ≥ 0 αγ + βδ ≤ α + β −λ is strictly positive, we infer that and αδ = βγ. Since √ the kernel |x − y| 2 2 I[a] + I[b] ≤ I[ a + b ] for any real functions a, b with strict inequality unless a(x)a(y) + b(x)b(y) ≥ 0 and a(x)b(y) = a(y)b(x) for almost every x, y ∈ RN . From this one easily concludes that any optimizer is a complex multiple of a non-negative function. We denote a non-negative optimizer for problem (2.1) by h := f = g. Since h satisfies the Euler-Lagrange equation h(η) dη = c hp−1 (ω) , λ |ω − η| N S we see that h is strictly positive.


61

Step 2. As in the proof of Theorem 3.1, we may assume that the center of mass of hp vanishes, that is, ωj h(ω)p dω = 0 for j = 1, . . . , N + 1 . (4.1) SN

We shall prove that the only non-negative optimizer satisfying (4.1) is the constant function. Then, for exactly the same reason as in the proof of Theorem 3.1, the only optimizers without condition (4.1) are the ones stated in (2.2). We also note that, once we know that a constant is the optimizer, the expression for the sharp constant follows by a computation (see the l = 0 case of Corollary 4.3 below). Step 3. The second variation around the optimizer h shows that f (ω) f (η) h(ω) h(η) p dω dη h dω − (p − 1) dω dη hp−2 |f |2 dω ≤ 0 |ω − η|λ |ω − η|λ (4.2) for any f satisfying hp−1 f dω = 0. Note that the term hp−2 causes no problems (despite the fact that p < 2) since h is strictly positive. Because of (4.1) the functions f (ω) = ωj h(ω) satisfy the constraint p−1 h f dω = 0. Inserting them in (4.2) and summing over j we find h(ω) h(η) h(ω) ω · η h(η) dω dη − (p − 1) dω dη ≤ 0 . (4.3) λ |ω − η| |ω − η|λ Step 4. This is the crucial step! The proof of Theorem 2.2 is completed by showing that for any (not necessarily maximizing) h the inequality opposite to (4.3) holds and is indeed strict unless the function is constant. This is the statement of the following theorem with α = λ/2, noting that p − 1 = α/(N − α). Proposition 4.1. Let 0 < α < N/2. For any f on SN one has f (ω) ω · η f (η) f (ω) f (η) α dω dη ≥ dω dη |ω − η|2α N −α |ω − η|2α

(4.4)

with equality iff f is constant. This proposition will be proved in Subsection 4.3. 4.2. The Funk–Hecke theorem We decompose L2 (SN ) into its O(N + 1)-irreducible components, ! Hl . L2 (SN ) =

(4.5)

l≥0

The space Hl is the space of restrictions to SN of harmonic polynomials on RN +1 which are homogeneous of degree l. It is well known that integral operators on SN whose kernels have the form K(ω · η) are diagonal with respect to this decomposition and their eigenvalues can be computed explicitly. A proof of the following Funk–Hecke

62


formula can be found, e.g., in [9, Sec. 11.4]. It involves the Gegenbauer poly(λ) nomials Cl , see [1, Chapter 22]. Proposition 4.2. Let K ∈ L1 ((−1, 1), (1 − t2 )(N −2)/2 dt). Then the operator on SN with kernel K(ω · η) is diagonal with respect to decomposition (4.5), and on the space Hl its unique eigenvalue is given by 1 (N −1)/2 κN,l K(t)Cl (t)(1 − t2 )(N −2)/2 dt , (4.6) −1

where κN,l

⎧ ⎪ ⎨2 = l ⎪ ⎩(4π)(N −1)/2

l! Γ((N −1)/2) (l+N −2)!

if N = 1 , l = 0 , if N = 1 , l ≥ 1 , if N ≥ 2 .

This proposition allows us to compute the eigenvalues of the family of operators appearing in Proposition 4.1. Corollary 4.3. Let −1 < α < N/2. The eigenvalue of the operator with kernel (1 − ω · η)−α on the subspace Hl is El = κN 2−α (−1)l where

κN =

Γ(1 − α) Γ(N/2 − α) , Γ(−l + 1 − α) Γ(l + N − α)

2π 1/2 22(N −1) π (N −1)/2

Γ((N −1)/2) Γ(N/2) (N −2)!

(4.7)

if N = 1 , if N ≥ 2 .

When α is a non-negative integer, formula (4.7) is to be understood by taking limits with fixed l. This result appears already (without proof) in [4]. Proof. By Proposition 4.2 we have to evaluate the integral (4.6) for the choice K(t) = (1 − t)−α . Our assertion follows from the β = (N − 2)/2 − α case of the formula 1 (N −1)/2 (1 + t)(N −2)/2 (1 − t)β Cl (t) dt (4.8) −1

= (−1)l

2N/2+β Γ(1 + β) Γ(N/2) Γ(l + N − 1) Γ(−N/2 + 2 + β) . l! Γ(N − 1) Γ(−l − N/2 + 2 + β) Γ(l + N/2 + 1 + β)

This formula, which is valid for β > −1, follows from [12, (7.311.3)] together (λ) (λ) with the fact that Cl (−t) = (−1)l Cl (t). As it stands, (4.8) is only valid for N ≥ 2. For N = 1 and l = 0, the (divergent) factors Γ(l + N − 1) and Γ(N − 1) need to be omitted, and for N = 1 and l ≥ 1, the divergent factor Γ(N − 1) in the denominator needs to be replaced by 12 .


63

4.3. Proof of Proposition 4.1 Using the fact that |ω − η|2 = 2(1 − ω · η), we see that the assertion is equivalent to f (ω) f (η) f (ω) f (η) N − 2α dω dη ≤ dω dη . (1 − ω · η)α−1 N −α (1 − ω · η)α Both quadratic forms are diagonal with respect to decomposition (4.5) and their eigenvalues on the subspace Hl are given by Corollary 4.3. For simplicity, we first assume that α = 1. The eigenvalue of the right side is (N −2α)El /(N − ˜l , which is α), with El given by (4.7), and the eigenvalue of the left side is E El with α replaced by α − 1. Noting that ˜l = El E

(α − 1)(N − 2α) (l − 1 + α)(l + N − α)

and that El > 0 and α < N/2, we see that the conclusion of the theorem is equivalent to the inequality 1 α−1 ≤ (l − 1 + α)(l + N − α) N −α for all l ≥ 0. This inequality is elementary to prove, distinguishing the cases α > 1 and α < 1. Finally, the case α = 1 is proved by letting α → 1 for fixed l. Strictness of inequality (4.4) for non-constant f follows from the fact that the above inequalities are strict unless l = 0. This completes the proof of Proposition 4.1.

Appendix A. Equivalence of Theorems 2.1 and 2.2 In this appendix we consider the stereographic projection S : RN → SN and its inverse S −1 : SN → RN given by 2x 1 − |x|2 ω1 ωN −1 S(x) = , (ω) = , . . . , , S . 1 + |x|2 1 + |x|2 1 + ωN +1 1 + ωN +1 The Jacobian of this transformation (see, e.g., [19, Thm. 4.4]) is N 2 , JS (x) = 1 + |x|2 which implies that

ϕ(ω) dω = SN

RN

ϕ(S(x))JS (x) dx

(A.1)

for any integrable function ϕ on SN . We now explain the equivalence of (1.1) and (2.1) for each fixed pair of parameters λ and p with p = 2N/(2N − λ). There is a one-to-one correspondence between functions f on SN and functions F on RN given by F (x) = |JS (x)|1/p f (S(x)) .

(A.2)

64


It follows immediately from (A.1) that f ∈ Lp (SN ) if and only if F ∈ Lp (RN ), and in this case f p = F p . Moreover, we note the fact that 2 2 2 2 |ω − η| = |x − y| 1 + |x|2 1 + |y|2 for ω = S(x) and η = S(y), where |ω − η| is the chordal distance between ω and η, i.e., the Euclidean distance in RN +1 . With the help of this relation one easily verifies that F (x) F (y) f (ω) f (η) dx dy = dω dη . λ |x − y| |ω − η|λ N N N N R ×R S ×S This shows that the sharp constants in (1.1) and (2.1) coincide and that there is a one-to-one correspondence between optimizers. In particular, the function f ≡ 1 on SN corresponds to the function |JS (x)|1/p = 2N/p H(x) on RN with H given in (1.2). Similarly, when p = 2N/(N − 2), and F and f are related via (A.2), then N (N − 2) 2 2 2 |f | |∇F | dx = |∇f | + dω , (A.3) 4 RN SN as can be checked by a direct computation.

Appendix B. The center of mass condition Here, we prove that by a suitable inequality preserving transformation of SN we may assume the center of mass conditions given in (3.4) and (4.1). We shall define a family of maps γδ,ξ : SN → SN depending on two parameters δ > 0 and ξ ∈ SN . To do so, we denote dilation on RN by Dδ , that is, Dδ (x) = δx. Moreover, for any ξ ∈ SN we choose an orthogonal (N + 1) × (N + 1) matrix O such that Oξ = (0, . . . , 0, 1) and we put γδ,ξ (ω) := OT S Dδ S −1 (Oω) for all ω ∈ SN \ {−ξ} and γδ,ξ (−ξ) := −ξ. This transformation depends only on ξ (and δ) and not on the particular choice of O. Indeed, a straightforward computation shows that 2δ (ω − (ω · ξ) ξ) γδ,ξ (ω) = (1 + ω · ξ) + δ 2 (1 − ω · ξ) (1 + ω · ξ) − δ 2 (1 − ω · ξ) + ξ. (1 + ω · ξ) + δ 2 (1 − ω · ξ) Lemma B.1. Let f ∈ L1 (SN ) with SN f (ω) dω = 0. Then there is a transformation γδ,ξ of SN such that γδ,ξ (ω)f (ω) dω = 0 . SN


65

Proof. We may assume that f ∈ L1 (SN ) is normalized by SN f (ω) dω = 1. We shall show that the RN +1 -valued function γ1−r,ξ (ω)f (ω) dω , 0 < r < 1 , ξ ∈ SN , F (rξ) := SN

has a zero. First, note that because of γ1,ξ (ω) = ω for all ξ and all ω, the limit of F (rξ) as r → 0 is independent of ξ. In other words, F is a continuous function on the open unit ball of RN +1 . In order to understand its boundary behavior, one easily checks that for any ω = −ξ one has limδ→0 γδ,ξ (ω) = ξ, and that this convergence is uniform on {(ω, ξ) ∈ SN × SN : 1 + ω · ξ ≥ ε} for any ε > 0. This implies that lim F (rξ) = ξ

r→1

uniformly in ξ .

Hence F is a continuous function on the closed unit ball, which is the identity on the boundary. The assertion is now a consequence of Brouwer’s fixed point theorem. In the proof of Theorem 3.1 we use Lemma B.1 with f = |u|q . Then the new function u ˜(ω) = |Jγ −1 (ω)|1/q u(γ −1 (ω)), with γ = γδ,ξ of Lemma B.1, satisfies the center of mass condition (3.4). Moreover, since rotations of the sphere, stereographic projection S and the dilations Dδ leave the inequality invariant, u can be replaced by u ˜ in (3.1) without changing the values of each side. In particular, if U is an optimizer, our proof in Section 3 shows that the ˜ is a constant, which means that the original U is a constant corresponding U 1/q times |Jγ | . It is now a matter of computation, using the explicit form of γδ,ξ , to verify that all such functions have the form of (3.2). Conversely, let us verify that all the functions given in (3.2) are optimizers. By the rotation invariance of inequality (3.1), we can restrict our attention to the case ξ = (0, . . . , 0, r) with 0 < r < 1. These functions correspond via stereographic projection, (A.2), to dilations of a constant times the function H in (1.2). Because of the dilation invariance of inequality (1.4) and because of the fact that we already know that H, which corresponds to the constant on the sphere, is an optimizer, we conclude that any function of the form (3.2) is an optimizer. We have discussed the derivative (Sobolev) version of the λ = N − 2 case of (2.1). Exactly the same considerations show the invariance of the fractional integral for all 0 < λ < N . Acknowledgement We thank Richard Bamler for valuable help with Appendix B.

References [1] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 edition. Dover Publications, New York, 1992.

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[2] Th. Aubin, Problèmes isoperimétriques et espaces de Sobolev. J. Differ. Geometry 11 (1976), 573–598. [3] W. Beckner, Sobolev inequalities, the Poisson semigroup, and analysis on the sphere Sn . Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 11, 4816–4819. [4] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. of Math. (2) 138 (1993), no. 1, 213–242. [5] E. A. Carlen, J. A. Carrillo, M. Loss, Hardy–Littlewood–Sobolev inequalities via fast diffusion flows, Proc. Nat. Acad. USA 107 (2010), no. 46, 19696–19701. [6] E. A. Carlen, M. Loss, Extremals of functionals with competing symmetries. J. Funct. Anal. 88 (1990), no. 2, 437–456. [7] E. A. Carlen, M. Loss, Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on Sn . Geom. Funct. Anal. 2 (1992), no. 1, 90–104. [8] S-Y. A. Chang, P. C. Yang, Prescribing Gaussian curvature on S 2 . Acta Math. 159 (1987), no. 3–4, 215–259. [9] A. Erdélyi, M. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions. Vol. II. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Melbourne, FL, 1981. [10] R. L. Frank, E. H. Lieb, Inversion positivity and the sharp Hardy–LittlewoodSobolev inequality. Calc. Var. PDE 39 (2010), no. 1–2, 85–99. [11] R. L. Frank, E. H. Lieb, Sharp constants in several inequalities on the Heisenberg group. Preprint (2010), arXiv:1009.1410. [12] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products. Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. [13] G. H. Hardy, J. E. Littlewood, Some properties of fractional integrals (1). Math. Z. 27 (1928), 565–606. [14] G. H. Hardy, J. E. Littlewood, On certain inequalities connected with the calculus of variations. J. London Math. Soc. 5 (1930), 34–39. [15] J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sr. A-B 270 (1970), A1645–A1648. [16] Y. Y. Li, M. Zhu, Uniqueness theorems through the method of moving spheres. Duke Math. J. 80 (1995), no. 2, 383–417. [17] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Appl. Math. 57 (1977), no. 2, 93–105. [18] E. H. Lieb, Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. of Math. (2) 118 (1983), no. 2, 349–374. [19] E. H. Lieb, M. Loss, Analysis. Second edition. Graduate Studies in Mathematics 14, Amer. Math. Soc., Providence, RI, 2001. [20] P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), no. 2, 45–121. [21] E. Onofri, On the positivity of the effective action in a theory of random surfaces. Comm. Math. Phys. 86 (1982), no. 3, 321–326. [22] G. Rosen, Minimum value for c in the Sobolev inequality ϕ3 ≤ c∇ϕ3 . SIAM J. Appl. Math. 21 (1971), 30–32. [23] S. L. Sobolev, On a theorem of functional analysis. Mat. Sb. (N.S.) 4 (1938), 471–479; English transl. in Amer. Math. Soc. Transl. Ser. 2 34 (1963), 39–68.


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[24] G. Talenti, Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353–372. Rupert L. Frank Department of Mathematics Princeton University Princeton NJ 08544 USA e-mail: [email protected] Elliott H. Lieb Department of Mathematics Princeton University Princeton NJ 08544 USA e-mail: [email protected]


Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example Dorothee D. Haroske Dedicated to Professor David E. Edmunds and Professor W. Desmond Evans on the occasion of their 80th and 70th birthday, respectively.

Abstract. We study dichotomy questions for weighted function spaces of Besov and Triebel–Lizorkin type where the weight is related to the distance of some point to a d-set Γ, 0 < d < n, and the trace is taken on Γ. We can prove that – depending on the function space, the weight and the set Γ – there occurs an alternative: either the trace on Γ exists, or smooth functions compactly supported outside Γ are dense in the space. Such a phenomenon is called dichotomy. The paper is based on trace results in [33] and the atomic decomposition in [18] and perfectly fits together with corresponding unweighted results in [51]. Mathematics Subject Classification (2010). 46E35. Keywords. Muckenhoupt weights, function spaces, traces, density, d-sets, atoms.

1. Introduction The main goal of this paper is to demonstrate the strength of the recently developed atomic approach in weighted function spaces in view of trace results. In particular, we concentrate on spaces of Besov and Triebel–Lizorkin type, thus including Sobolev spaces, where the weight function belongs to some Muckenhoupt class. Such questions are of particular interest in view of boundary value problems of elliptic operators, where some singular behaviour near the boundary (characterised by the appropriate Muckenhoupt class) is admitted. Usually one starts with assertions about traces on hyperplanes and tries to transfer these results to bounded domains with sufficiently smooth boundary afterwards. Further studies may concern compactness or regularity results, leading to the investigation of spectral properties. However, the problem is not so simple and little is known so far. First partial results can be found in [45] and [30] for domains Ω with smooth boundaries ∂Ω and

70

D.D. Haroske γ

Muckenhoupt weights of type w(x) = (dist(x, ∂Ω)) , γ > −1. This was further extended to fractal d-sets Γ in [34], using the atomic approach [18] and based on ideas for the unweighted case in [48]. Recently we noticed renewed interest in trace questions in weighted spaces which led to the papers [1, 19] dealing with some modification of the most prominent Muckenhoupt weight function, w(x) = |x| . In this paper we stick to the situation where the trace is taken on a compact d-set Γ ⊂ Rn , 0 < d < n. We rely on results obtained in [33, 34] and combine it with the recently studied dichotomy question explained in more detail below. Roughly speaking, one links two phenomena: the existence of a trace on Γ (by completion of pointwise traces) on the one hand, and the density of smooth functions compactly supported outside Γ, denoted by D(Rn \ Γ), on the other hand. Though it is rather obvious that the density of D(Rn \ Γ) in some space prevents the existence of a properly defined trace, it is not clear (and also not true in general) that there is some close connection vice versa. However, in our setting there appears an alternative that either we have an affirmative answer to the density question or traces exist. The criterion which case occurs naturally depends on the function spaces, the weight and the set Γ. More precisely, for a d-set Γ, 0 < d < n, we deal with the weight dist (x, Γ)κ , if dist (x, Γ) ≤ 1, wκ,Γ (x) = 1, if dist (x, Γ) ≥ 1, when κ > −(n − d). Then wκ,Γ is a Muckenhoupt weight and fits in our s scheme for weighted spaces of Besov type Bp,q (Rn , w) or Triebel–Lizorkin s n type Fp,q (R , w). As mentioned above, corresponding trace results were obtained in [33], complemented by some first (sufficient) conditions when D(Rn \ s s Γ) is dense in Bp,q (Rn , wκ,Γ ) or Fp,q (Rn , wκ,Γ ). We can strengthen this now as follows: in case of Besov spaces we obtain that s > n−d+κ , 0 < q < ∞, s n p trΓ Bp,q (R , wκ,Γ ) exists for , 0 < q ≤ min(p, 1), s = n−d+κ p and

D(R \ Γ) is dense in n

s Bp,q (Rn , w)

for

s= s
0 such that c1 αk ≤ βk ≤ c2 αk for all k ∈ N; similarly for positive functions. Given two (quasi-) Banach spaces X and Y , we write X → Y if X ⊂ Y and the natural embedding of X in Y is continuous. Let for m ∈ Zn and ν ∈ N0 , Qν,m denote the n-dimensional cube with sides parallel to the axes of coordinates, centered at 2−ν m and with side length 2−ν . For x ∈ Rn and r > 0, let B(x, r) denote the open ball B(x, r) = {y ∈ Rn : |y − x| < r}. For convenience, let both dx and |·| stand for the (n-dimensional) Lebesgue measure in the sequel. All unimportant positive constants will be denoted by c, occasionally with subscripts. As we shall always deal with function spaces on Rn , we may often omit the ‘Rn ’ from their notation for convenience. 2.1. Muckenhoupt weights We briefly recall some fundamentals on Muckenhoupt classes Ap . By a weight n w we shall always mean a locally integrable function w ∈ Lloc 1 (R ), positive a.e. in the sequel. Let M stand for the Hardy-Littlewood maximal operator given by 1 |f (y)| dy, x ∈ Rn , (2.1) M f (x) = sup B(x,r)∈B |B(x, r)| B(x,r) where B is the collection of all open balls B(x, r) centred at x ∈ Rn , r > 0. Definition 2.1. Let w be a weight on Rn . (i) Then w belongs to the Muckenhoupt class Ap , 1 < p < ∞, if there exists a constant 0 < A < ∞ such that for all balls B the following inequality holds, 1/p 1/p 1 1 −p /p w(x) dx · w(x) dx ≤ A. (2.2) |B| B |B| B (ii) Then w belongs to the Muckenhoupt class A1 if there exists a constant 0 < A < ∞ such that the inequality M w(x) ≤ Aw(x) holds for almost all x ∈ R . n

(2.3)

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D.D. Haroske

(iii) The Muckenhoupt class A∞ is given by A∞ = Ap .

(2.4)

p>1

Since the pioneering work of Muckenhoupt [27–29], these classes of weight functions have been studied in great detail, we refer, in particular, to the monographs [16], [42, Ch. V], [43], and [44, Ch. IX] for a complete account on the theory of Muckenhoupt weights. As usual, we use the abbreviation w(x) dx, (2.5) w(Ω) = Ω

where Ω ⊂ Rn is some bounded, measurable set. Then a weight w on Rn belongs to Ap , 1 ≤ p < ∞, if, and only if, 1/p c 1 f (y) dy ≤ f p (x)w(x) dx |B| B w(B) B holds for all non-negative f and all balls B. In particular, with E ⊂ B and f = χE , this implies that 1/r |E| w(E) ≤c , E ⊂ B, w ∈ Ar , r ≥ 1. (2.6) |B| w(B) Another property of Muckenhoupt weights that will be used in the sequel is that w ∈ Ap , p > 1, implies the existence of some number r < p such that w ∈ Ar . This is closely connected with the so-called ‘reverse Hölder inequality’, see [42, Ch. V, Prop. 3, Cor.]. In our case this fact will re-emerge in the number (2.7) rw = inf{r ≥ 1 : w ∈ Ar }, w ∈ A∞ , that plays an essential rˆ ole later on. Obviously, 1 ≤ rw < ∞, and w ∈ Arw if, and only if, rw = 1. Example 2.2. We restrict ourselves to a ‘fractal’ example studied in [18], and refer for further examples to [12, 20, 21]. Let Γ ⊂ Rn be a d-set, 0 < d < n, in the sense of [48, Def. 3.1], see also [23, 24] (which is different from [11]), i.e., there exists a Borel measure μ in Rn such that supp μ = Γ compact, and there are constants c1 , c2 > 0 such that for arbitrary γ ∈ Γ and all 0 < r < 1 holds c1 rd ≤ μ(B(γ, r) ∩ Γ) ≤ c2 rd . Recall that some self-similar fractals are outstanding examples of d-sets; for instance, the usual (middle-third) Cantor set in R1 is a d-set for d = ln 2/ ln 3, and the Koch curve in R2 is a d-set for d = ln 4/ ln 3. We proved in [18] that the weight wκ,Γ , given by dist (x, Γ)κ , if dist (x, Γ) ≤ 1, wκ,Γ (x) = (2.8) 1, if dist (x, Γ) ≥ 1, satisfies wκ,Γ ∈ Ap

if, and only if,

− (n − d) < κ < (n − d)(p − 1),

1 < p < ∞,

Dichotomy in Muckenhoupt Weighted Function Space and wκ,Γ ∈ A1 if −(n − d) < κ ≤ 0. Consequently, rwκ,Γ = 1 +

73

max(κ,0) n−d .

Remark 2.3. For a refined study of the singularity behaviour of Muckenhoupt A∞ weights we introduced in [21] the notion of the set of singularities Ssing (w) for w ∈ A∞ by " w(Qν,m ) n Ssing (w) = x0 ∈ R : =0 inf Qν,m x0 |Qν,m | w(Q ) ν,m ∪ x0 ∈ Rn : sup =∞ . (2.9) Qν,m x0 |Qν,m | This is a special case of Ssing (w1 , w2 ) defined in [21] with w2 ≡ 1, w1 ≡ w. In case of the weight wκ,Γ introduced in (2.8) where Γ is a d-set in Rn with 0 < d < n and κ > −(n − d), one can prove that Γ, if κ = 0, Ssing (wκ,Γ ) = ∅, if κ = 0, based on the estimate wκ,Γ (Qν,m ) ∼ |Qν,m |

if 2Qν,m ∩ Γ = ∅,

1, 2

−νκ

(2.10)

, otherwise,

see [18]. Note that we have |Ssing (wκ,Γ )| = 0 which reflects the general fact |Ssing (w)| = 0 for all w ∈ A∞ , cf. [21]. s s 2.2. Function spaces of type Bp,q (Rn , w) and Fp,q (Rn , w) with w ∈ A∞ Let w ∈ A∞ be a Muckenhoupt weight and 0 < p < ∞. Then the weighted Lebesgue space Lp (Rn , w) contains all measurable functions such that 1/p

f |Lp (Rn , w) = |f (x)|p w(x) dx (2.11) Rn

is finite. For p = ∞ one obtains the classical (unweighted) Lebesgue space, L∞ (Rn , w) = L∞ (Rn ),

w ∈ A∞ ;

(2.12)

we thus mainly restrict ourselves to p < ∞ in what follows. The Schwartz space S(Rn ) and its dual S (Rn ) of all complex-valued tempered distributions have their usual meaning here. Let ϕ0 = ϕ ∈ S(Rn ) be such that supp ϕ ⊂ {y ∈ Rn : |y| < 2}

and ϕ(x) = 1 if |x| ≤ 1 ,

−j

and for each j ∈ N let ϕj (x) = ϕ(2 x) − ϕ(2−j+1 x). Then {ϕj }∞ j=0 forms a smooth dyadic resolution of unity. Given any f ∈ S (Rn ), we denote by F f and F −1 f its Fourier transform and its inverse Fourier transform, respectively. Let f ∈ S (Rn ), then the Paley–Wiener–Schwartz theorem implies that F −1 (ϕj F f ) is an entire analytic function on Rn . Definition 2.4. Let w ∈ A∞ , 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and {ϕj }j∈N0 a smooth dyadic resolution of unity.

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D.D. Haroske

s (i) The weighted Besov space Bp,q (Rn , w) is the set of all distributions f ∈ n S (R ) such that s f |Bp,q (2.13) (Rn , w) = 2js F −1 (ϕj F f )|Lp (Rn , w) j∈N0 |q

is finite. s (ii) The weighted Triebel–Lizorkin space Fp,q (Rn , w) n butions f ∈ S (R ) such that js −1 s 2 |F (ϕj F f )(·)| j∈N0 |q f |Fp,q (Rn , w) =

is the set of all distri Lp (Rn , w)

(2.14)

is finite. s s Remark 2.5. The spaces Bp,q (Rn , w) and Fp,q (Rn , w) are independent of the particular choice of the smooth dyadic resolution of unity {ϕj }j appearing in their definitions. They are quasi-Banach spaces (Banach spaces for p, q ≥ 1), s (Rn , w) → S (Rn ), where the first embedding is dense if and S(Rn ) → Bp,q q < ∞, similarly for the F -case; cf. [4]. Moreover, for w0 ≡ 1 ∈ A∞ these are the usual (unweighted) Besov and Triebel–Lizorkin spaces; we refer, in particular, to the series of monographs [46–50] for a comprehensive treatment of the unweighted spaces. The above spaces with weights of type w ∈ A∞ have been studied systematically in [4,5], with subsequent papers [6,7]. It turned out that many of the results from the unweighted situation have weighted counterparts: e.g., we 0 (Rn , w) = hp (Rn , w), 0 < p < ∞, where the latter are Hardy spaces, have Fp,2 0 (Rn , w), see [4, Thm. 1.4], and, in particular, hp (Rn , w) = Lp (Rn , w) = Fp,2 1 < p < ∞, w ∈ Ap , see [43, Ch. VI, Thm. 1]. Concerning (classical) Sobolev spaces Wpk (Rn , w) (built upon Lp (Rn , w) in the usual way) it holds that k (Rn , w), Wpk (Rn , w) = Fp,2

k ∈ N0 ,

1 < p < ∞,

w ∈ Ap ,

(2.15)

cf. [4, Thm. 2.8]. Further details can be found in [2–5,15,16,36,37]. In [38] the above class of weights was extended in order to incorporate locally regular weights, too, creating in that way the class A oc p . We partly rely on our approaches in [18, 20–22]. We briefly recall the definition of atoms. Definition 2.6. Let K ∈ N0 and b > 1. (i) The complex-valued function a ∈ C K (Rn ) is said to be an 1K -atom if supp a ⊂ bQ0,m for some m ∈ Zn , and |Dα a(x)| ≤ 1 for |α| ≤ K, x ∈ Rn . (ii) Let s ∈ R, 0 < p ≤ ∞, and L + 1 ∈ N0 . The complex-valued function a ∈ C K (Rn ) is said to be an (s, p)K,L -atom if for some ν ∈ N0 , supp a ⊂ bQν,m for some m ∈ Zn , |Dα a(x)| ≤ 2−ν(s− p )+|α|ν for |α| ≤ K, x ∈ Rn , n

Rn

xβ a(x) dx = 0 for |β| ≤ L.

Dichotomy in Muckenhoupt Weighted Function Space

75

We shall denote an atom a(x) supported in some Qν,m by aν,m in the sequel. Choosing L = −1 in (ii) means that no moment conditions are required. (p) For 0 < p < ∞, ν ∈ N0 , m ∈ Zn , let χνm denote the p-normalised characteristic function of Qνm , that is νn νn 2 p , for x ∈ Qνm , p χ χ(p) (2.16) νm (x) = νm (x) = 2 0, for x ∈ / Qνm , (p)

such that χνm |Lp (Rn ) = 1. Now we can introduce suitable sequence spaces bpq (w) and fpq (w) for 0 < p < ∞, 0 < q ≤ ∞, w ∈ A∞ , by bpq (w) =

λ = {λν,m }ν,m : λν,m ∈ C,

λ|bpq (w) =

n λνm χ(p) νm |Lp (R , w)

m∈Zn

and

ν∈N0

|q < ∞

(2.17)

fpq (w) =

λ = {λν,m }ν,m : λν,m ∈ C,

∞ q 1/q (p) n λ |fpq (w) = (2.18) λνm χνm (·) Lp (R , w) < ∞ ν=0 m∈Zn

(with the usual modification if q = ∞). For convenience we adopt the usual notation 1 1 −1 −1 , σp,q = n , (2.19) σp = n p min(p, q) + + where 0 < p, q ≤ ∞. Then the atomic decomposition result used below reads as follows. Theorem 2.7. Let 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and w ∈ A∞ be a weight with rw given by (2.7). (i) Let K, L + 1 ∈ N0 with K ≥ (1 + s)+

and

L ≥ max −1, σp/rw − s .

(2.20)

Then f ∈ S(R ) belongs to if, and only if, it can be written as a series ∞

λνm aν,m (x), converging in S (Rn ), (2.21) f= n

s Bp,q (Rn , w)

ν=0 m∈Zn

where aν,m (x) are 1K -atoms (ν = 0) or (s, p)K,L -atoms (ν ∈ N) and λ ∈ bpq (w). Furthermore, inf λ|bpq (w)

(2.22)

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D.D. Haroske

s is an equivalent quasi-norm in Bp,q (Rn , w), where the infimum ranges over all admissible representations (2.21). (ii) Let K, L + 1 ∈ N0 with (2.23) K ≥ (1 + s)+ and L ≥ max −1, σp/rw ,q − s . s Then f ∈ S(Rn ) belongs to Fp,q (Rn , w) if, and only if, it can be written as a series (2.21) where aν,m (x) are 1K -atoms (ν = 0) or (s, p)K,L atoms (ν ∈ N) and λ ∈ fpq (w). Furthermore,

inf λ|fpq (w)

(2.24)

s is an equivalent quasi-norm in Fp,q (Rn , w), where the infimum ranges over all admissible representations (2.21).

Remark 2.8. The above result coincides with [18, Thm. 3.10], cf. also [2, Theorem 5.10]. Notational agreement. We adopt the nowadays usual custom to write Asp,q s s instead of Bp,q or Fp,q , respectively, when both scales of spaces are meant simultaneously in some context. 2.3. Continuous Embeddings We collect some embedding results for weighted spaces that will be used later. Recall that we deal with function spaces on Rn exclusively, and will thus omit the ‘Rn ’ from their notation. Proposition 2.9. Let w1 and w2 be two A∞ weights and let −∞ < s2 ≤ s1 < ∞, 0 < p1 , p2 ≤ ∞, 0 < q1 , q2 ≤ ∞. We put 1 1 1 1 1 1 = − and = − . (2.25) p∗ p2 p1 + q∗ q2 q1 + Then id : Bps11 ,q1 (w1 ) → Bps22 ,q2 (w2 ) is continuous if, and only if, " w2 (Qν,m )1/p2 −ν(s1 −s2 ) |p∗ 2 w1 (Qν,m )1/p1 m∈Zn

(2.26)

∈ q ∗ .

(2.27)

ν∈N0

Remark 2.10. For the proof and further details, also concerning questions of compactness, we refer to [20]. In view of (2.12) it is clear that we obtain unweighted Besov spaces if p1 = p2 = ∞. Then by (2.5), w1 (Qν,m ) = w2 (Qν,m ) = 2−νn for all ν ∈ N0 and m ∈ Zn , such that (2.27) leads to p∗ = ∞, i.e., p1 ≤ p2 , and n n δ∗ = s1 − − s2 + > 0, (2.28) p1 p2 with the extension to δ∗ = 0 if q1 ≤ q2 , i.e., q ∗ = ∞. In [20,21] we concentrated on the interplay between smoothness parameters and properties of the weight. We exemplify it for our special weight wκ,Γ .


77

Corollary 2.11. Let Γ ⊂ Rn be a d-set, 0 < d < n, and wκ,Γ be given by (2.8) with κ > −(n − d). Let the parameters satisfy −∞ < s2 ≤ s1 < ∞,

0 < p1 < ∞,

0 < p2 ≤ ∞,

0 < q1 , q2 ≤ ∞. (2.29)

The embedding idκ,Γ : Bps11 ,q1 (wκ,Γ ) → Bps22 ,q2 is continuous if, and only if, p1 ≤ p2

and

2

−ν(δ∗ − max(κ,0) ) p

1

ν∈N0

(2.30)

∈ q ∗ .

(2.31)

In [20, 21] we also considered situations where both source and target space are weighted with the same w ∈ A∞ . Here we shall only need the following basic observation. Proposition 2.12. Let 0 < q ≤ ∞, 0 < p < ∞, s ∈ R and w ∈ A∞ . (i) Let −∞ < s1 ≤ s0 < ∞ and 0 < q0 ≤ q1 ≤ ∞. Then 0 1 Asp,q (w) → Asp,q (w)

and

Asp,q0 (w) → Asp,q1 (w).

(2.32)

(ii) We have s s s (w) → Fp,q (w) → Bp,max(p,q) (w). Bp,min(p,q)

(2.33)

(iii) Assume that there are numbers c > 0, d > 0 such that for all cubes, w (Qν,m ) ≥ c2−νd ,

ν ∈ N0 ,

m ∈ Zn .

(2.34)

Let 0 < p0 < p < p1 < ∞, −∞ < s1 < s < s0 < ∞ satisfy s0 −

d d d = s − = s1 − . p0 p p1

(2.35)

Then Bps00 ,q (w) → Bps11 ,q (w)

and

s Bps00 ,p (w) → Fp,q (w) → Bps11 ,p (w). (2.36)

Remark 2.13. These embeddings are natural extensions from the unweighted case w ≡ 1, see [46, Prop. 2.3.2/2, Thm. 2.7.1] and [41, Thm. 3.2.1]. The above result essentially coincides with [4, Thm. 2.6] and can be found in [20, Prop. 1.8]. Example 2.14. Assume that inf m∈Zn w(Q0,m ) ≥ c > 0, then (2.6) implies d ≥ nrw in (2.34). In particular, for our model weight wκ,Γ the embeddings (2.36) can be exemplified as follows, recall also (2.10). Let 0 < p0 −(n − d). Assume that s0 −

max(κ, 0) + n max(κ, 0) + n max(κ, 0) + n = s1 − =s− . p0 p p1

Then (2.36) holds for w = wκ,Γ .

(2.37)

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D.D. Haroske

3. Dichotomy 3.1. Traces on hyperplanes and d-sets Let Γ ⊂ Rn , 0 σp , w ∈ A∞ . Suppose there exists some c > 0 such that for all ϕ ∈ D(Rn ), ϕ|Γ |Lp (Γ) ≤ c ϕ|Asp,q (Rn , w) ,

(3.1)

where the trace on Γ is taken pointwise. By the density of D(Rn ) in Asp,q (Rn , w) for p, q < ∞ and the completeness of Lp (Γ) one can thus define for f ∈ Asp,q (Rn , w) its trace trΓ f = f |Γ on Γ in the standard way. Remark 3.1. Since our main goal in this paper are density questions in contrast to the existence of traces, we shall not further discuss possible extensions of the above approach to q = ∞. But something can be done as far as traces are concerned; we refer to [48, Cor. 18.12], [50, Prop. 1.172]. We recall what is known in the unweighted situation for hyperplanes and d-sets. For m ∈ N, 1 ≤ m ≤ n − 1, we adopt the usual convention to identify an m-dimensional hyperplane in Rn with Rm . Proposition 3.2. Let 0 

n−m p

A = B, A = F,

+ m( p1 − 1)+ ,

s s− n−m trRm Bp,q (Rn ) = Bp,q p (Rm ), s s− n−m s− n−m trRm Fp,q (Rn ) = Fp,p p (Rm ) = Bp,p p (Rm ). (ii) Let Γ be a compact d-set, 0 < d < n. Then n−d 0 < q ≤ min(p, 1), p n trΓ Ap,q (R ) = Lp (Γ) if 0 

A = B, A = F,

n−d p ,

s s− n−d (Rn ) = Bp,q p (Γ). trΓ Bp,q Remark 3.3. The above trace results have a long history, we refer to [8, 17, 31, 32] for the Besov cases p ≥ 1, and [13], whereas their F -counterparts can be found in [14,47]. The situation (ii) was essentially solved in [48] with some later additions, see also [50].


79

In case of Muckenhoupt weighted spaces of the above type and traces on hyperplanes one can find first trace results in [45, Thm. 3.6.4/2], and recently for spaces of type Asp,q (wα,β ) with |x|α , if |x| ≤ 1 , wα,β (x) = with α > −n, β > −n, |x|β , if |x| > 1 , in [1, 19], where the latter is also based on the atomic approach. We study traces of weighted spaces on d-sets and rely on the following related results in [33, 34]. Proposition 3.4. Let 0 −(n − d). Then κ+n−d 0 < q ≤ min(p, 1), A = B, p trΓ Ap,q (Rn , wκ,Γ ) = Lp (Γ) if (3.2) 0 

κ+n−d , p

s s− κ s− κ+n−d (Rn , wκ,Γ ) = trΓ Bp,q p (Rn ) = Bp,q p (Γ), trΓ Bp,q s s− κ s− κ+n−d (Rn , wκ,Γ ) = trΓ Bp,p p (Rn ) = Bp,p p (Γ). trΓ Fp,q

(3.3) (3.4)

3.2. Dichotomy If Γ is a compact d-set in Rn or the hyperplane Rm in Rn , then we abbreviate now (3.5) DΓ = D(Rn \ Γ). Recall that D(Rn ) is dense in all spaces Asp,q (Rn , w) with p < ∞, q < ∞, independent of s ∈ R and w ∈ A∞ . So removing from Rn only ‘small enough’ Γ one can ask whether (still) DΓ is dense in Asp,q (Rn , w).

(3.6)

Conversely, we have the affirmative trace results mentioned above, but one can also ask for what (‘thick enough’) Γ there exists a trace of Asp,q (Rn , w) in Lp (Γ)

(3.7)

(for sufficiently high smoothness). Though these questions may arise independently, it is at least clear that whenever DΓ is dense in Asp,q (Rn , w), then there cannot exists a trace according to (3.1); see [51] and [52, Chapter 6.4] for the corresponding argument in the unweighted case. However, it is not always clear that one really has an alternative in the sense that either there is a trace or DΓ is dense in Asp,q (Rn , w). More precisely, there is described a situation in [51] in the unweighted context where a gap remains: traces can only exist for spaces Asp,q with smoothness s ≥ s0 , whereas density requires s ≤ s1 and s1 < s0 . However, in the setting described here, we are in the lucky situation that we have an alternative between (3.6) and (3.7); following Triebel in the unweighted setting we call this phenomenon dichotomy.

80

D.D. Haroske We introduce the following notation. Let 0 < p < ∞, w ∈ A∞ . Then Ap (Rn , w) = {Asp,q (Rn , w) : 0 < q < ∞, s ∈ R}.

(3.8)

In the spirit of our notational agreement above we shall also use Bp (Rn , w) and Fp (Rn , w) occasionally. Moreover, when w ≡ 1, we shall write Ap (Rn ) = Ap (Rn , 1) for the unweighted situation. Let trΓ : Asp,q (Rn , w) → Lp (Γ)

(3.9)

be the trace operator defined by completion from the pointwise trace according to (3.1). Definition 3.5. Let n ∈ N, Γ ⊂ Rn , 0 sΓ , 0 < q < ∞, trΓ exists for s = sΓ , 0 < q ≤ qΓ ,

and DΓ is dense in

Furthermore, D Ap (Rn , w), Lp (Γ) = (sΓ , 0) means that

trΓ DΓ

trΓ DΓ

(3.11)

s = sΓ , qΓ < q < ∞, for s < sΓ , 0 < q < ∞. (3.12)

(3.13)

exists for s > sΓ , 0 < q < ∞, is dense in Asp,q (Rn , w) for s ≤ sΓ , 0 < q < ∞, (3.14)

and D Ap (Rn , w), Lp (Γ) = (sΓ , ∞) means that

Asp,q (Rn , w)

(3.10)

(3.15)

exists for s ≥ sΓ , 0 < q < ∞, is dense in Asp,q (Rn , w) for s < sΓ , 0 < q < ∞. (3.16)


81

Remark 3.6. Recall that D(Rn ) is dense in Asp,q (Rn , w) with p < ∞, q < ∞. In view of the continuous embeddings in Proposition 2.12, in particular (2.32), this definition makes sense; as discussed in [52, Sect. 6.4.3] already, it might be more reasonable in general, to exclude the limiting case q = qΓ in (3.11) or shift it to (3.12). But as will turn out below, in our context the breaking point q = qΓ is always on the trace side. For convenience, we first collect what is known in the unweighted situation, w ≡ 1, for hyperplanes Γ = Rm or d-sets Γ, 0 < d < n. Proposition 3.7. Let 0 1, p ≤ 1.

(ii) Let Γ be a compact d-set, 0 < d < n. Then n−d n , min(p, 1) D(Bp (R ), Lp (Γ)) = p and

⎧ ⎨ n−d , 0 , p D(Fp (Rn ), Lp (Γ)) = ⎩ n−d , ∞ , p

p > 1, p ≤ 1.

Remark 3.8. The result (i) is proved in this explicit form in [51], see also [52, Cor. 6.69], for a different approach see also [39,40] and Remark 3.3 for further literature. The second part (ii) can be found in [51] and [52, Thm. 6.68]. Now we state our main result in this paper. Theorem 3.9. Let Γ be a compact d-set, 0 < d < n, 0 −(n − d). Then n−d+κ , min(p, 1) (3.17) D(Bp (Rn , wκ,Γ ), Lp (Γ)) = p and ⎧ ⎨ n−d+κ ,0 , p D(Fp (Rn , wκ,Γ ), Lp (Γ)) = ⎩ n−d+κ , ∞ , p

p > 1, p ≤ 1.

(3.18)

82

D.D. Haroske

Proof. Step 1. In view of Proposition 3.4, in particular the transfer formulas between trace spaces in the weighted and unweighted setting (3.3), (3.4), together with the (unweighted) dichotomy result Proposition 3.7 it is clear that all the corresponding trace assertions in Definition 3.5, that is, (3.11) and the upper lines in (3.14), (3.16), are already covered for our situation. Furthermore, in view of the embeddings (2.32) and the density of D(Rn ) in all spaces Asp,q (Rn , wκ,Γ ) it is only left to prove that DΓ

n−d+κ

is dense in Bp,q p

(Rn , wκ,Γ ) if 0 min(1, p),

(3.19)

(Rn , wκ,Γ ) if 1 < p < ∞, 0 < q < ∞.

(3.20)

and DΓ

n−d+κ

is dense in Fp,q p

This will immediately imply the density of DΓ in Asp,q (Rn , wκ,Γ ) with s < sΓ = n−d+κ and thus already finish the F -case with p ≤ 1. p The plan is as follows. We adapt ideas presented in [51] for the unweighted case, see also [52, Thm. 6.68]. Roughly speaking, the clue is to n−d+κ

construct for given f ∈ Bp,q p fies n−d+κ

f − fJ |Bp,q p

(Rn , wκ,Γ ) a sequence {fJ }J∈N which satis-

(Rn , wκ,Γ ) −−−−→ 0 J→∞

and fJ ∈ DΓ .

These approximating functions fJ are based on special atomic decompositions which are appropriately adapted to the different q-cases in (3.19): q > 1 (studied in Steps 2 and 3) and q > p (postponed to Step 5). This will cover the B-case. For simplicity we shall describe the easiest case (that is, when no moment conditions in (2.20) are required) in some detail; the extension to all parameters needs some further tricky technical refinement, but is essentially covered by parallel arguments in the unweighted situation. Here one essentially benefits from the so-called porosity of d-sets, cf. [24, p.156] and [50, Sects. 9.16–9.19], to circumvent this difficulty. ∞ Step 2. We begin with a preparation and construct a sequence ϕJ J=1 ⊂ D(Rn ) with ϕJ (x) = 1

in an open neighbourhood of Γ

(3.21)

(depending on J) and n−d+κ

ϕJ −−−−→ 0 in Bp,q p J→∞

(Rn , wκ,Γ ),

p ≥ rwκ,Γ ,

q > 1.

(3.22)

We proceed as in [51] and cover for given j ∈ N a neighbourhood of Γ with balls Bj,m centred at Γ and of radius 2−j , where m = 1, . . . , Mj with Mj ∼ Mj , with ϕj,m ∈ 2jd . Accordingly we choose a resolution of unity {ϕj,m }m=1


83

D(Bj,m ), ϕj,m ≥ 0, |Dα ϕj,m (x)| ≤ cα 2j|α| , α ∈ Nn0 , and Mj

ϕj,m (x) = 1 near Γ.

m=1

J0 1 J0 +1 1 Assume J ≥ 2 and determine J0 ∈ N such that j=J j=J j . j < 1 ≤ Thus we obtain that J0 1

, j = J, . . . , J0 − 1, rj = 1 with rj = j J0 1 1 − j=J j , j = J0 . j=J Then J

ϕ (x) =

J0

rj 2

−j d−κ p

Mj

2

j(d−κ) p

x ∈ Rn ,

ϕj,m (x),

(3.23)

m=1

j=J

n−d+κ

is an atomic decomposition in Bp,q p (Rn , wκ,Γ ) according to Theorem 2.7(i) and Definition 2.6 for K > 1 + n−d+κ and L = −1 (no moment conditions p needed). Thus Theorem 2.7(i) together with (2.17) and (2.10) implies that n−d+κ

ϕJ |Bp,q p

(Rn , wκ,Γ )

q

≤ c1

J0

rjq 2−j

d−κ p

q

Mj

≤ c2 ≤ c3

j=J ∞

q/p

m=1

j=J J0

2jn wκ,Γ (Bj,m )

rjq 2−j

d−κ p

q −j κq p

2

q

Mjp

j −q ∼ J 1−q .

(3.24)

j=J

This gives (3.22). Step 3. We show (3.19) for p > rwκ,Γ ≥ 1 and q > min(p, 1) = 1. Note that by density arguments it is sufficient to approximate f ∈ D(Rn ) in n−d+κ

Bp,q p (Rn , wκ,Γ ) by functions fJ ∈ DΓ . Let ϕJ be the functions according to (3.21), (3.22), put f J = ϕJ f such that f can be decomposed into f = f J + fJ

with

f J = ϕJ f

and fJ = (1 − ϕJ )f ∈ DΓ .

By an appropriately adapted weighted counterpart of the pointwise multiplier theorem in [47, Sect. 4.2.2] one has for sufficiently large smoothness that there is some c > 0, such that for all f ∈ D(Rn ) and all ϕJ , n−d+κ

f J |Bp,q p

n−d+κ

(Rn , wκ,Γ ) ≤ c f |C (Rn ) · ϕJ |Bp,q p

(Rn , wκ,Γ ) −−−−→ 0 J→∞

according to (3.24). This completes the argument for (3.19) when p > rwκ,Γ , q > 1. Step 4. We prove (3.20) and use the fact that d-sets are porous for d < n, we refer to [24, p.156] and [50, Sects. 9.16-9.19] for further details. Let first q ≥ 1. Then by the same reasoning as above we consider the atomic

84

D.D. Haroske n−d+κ

decomposition (3.23) in Fp,q p (Rn , wκ,Γ ) with p > rwκ,Γ , q ≥ 1 (where we may choose L = −1 in Theorem 2.7(ii), i.e., no moment conditions needed) leading to n−d+κ

lim ϕJ |Fp,q p

J→∞

n−d+κ

(Rn , wκ,Γ ) = lim ϕJ |Bp,p p J→∞

(Rn , wκ,Γ ) = 0.

For arguments of this type we refer to [49, pp. 142/143]. In view of (2.23) the situation is more complicated when 0 < q < 1 or 1 < p ≤ rwκ,Γ , since one needs moment conditions L ≥ 0 in that case. So it may happen that (3.23) n−d+κ

is no longer an atomic decomposition in Fp,q p (Rn , wκ,Γ ) and we cannot apply Theorem 2.7(ii) as above. But the porosity of (the d-set) Γ ensures that one can complement ϕj,m outside of Γ in an appropriate way in order to obtain the needed moment conditions. For details and further discussion one may consult [48, p. 143] based on [53]. This completes the proof of (3.20) and thus of (3.18). Moreover, by the same reasoning we can also cover the B-case in (3.19) when 1 min(p, 1) = 1, that is, when moment conditions are needed in (2.20), which we excluded in Step 3 for simplicity. Step 5. It remains to verify (3.19) in case of 0 min(p, 1) = p. (In fact, the proof below will cover all situations when q > p.) For this reason we have to refine our covering and decomposition argument from Step 2. Let L ∈ N and assume, for convenience, that μ(Γ) = 1. We use the same decomposition of Γ by other d-sets Γl , l = L, . . . , L0 , as described in [51], resulting in Γ=

L0

Γl ,

μ(Γl ) ∼ l−1 ,

l=L

L0

μ(Γl ) ∼ μ(Γ) = 1,

l=L

where L0 ∈ N with L0 > L is chosen appropriately. Moreover, according to [52, p. 224] this can be done in such a way that there are non-negative functions ψl ∈ D(Rn ) with L0

ψl (γ) = 1 if γ ∈ Γ,

Γl ⊂ supp ψl ⊂ {y ∈ Rn : dist (y, Γl ) < εl }

l=L

for some εl > 0. For given l ∈ N between L and L0 and appropriately chosen j(l) ∈ N we mimic the construction from Step 2 and arrive at

Mj(l)

ϕj(l),m (x) = 1 near Γ,

0 ≤ ϕj(l),m ∈ D Bj(l),m ,

m=1

with Mj(l) ∼ 2j(l)d . We assume that j(L) < · · · < j(l) < j(l + 1) < · · · < j(L0 ), and in analogy to (3.23), L

ϕ (x) =

L0

l=L

ψl (x) 2

− j(l)(d−κ) p

Mj(l)

m=1

2

j(l)(d−κ) p

ϕj(l),m (x),

x ∈ Rn . (3.25)


85

n−d+κ + d−κ > σp/rwκ,Γ , we do not If p > pκ = max(κ,0) n−d n , i.e., when s = p need moment conditions in (2.20) and can apply Theorem 2.7(i): for large j(l) (3.25) represents an atomic decomposition,

ϕL (x) =

L0

2−

j(l)(d−κ) p

Nj(l)

x ∈ Rn ,

aj(l),m (x),

m=1

l=L −1

with Nj(l) ∼ μ(Γl ) 2 ∼ l 2 , counting only non-vanishing terms, where the equivalence constants are independent of L. Clearly, ϕL (x) = 1 near Γ, and by Theorem 2.7(i) for q > p, j(l)d

n−d+κ p

ϕ |Bp,q L

q

n

(R , wκ,Γ )

≤ c1

j(l)d

L0

2

−j(l) d−κ p q

m=1

l=L

≤ c2

L0

q/p 2jn wκ,Γ (Bj(l),m )

j(l) N

2−j(l)

d−κ p q

κ

q

p 2−j(l) p q Nj(l)

l=L

≤ c3

∞

q

l−q/p ∼ L1− p .

l=L

This replaces (3.24) in this case and the counterpart of Step 3 concludes the argument. In case of p ≤ pκ we need moment conditions in (2.20), but the same type of modification as mentioned in Step 4 works here as well, where we benefit from the porosity of the d-set Γ. This completes the proof. Remark 3.10. The above theorem extends partial results in [34] related to the case 1 < p < ∞, cf. [34, Corollars 4.8, 4.13]. Moreover, in view of (2.31) we know that s+ max(κ,0) p

Bp,q

s (Rn , wκ,Γ ) → Bp,q (Rn ),

that is, for κ ≥ 0, s+ κ

s (Rn ). Bp,q p (Rn , wκ,Γ ) → Bp,q

Though these spaces are only continuously embedded (but differ from each other), their dichotomy parameters just scale as expected: by Proposition 3.7(ii) and Theorem 3.9 we have that D(Bp (Rn ), Lp (Γ)) = sΓ , qΓ and κ D(Bp (Rn , wκ,Γ ), Lp (Γ)) = sΓ + , qΓ . p A similar phenomenon occurs in view of the continuously embedded (but different) spaces from Besov and Triebel–Lizorkin type, see Proposition 2.12 together with Example 2.14 in contrast to Theorem 3.9, in particular (3.17) and (3.18). The stronger assumption in (2.31) and Example 2.14 for negative κ > −(n − d) is due to the embedding of the whole Rn ‘far away’ which

86

D.D. Haroske

cannot be better than in the unweighted situation. However, since our trace questions are of purely local nature in a close neighbourhood of the compact d-set Γ, it is less astonishing that we have no extra assumption. Just to the contrary we have the meanwhile well-known phenomenon, that for κ < 0 (that is, where the weight tends to infinity when approaching Γ) we have an increased smoothness in the sense of (3.3), (3.4): the trace of the weighted space results in the same space on Γ like the trace of an even smoother unweighted space, such that the special weight wκ,Γ can compensate some lack of smoothness in this situation. Remark 3.11. The concept of d-sets Γ can be generalised in a natural way to (d, Ψ)-sets studied in [10,26], where Ψ is a so-called ‘admissible’ function or a slowly varying function; we also refer to [24, 35] for more general background material. Then it turns out that appropriate spaces in the context of trace or dichotomy questions as discussed above are spaces of generalised smoothness (s,Ψ) Ap,q (Rn ) studied in [25, 26] in great detail. Based on these observations in the unweighted setting there are some first consequences in the weighted case in [34, Chapter 5], where the idea is to modify the weight function wκ,Γ appropriately such as to obtain the counterpart of (3.3), (3.4) in this context: Ψ is this modified weight function corresponding to the admissible if vκ,Γ Ψ function Ψ and the (d, Ψ)-set ΓΨ , then s (s− κ ,Ψ1/p ) Ψ trΓΨ Bp,q (Rn , vκ,Γ ) = trΓΨ Bp,q p (Rn ) Ψ and one can use unweighted trace results from [25,26]. It would be interesting to determine the influence of the function Ψ one the dichotomy results. Some first approach in the unweighted setting, but dealing with the even more general h-sets Γ (and thus requiring more general spaces, too) is suggested in [9]; one can find further discussions, curiosities and applications in [52, Chapter 6.4]. Remark 3.12. Instead of playing with the set Γ where the trace is taken, one can likewise inquire into the influence of the weight function w ∈ A∞ in a more qualitative way. Certainly one can expect that the singular set Ssing (w) introduced in (2.9) should have some essential influence on trace questions, and possibly also on dichotomy matters. But nothing seems to be done yet in this direction, though in view of some application to sampling numbers as explained in [52, Chapter 6.4] this could also be of wider interest.

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[43] J.-O. Str¨ omberg and A. Torchinsky. Weighted Hardy spaces, volume 1381 of Lecture Notes in Mathematics. Springer, Berlin, 1989. [44] A. Torchinsky. Real-variable methods in harmonic analysis, volume 123 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1986. [45] H. Triebel. Interpolation theory, function spaces, differential operators. NorthHolland, Amsterdam, 1978. [46] H. Triebel. Theory of function spaces. Birkh¨ auser, Basel, 1983. [47] H. Triebel. Theory of function spaces II. Birkh¨ auser, Basel, 1992. [48] H. Triebel. Fractals and spectra. Birkh¨ auser, Basel, 1997. [49] H. Triebel. The structure of functions. Birkh¨ auser, Basel, 2001. [50] H. Triebel. Theory of function spaces III. Birkh¨ auser, Basel, 2006. [51] H. Triebel. The dichotomy between traces on d-sets Γ in Rn and the density of D(Rn \ Γ) in function spaces. Acta Math. Sinica, 24(4):539–554, 2008. [52] H. Triebel. Function Spaces and Wavelets on domains. EMS Tracts in Mathematics (ETM). European Mathematical Society (EMS), Z¨ urich, 2008. [53] H. Triebel and H. Winkelvoß. Intrinsic atomic characterizations of function spaces on domains. Math. Z., 221(4):647–673, 1996. Dorothee D. Haroske Mathematical Institute Friedrich-Schiller-University Jena D-07737 Jena Germany e-mail: [email protected]


Lavrentiev’s Theorem and Error Estimation in Elliptic Inverse Problems Ian Knowles and Mary A. LaRussa Abstract. Lavrentiev’s theorem provides bounds for analytic functions known to be small at a finite number of points in a bounded region. An analogous result is established for solutions of elliptic equations on bounded regions in R2 and applied to estimating non-uniqueness error in elliptic inverse problems. Mathematics Subject Classification (2010). 35R30, 35J25, 86A22. Keywords. Quasiconformal mapping, Beltrami equation, Stoilow factorization, Beurling and Cauchy transforms, elliptic error estimates.

1. Introduction It has long been known in the complex analysis world that surfaces defined by analytic functions have a certain “rigidity” in that at points where the derivative is non-zero (so the analytic function becomes a conformal mapping), the surface is locally close to its linearization in a very confining manner (see [1, §2.10]). It is also known that an analytic function that is zero on a set with a limit point (which can be a rather small set in the plane) must be identically zero. Of course, just knowing that an analytic function is zero on a finite set does not make it zero everywhere, but when one adds these rigidity properties, it is natural to speculate that if such a function is zero (or small) at a few points, then it should not be able to “wobble” much in between, with the wobble becoming less as the number of such points increases. The remarkable theorem of Lavrentiev (see Theorem 2.3 below) indicates, in a precise and quantitative fashion, that analytic functions do indeed behave in this way. Now, if f is analytic and f = σ + iω, it is known that the real functions σ and ω are solutions of the Laplace equation. It follows from the Lavrentiev theorem that these conjugate harmonic functions inherit the same rigidity properties. In this paper, using the Lavrentiev theorem and ideas from the

92

I. Knowles and M.A. LaRussa

theory of quasi-conformal mappings in the plane, our primary goal is to show that solutions u of the elliptic equation ∇ · p(x)∇u = 0,

x ∈ Ω ⊂ R2 .

(1.1)

have similar behaviour. In the final sections we apply this theory to the estimation of the “non-uniqueness” error, an important component of the modelling error in the inverse groundwater problem.

2. Some complex function theory We assume throughout that Ω ⊂ R2 is a bounded domain with a connected complement, and that p : Ω → (0, ∞) is measurable and bounded away from zero and infinity. Given the famous connection between analytic functions and harmonic functions afforded by the Cauchy–Riemann equations, it is perhaps not surprising that the solutions of the more general elliptic equation (1.1) belong to a similar framework. In particular we have [2, Lemma 2.1] Theorem 2.1. Assume that the function u lies in the Sobolev space H1 (Ω), is real-valued, and satisfies (1.1). Then there exists a function v ∈ H1 (Ω), called the p-harmonic conjugate of u, unique up to an additive constant, such that (2.1) ∂x v = −p∂y u, ∂y v = p∂x u, and 1 (2.2) ∇ · ∇v = 0. p Also f = u + iv satisfies the R-linear Beltrami equation ∂f = μ∂f , where ∂ =

1 2 (∂x

− i∂y ), ∂ =

1 2 (∂x

(2.3)

+ i∂y ), and μ = (1 − p)/(1 + p).

H1loc (Ω)

that satisfies (2.3) is called a quasi-regular A function f ∈ mapping, and if it is also a homeomorphism it is called quasiconformal. From the Stoilow factorization [1, Theorem 5.5.1] we know that H1loc (Ω) solutions f of (2.3) may be written in the form f = ψ ◦ h where ψ is C-analytic and h is a quasiconformal homeomorphism. Let E be a set of points of the real axis in C, and let Φn = {Δk : 1 ≤ k ≤ n} be a collection of n intervals whose union contains the set E, and denote by k the length of Δk . Define the Lavrentiev set function, μn by n

μn (E) = inf k . Φn

k=1

Let D = {z ∈ C : |z| < 1},

D1/4 = {z ∈ C : |z| < 1/4}

and consider the finite set of points (i)

(i)

A = {x(i) = (x1 , x2 ) : 1 ≤ i ≤ m} ⊂ D1/4 ,

Lavrentiev’s Theorem and Error Estimation

93

where we assume that (1)

(2)

(m)

x1 < x1 < · · · < x1 . For later use we have, more or less directly from the definition of the Lavrentiev set function, Theorem 2.2. Let A˜ denote the projection of the finite set A onto the real axis in C. Let m denote the number of points in A. ˜ = 0 for all k ≥ 1. (i) If m = 1, then μk (A) (ii) If m = 2, then ˜ = x(2) − x(1) ; μ1 (A) 1 1 ˜ = 0 for k ≥ 2. μk (A) (iii) If m = 3, then ˜ = x(3) − x(1) ; μ1 (A) 1 1 (2)

(1)

(3)

(2)

˜ = min{x − x , x − x }; μ2 (A) 1 1 1 1 ˜ μk (A) = 0 for k ≥ 3. (iv) If m = 4, then (4)

(1)

˜ =x −x ; μ1 (A) 1 1 ˜ = min{x(4) − x(2) , x(3) − x(1) , (x(4) − x(3) ) + (x(2) − x(1) )}; μ2 (A) 1 1 1 1 1 1 1 1 (j+1)

(j)

˜ = min{x − x1 : 1 ≤ j ≤ 3}, μ3 (A) 1 ˜ = 0, for k ≥ 4. μk (A) Note in particular that these measure values are either zero or can be made as small as we like, depending on the placement of the points x(i) , (i) 1 ≤ i ≤ m, in the region Ω. In particular, if the projected coordinates x1 , 1 ≤ i ≤ m, are all close together on the real axis, these measures would all be small, even with the wells being well spaced in the y-direction. We comment on this later in §5. Of central interest to us here is the following theorem of Lavrentiev [10, p. 58, Theorem 1]: Theorem 2.3. Suppose that the set A ⊂ D1/4 and let A˜ be the projection of ˜ A onto the real axis in C. Assume that n ∈ N and , where 0 < ≤ μ1 (A)/4, are related by the inequalities n+1 n ˜ ˜ 1 μn+1 (A) 1 μn (A) 0 Ω = {x ∈ R2 : |x| < 1 + η}, and that the coefficient function p is bounded and measurable on Ω, and satisfies the ellipticity condition 1 ≥ p(x) ≥ ν > 0, ν for some constant ν, and all x ∈ Ω. We now establish

(3.1)

Theorem 3.1. Let A be a finite subset of D1/4 and let u be an H1 (Ω) solution of (1.1) satisfying the conditions |u(x)| ≤ 1, for all x ∈ Ω, and |u(x)| ≤ for ˜ and all x ∈ A. Let n be defined by the some for which 0 ≤ ≤ μ1 (A)/4, inequality (2.4). Then, for all x in the set D1/4 , one of the inequalities |u(x)| ≤ 4/25 ,

|u(x)| ≤ (6/7)n

(3.2)

holds. Proof. Let v be a p-harmonic conjugate of u. From standard elliptic regularity results (for example [9, Chapter 3, §14]) we know that v, which satisfies (2.2), is continuous on the interior closed unit ball D ⊂ Ω. So |v(x)| < M1 for some constant M1 > , and all x ∈ D. Let α > 0 be chosen arbitrarily, and consider the complex function fα =

1 (u + iαv). 1 + αM1

One can see after multiplying both equations in (2.1) by α, that the function fα lies in H1loc (D) and satisfies the Beltrami equation (2.3) with μ = μα = (1 − αp)/(1 + αp). So, by the Stoilow factorization theorem [1, Theorem 5.5.1] (taking Ω = D in that theorem) fα may be written in the form fα = ψα ◦ hα


95

where hα : D → hα (D) is a quasiconformal homeomorphism and ψα is Canalytic on hα (D). Further, as |u(x)| + α|v(x)| < 1, 1 + αM1 for all y ∈ hα (D), we have ψα : hα (D) → D here. Fix α. For all y ∈ Bα = hα (A) ∩ D1/4 , we have |ψα (y)| = |ψα (hα (x))| = |fα (x)| ≤

+ αM1 . 1 + αM1 one of the inequalities

|ψα (y)| ≤ α = So, by Theorem 2.3, for each y ∈ D1/4 , |ψα (y)| ≤ 4/25 α

|ψα (y)| ≤ (6/7)m

holds, where m = m(α) is defined by m+1 m ˜α ) ˜α ) 1 μm+1 (B 1 μm (B < α ≤ . 2 2 2 2

(3.3)

In particular, as 1 |u(x)| ≤ |fα (x)| = |ψα (hα (x))|, 1 + αM1 it follows that, for all x for which hα (x) ∈ D1/4 , one of the inequalities 1 |u(x)| ≤ 4/25 , α 1 + αM1

1 |u(x)| ≤ (6/7)m 1 + αM

(3.4)

holds. We show below that as α → 0, hα → h pointwise uniformly, where the limit function h : D → D is the projection onto the real axis in D. Let us assume this for the moment. So, if x ∈ D1/4 , then, as D1/4 is open and hα → h pointwise, for all α small enough hα (x) ∈ D1/4 . Hence, for all x ∈ D1/4 u(x) satisfies (3.4), where m = m(α) and α satisfy (3.3), for all α small enough. ˜ α = h(hα (A)) for all α small enough. Define Also, as A is finite, B n ˜ 1 μn (A) . qn = 2 2 We also have that, as α → 0, n ˜α ) 1 μn (B → qn , 2 2

1 2

˜α ) μn+1 (B 2

n+1 → qn+1 .

(3.5)

This follows from the pointwise convergence of the functions hα and the finiteness of A, via Theorem 2.2. If we set + αM1 α = g(α) = , 1 + αM1 then M1 (1 − ) g (α) = > 0. (1 + αM1 )2

96


So we see that α decreases to as α decreases to 0. Choose α0 so that for 0 < α < α0 we have 1 < α < ( + qn ). 2 In addition, by making α0 smaller if necessary, we have from (3.5) that n 1 1 μn (B˜α ) − qn | < (qn − ) | 2 2 2 and 1 | 2

μn+1 (B˜α ) 2

n+1 − qn+1 |
0. Now, note from [1, (4.24)] that 1 χ , z 2 C \D

(3.11)

z, |z| ≤ 1, 1 z , |z| > 1.

(3.12)

S χD = − and from [1, (4.38)]

C χD =

So as μα is a constant, 2

μ α χ χ = 0. z 2 D C \D So, from (3.9), (3.10), (3.12) and (3.13), z, |z| ≤ 1, ωα = C(μα χD ) = μα 1 z , |z| > 1, μα χD S(μα χD ) = −

(3.13)

(3.14)

and consequently, as α → 0, we have μα → 1 and hα → h0 where z z/2, |z| ≤ 1, h0 (z) = + 1/2z, |z| > 1. 2 Notice that h = h0 |D : D → D is the projection map of D onto the real axis in D. Next, subtracting equations (3.7) and (3.8) we get ∂(ωα − ωα ) ∂ωα ∂(ωα − ωα ) = μα χ D + .(μα −μα )χD +(μα −μα )χD . (3.15) ∂z ∂z ∂z Observe that from (3.14) we have ∂ωα 0, |z| ≤ 1, = μα , 1 , |z| > 1, − ∂z z2 so that the second term on the right of (3.15) vanishes identically. Consequently (3.16) ωα − ωα = C (I − μα χD S)−1 ((μα − μα )χD ) .

98


Now by (3.11) μα χD S((μα − μα )χD ) = μα χD S(μα χD ). From (3.1) and (3.6) we have 2α 2αν ≤ μα ≤ 1 − = μα , μα = 1 − α+ν 1 + αν so that μα χD μα χ D μα χD dτ ≤ dτ ≤ dτ. 2 2 2 |z−τ |> (z − τ ) |z−τ |> (z − τ ) |z−τ |> (z − τ ) If we multiply by −1/π and let → 0, it follows from (3.11) that 1 1 −μα 2 χC\D ≥ S(μα χD ) ≥ −μα 2 χC\D , z z so that μα χD S(μα χD ) = 0. This means that, by (3.16) ωα − ωα = C (μα − μα )χD ν −p χ , = 2αC (1 + αp)(1 + αν) D which approaches zero pointwise uniformly as α → 0 because the Cauchy transform of a function in L3 (C) lies in the Hölder space C 1/3 (C) by [1, Theorem 4.3.13]. Finally, we have hα − h0 = (hα − hα ) + (hα − h0 ) = (ωα − ωα )/2 + (hα − h0 ) → 0 pointwise uniformly as α → 0, which completes the proof.

4. Error estimation in the inverse groundwater problem Underground aquifer systems (as well as oil reservoirs) are often modeled by the diffusion equation: ∂w q(x) = ∇ · [p(x)∇w(x, t)] + R(x, t), (4.1) ∂t for time 0 ≤ t ≤ T , and for x = (x1 , x2 ) in a two-dimensional aquifer region Ω, which we assume to be equal to D for notational convenience. Here, w represents the piezometric head, p the aquifer transmissivity, R the recharge, and q ≥ 0 the storativity of the aquifer (see, for example, [3]). It is well known among hydrologists that the inability to obtain reliable values for the coefficients in (1.1) from measured values of the water levels w taken over time at a collection of wells in the aquifer, together with reasonable estimates of the associated inverse recovery error, is a serious impediment to the confident use of such models. While new methods for the simultaneous recovery of p, q, and R (including its time dependence) from a known w have recently been developed [6, 7, 11], the nagging and fundamentally important problem of properly quantifying the associated error remains. In discussing the recovery error for the inverse groundwater problem we assume in advance that one is using a recovery algorithm that is provably


99

well-posed (see for example [5]) so that in theory at least, the associated recovery error may be properly quantified. With this in mind, the recovery error that we seek to quantify centers on the error in the head data function w(x, t) that we use, given that the theorems guaranteeing unique recovery of the coefficient functions in (1.1) [4] require that we know the head function w(x, t) at all points x ∈ Ω and for all 0 ≤ t ≤ T . We restrict attention here to a choice of w(x, t) obtained by linear interpolation in space and time from the finite set of well data values. The error in our interpolated head function w now has basically three components. First, the head data may have measurement error. Second, we observe that in practice one only knows w at discrete times and at a finite (often small) number, m, of wells located at points {x(i) : 1 ≤ i ≤ m} in Ω. The solution w is certainly not specified uniquely by a knowledge of its values at a finite number of interior points in Ω. In general one needs to know something like the solution values at all points of the boundary at a particular time. So we have a “non-uniqueness” error to contend with, and in particular, while our solution is tied-down nicely at the well points {x(i) : 1 ≤ i ≤ m} we need to know how much “non-uniqueness wobble” in w is possible in between the well points. Finally, as aquifers change only slowly over time, interpolation in t is generally not a source of significant error. However, as a practical matter, and given that one typically does not have very many measurement wells available in a given aquifer, the interpolation of w values in x at fixed t is a third, and possibly large, source of error in the inverse recovery process. We outline how to quantify the first and second type of error below, and a method for estimating the third type of error may be found in [8]. Specifically, assume that we are given measured values of the solution w of (4.1) at the well points {x(i) : 1 ≤ i ≤ m} at times t satisfying 0 ≤ t ≤ 1. We assume that the term q(x) ∂w ∂t in (4.1) contributes little to this discussion, because the storativity values q(x) are typically quite tiny relative to the conductivity term, and in any event, aquifers tend to change only slowly in time, so the factor ∂w/∂t is also rather small. There are infinitely many solutions of (4.1) equal to the measured values w at the points {x(i) : 1 ≤ i ≤ m}. Let w1 and w2 be two such solutions, and set w∗ = w1 − w2 . Then for any given time t, w∗ (x(i) , t) = 0

(4.2)

for all 1 ≤ i ≤ m. Also as the two solutions wi share the right-hand side function R in (4.1), and we are ignoring the term q(x) ∂w ∂t , for fixed t the function w∗ is a solution of our elliptic equation (1.1). We are interested in how large |w∗ | can be in a connected neighbourhood N ⊂ Ω of the set of well points {x(i) : 1 ≤ i ≤ m} as w1 and w2 range over all possible solution choices. With no further restrictions, bounds on |w∗ | are problematical. However, for any given aquifer, in practice there are physical limits on how much the heads can vary over the entire aquifer over time. So, guided by these physical

100


considerations, we make the assumption that for any choice of the solutions w1 and w2 the error w∗ is bounded, |w∗ (x, t)| < M,

for all x ∈ Ω and 0 ≤ t ≤ 1,

(4.3)

for some known constant M . In [8], a method for estimating M from groundwater well data is given; we return to this later. It is convenient to consider the normalized error function u = w∗ /M , so we actually assume that |u(x, t)| < 1,

for all x ∈ Ω,

(4.4)

at all points x in the aquifer region Ω, and at all times 0 ≤ t ≤ 1. As we consider the time t to be fixed in this discussion, we henceforth omit reference to this variable. If there are measurement errors in the head data, the equality (4.2) may be modified to (4.5) |u(x(i) )| ≤ for some 0 ≤ < 1 and all 1 ≤ i ≤ m. We then have, from Theorem 3.1, an explicit bound on the scaled head error, u, in terms of m and the scaled measurement error , over the quarter disk N = D1/4 . This allows us to estimate the measurement and “non-uniqueness” errors in a groundwater model as we see next.

5. Estimating “non-uniqueness” head error We are interested in situations in which the “non-uniqueness” head error, |u|, is small on D1/4 in situations when m, the number of wells, is relatively small as well. Using Theorem 2.2 one sees immediately that one should not consider m = 1, as there is no estimate in this case. For each m > 1 we # wells m n 2 1 3 2 4 3 5 4 6 5 7 6 8 7 11 10

max scaled absolute head error 0.38 0.86 0.15 0.73 0.06 0.63 0.02 0.54 0.008 0.46 0.003 0.40 0.001 0.34 0.00007 0.21

˜ = 0 henceforth. For certain exceptional assume for simplicity that μm−1 (A) arrangements of the wells one would need an appropriate modification of the following discussion. By way of example, if all the wells were on a line parallel to the vertical axis, then by Theorem 2.2 (and analoguous formulae ˜ = 0 for all k and the method fails. Of course, in this case for m > 4), μk (A) one can rotate the plane by π/2 and successfully re-apply the method.


101

For m = 2 one is restricted to choosing n = m − 1 = 1 in applying (3.2); here, for small enough, the second estimate in (3.2) dominates, and we see that |u| ≤ 6/7 ≈ 0.86. For m = 3, from Theorem 2.2 the largest feasible n is n = 2, and we see that |u| < (6/7)2 ≈ 0.73 provided that < 0.14. In general, for small enough, one chooses n = m − 1, and the first few cases for the largest compatible , i.e. the that solves 6 n , (5.1) 4/25 = 7 and the resultant maximum scaled head errors |u| = (6/7)n on D1/4 are given in the table above. These are hard estimates for the worst case scenario for the “non-uniqueness” head error. If, , the absolute head measurement error at the wells, is larger than that listed in the “max ” column, one needs to substitute the value 4/25 instead of (6/7)n in the last column. In particular, if one fixes the number of wells, m, and checks the variation of error with a somewhat different calculation ensues. We note that the specific numbers, such as 6/7 and 4/25, appearing in Theorem 3.1 (and, consequently, (5.1)) are unlikely to be optimal for this problem. Recall that u here represents the scaled absolute “non-uniqueness” head error. If the actual absolute error w∗ = w1 − w2 satisifies |w∗ | ≤ M , then u = w∗ /M . From the table one can see, for example, that with four wells

Figure 1. Positions of 4 wells. (m = 4), each with an absolute head measurement error of 0.06M , one should expect a worst case absolute head non-uniqueness error of 0.62M . In general the aquifer “wobble”, M , which is an estimate for the maximal head variation in a given aquifer region over time, would be known to a field hydrologist with local knowledge of a particular groundwater system. In particular, data from the Port Willunga aquifer in south-eastern South Australia over the 13 year

102


period January 1995 to December 2007 using the 4 wells WLG014, WLG051, WLG055, and WLG060 pictured in Figure 1, and assuming that the circle shown is scaled and translated to D1/4 , we have that M ≈ 3.4 meters. For the shorter one-year time period January 1998 to December 1998, M ≈ 2.2 meters. This means, for example, in the circular region around the four wells shown in Figure 1, during the year 1998 when the heads varied around 2.2 meters, (and provided the absolute measurement error for each of these head measurements is no more than 0.13 meters), we have a maximum head nonuniqueness error of 1.4 meters in a typical head measurement for this area of 40 to 80 meters. In practice, one might expect that using, for example, linear interpolation to create the “measured” head surface from the discrete well data, the actual interpolation error would be somewhat less than the figure obtained by the foregoing method. Acknowledgement The authors are grateful to the referee for a careful reading of the paper, and for catching a number of errors in the initial draft.

References [1] Kari Astala, Tadeusz Iwaniec, and Gaven Martin. Elliptic partial differential equations and quasiconformal mappings in the plane, volume 48 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2009. [2] Kari Astala and Lassi P¨ aiv¨ arinta. Calder´ on’s inverse conductivity problem in the plane. Ann. of Math. (2), 163(1):265–299, 2006. [3] J. Bear. Dynamics of Fluids in Porous Media. American Elsevier, New York, 1972. [4] Ian Knowles. Uniqueness for an elliptic inverse problem. SIAM J. Appl. Math., 59(4):1356–1370, 1999. [5] Ian Knowles and Mary A. LaRussa. Conditional well-posedness for an elliptic inverse problem. preprint. Available online at http://www.math.uab.edu/knowles/pubs.html. [6] Ian Knowles, Tuan A. Le, and Aimin Yan. On the recovery of multiple flow parameters from transient head data. J. Comp. Appl. Math., 169:1–15, 2004. [7] Ian Knowles, Michael Teubner, Aimin Yan, Paul Rasser, and Jong Wook Lee. Inverse groundwater modelling in the Willunga Basin, South Australia. Hydrogeology Journal, 15:1107–1118, 2007. [8] Mary A. La Russa. Conditional well-posedness and error estimation in the groundwater inverse problem. PhD thesis, University of Alabama at Birmingham, 2010. [9] Olga A. Ladyzhenskaya and Nina N. Uraltseva. Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York, 1968. [10] M. M. Lavrentiev, V. G. Romanov, and S. P. Shishatski˘ı. Ill-posed problems of mathematical physics and analysis, volume 64 of Translations of Mathematical


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Monographs. American Mathematical Society, Providence, RI, 1986. Translated from the Russian by J. R. Schulenberger, Translation edited by Lev J. Leifman. [11] Aimin Yan. An Inverse Groundwater Model. PhD thesis, University of Alabama at Birmingham, 2004. Ian Knowles and Mary A. LaRussa Department of Mathematics University of Alabama at Birmingham Birmingham, Alabama, 35294 USA e-mail: [email protected]


Two-weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Space Vakhtang Kokilashvili and Alexander Meskhi Dedicated to Professor D. E. Edmunds and Professor W. D. Evans to mark their 80th and 70th birthday, respectively.

Abstract. Two-weight norm estimates for double Hardy transforms and variable order strong fractional maximal functions are established in variable exponent Lebesgue spaces. Derived conditions are simultaneously necessary and sufficient in the case when the exponent function of the right–hand side space is constant. In the main statements the weak logarithmic condition for exponents of the spaces is not assumed. Mathematics Subject Classification (2010). 42B20, 46E30. Keywords. Variable exponent Lebesgue spaces, double Hardy transforms, strong fractional maximal operators, weights, two-weight inequality.

1. Introduction The paper deals with two-weight criteria for double Hardy transforms and strong fractional maximal functions in the framework of variable exponent Lebesgue spaces. Recently T. Kopaliani [14] showed that the Hardy–Littlewood strong maximal operator is bounded in Lp(·) space if and only if p is constant. The similar result occurs for fractional maximal functions. However, we discovered that situation with strong fractional maximal function of variable order and multiple Hardy transform is completely different. One of the novelties of this research is that two-weight problem for the double Hardy transform is studied for the first time in Lp(·) spaces. We treat the similar task for fractional maximal function of variable order and solve the trace problem. It should be emphasized that two-weight estimates are derived without the requirement of the weak logarithmic condition for exponents of spaces.

106

V. Kokilashvili and A. Meskhi

For the weighted results regarding the classical Hardy transform in Lp(·) spaces with power weights we refer to [12] and [4]. The papers [5] and [13] deal with the Hardy inequality with weights of general type. We notice that in [5] the weak logarithmic condition is not assumed for the exponents of spaces. The boundedness of the fractional maximal operator Mα from Lp(·) (Rn ) to Lq(·) (Rn ) was proved in [1], where p and q are related by the Sobolev condition and p satisfies the log-Hölder continuity condition. In [10] the authors of this paper derived criteria governing the boundedness of variable parameter q(·) fractional maximal operator Mα(·) from Lpw (Rn ) to Lv (Rn ) (see also [9] for two-weight criteria of Sawyer type). Let us recall some well-known results regarding the Hardy inequality in the classical Lebesgue spaces (see e.g., [16], [17]). The celebrated classical Hardy inequality states: Theorem A. Let p be constant satisfying the condition 1 < p < ∞ and let f be a measurable, non-negative function on (0, ∞). Then ∞ x p p1 p1 ∞ 1 p p f (y)dy dx ≤ f (x)dx . x 0 p−1 0 0 Two-weighted boundedness criteria for the Hardy transform x (H1 f )(x) =

f (y)dy, 0

reads as follows: Theorem B. Let p and q be constants satisfying the condition 1 < p ≤ q < ∞. Suppose that u and v are weight functions on R+ . Then each of the following conditions are necessary and sufficient for the inequality ∞

q1 ∞ p1 q p H1 f (x)v(x)dx ≤C f (x)w(x)dx

0

(1.1)

0

to hold for all non-negative and measurable functions f on R+ : a) The Muckenhoupt condition, ⎛∞ ⎞ q1 ⎛ x ⎞ 1 p 1−p ⎠ ⎝ ⎝ ⎠ AM := sup v(t)dt w(t) dt < ∞. x>0

x

0

Moreover, the best constant C in (1.1) can be estimated as follows: 1 1 q q p p AM . 1+ AM ≤ C ≤ 1 + p q

Two-weighted Norm Inequalities

107

b) The condition of L.-E. Persson and V. D. Stepanov, ⎛ x ⎞ q1 x − p1 ⎝ q ⎠ AP S := sup W (x) v(t)W (t) dt < ∞, W (x) := w(t)1−p dt. x>0

0

0

Moreover, the best constant C in (1.1) satisfies the following estimates: AP S ≤ C ≤ p AP S . In 1984 E. Sawyer [20] found a characterization of the two-weight inequality in terms of three independent conditions for the double Hardy transform x y f (t, τ )dtdτ. (H2 f )(x, y) = 0

0

The following statements gives two-weight criteria written by one condition when the weight on the right-hand side is a product of two univariate weights (see [18], [11], Ch. 1): Theorem C. Let p and q be constants such that 1 < p ≤ q < ∞ and let w(x, y) = w1 (x)w2 (y). Then the operator H2 is bounded from Lpw to Lqv (1 < p ≤ q < ∞) if and only if the Muckenhoupt type condition ⎛∞∞ ⎞ 1q ⎛ y1 y2 ⎞ 1 p 1−p ⎝ ⎠ ⎝ ⎠ v(x1 , x2 )dx1 dx2 w(x1 , x2 ) dx1 dx2 0

y1 y2

0

0

is fulfilled. It should be emphasized that from the results regarding the two-weight problem derived in this paper, as a corollary, we deduce trace inequality criteria for the double Hardy transform when the exponent of the initial Lebesgue space is a constant. Another remarkable corollary is that there exists a variable exponent p(x) for which the double average operator is bounded in Lp(·) . In the paper [8] the authors established trace inequality criteria for the strong fractional maximal operator 1 (Mα,β f )(x, y) := sup |f (t, τ )|dtdτ, 0 < α, β < 1, 1−α |J|1−β I×J(x,y) |I| I×J

in constant exponent Lebesgue spaces. In particular, the next statement holds (see [8], [11], Ch. 4): Theorem D. Let p, q, α and β be constants satisfying the conditions 1 < p < q < ∞, 0 < α, β < 1/p. Then the following statements are equivalent: (i) Mα,β is bounded from Lp (R2 ) to Lqv (R2 );

108


(ii)

v (x, y)dxdy |I|q(α−1/p) |J|q(β−1/p) < ∞, q

sup I,J I×J

where I and J are arbitrary bounded intervals in R. Exploring the two-weight problem for the strong fractional maximal function of variable order we prove an analog of Theorem D in Lp(·) spaces when the exponent of the initial Lebesgue space is constant. Let p be a non-negative measurable function on Rn . Suppose that E is a measurable subset of Rn . In the sequel we will use the following notation: p− (E) := inf p; p+ (E) := sup p; E

E

p− := p− (Rn ); p+ := p+ (Rn ).

Let Ω be an open set in Rn . Suppose that P (Ω) is the class of all measurable functions p : Ω → R satisfying the condition 1 < p− (Ω) ≤ p(t) ≤ p+ (Ω) < ∞, t ∈ Ω. By L such that

p(·)

(Ω) we denote the space of measurable functions f : Ω → R

f Lp(·) (Ω)

⎧ ⎨

⎫ ⎬ f (x) p(x) := inf λ > 0 : dx ≤ 1 < ∞. ⎩ ⎭ λ Ω

p(·) Lw (Ω)

we denote the weighted variable exponent Lebesgue In the sequel by space defined by the norm

f Lp(·) (Ω) := f w Lp(·) (Ω) . w

Sometimes we use the symbol Lp(·,·) (R2 ) (or Lp(x,y) (R2 )) when p is defined on R2 . It is known (see e.g., [15], [19], [12]) that Lp(·) (Ω) is a Banach space. For other essential properties of Lp(·) spaces we refer, e.g., to [22], [15], [19], [6]. Finally we point out that constants (often different constants in the same series of inequalities) will generally be denoted by c or C. Throughout the paper by the symbol p (x) we denote the function p(x)/(p(x) − 1). Under rectangle we mean a rectangle with sides parallel to the coordinate axes.

2. Strong fractional maximal functions in Lp(·) spaces. Unweighted case Let S |f (t, τ )|dtdτ, Mα f (x, y) = sup |R|α−1 R(x,y)

(x, y) ∈ R2 , 0 < α < 1,

R

be the fractional maximal function, where the supremum is taken over all rectangles R ⊂ R2 containing (x, y).


109

Theorem 2.1. Let p be a measurable function defined on R2 satisfying the condition p ∈ P (R2 ). Suppose that α is a constant for which the condition p(x) . Then MαS is bounded from 0 < α < p1− is satisfied. We set q(x) = 1−α·p(x) Lp(·) (R2 ) to Lq(·) (R2 ) if and only if p ≡ const.

Proof. Sufficiency can be obtained easily by using the Lp (R) → Lq (R) boundedness twice for the one-dimensional fractional maximal operator 1 (Mα f )(x) = sup 1−α |f (t)|dt, 0 < α < 1. Ix |I| I⊂R

I

Necessity. We follow T. Kopaliani [14] which proved the theorem for α = 0. First we observe that if MαS is bounded from Lp(·) (R2 ) to Lq(·) (R2 ), then 1

χR Lq(·) χR Lp (·) < ∞, sup AR := sup 1−α R R |R| where the supremum is taken over all rectangles R in R2 . Indeed, let f Lp(·) (R2 ) ≤ 1. Then for every rectangle R we have c ≥ MαS f Lq(·) (R2 ) ≥ MαS f Lq(·) (R) ≥ χR Lq(·) |R|α−1

|f (t, τ )|dtdτ. R

Taking now the supremum with respect to f , f Lp(·) ≤ 1, we find that |R|α−1 χR Lq(·) χR Lp (·) ≤ c for all R ⊂ R2 . Further, suppose the contrary: p is not constant, i.e. inf2 p(t) < sup p(t). R

R2

By using Luzin’s theorem we can conclude that there is a family of pairwise disjoint sets {Fi } satisfying the conditions: (i) |R2 \ ∪j Fj | = 0; (ii) functions p : Fi → R are continuous; (iii) for every fixed i, all points of Fi are points of density with respect to the basis consisting of all open rectangles in R2 . Indeed, let us represent R2 as follows R2 = ∪j Qj , where {Qj } is a family of pairwise disjoint half-open unit squares. Let us fixed j. Suppose that εk is a sequence converging to 0. By using Luzin’s theorem step by step, there is a family of pairwise disjoint sets Fkj in Qj such that |Qj \ (∪k Fkj )| = 0 and p is continuous on Fkj . Removing now sets of measure zero from Fkj we can assume that all points of Fki are points of density with respect to open rectangles. Further, we can find a pair of points of the type ((x0 , y1 ), (x0 , y2 )) or ((x1 , y0 ), (x2 , y0 )) in ∪Fi such that p(x0 , y1 ) = p(x0 , y2 ) or p(x1 , y0 ) = p(x2 , y0 ). Without loss of generality, assume that this pair is ((x0 , y1 ), (x0 , y2 )) such that (x0 , y1 ) ∈ F1 , (x0 , y2 ) ∈ F2 and y1 < y2 .

110


Let 0 < ε < 1 be a fixed number. Then there is a number δ > 0 such that for any rectangles Q1 (x0 , y1 ) and Q2 (x0 , y2 ) with diameters less than δ, the following inequalities hold: |Q1 ∩ F1 | > (1 − ε)|Q1 |, |Q2 ∩ F2 | > (1 − ε)|Q2 |,

(2.1)

p1 = sup p(x, y) < c1 < c2 < inf p(x, y) = p2 ,

(2.2)

Q2 ∩F2

Q1 ∩F1

where c1 and c2 are some positive constants. Let Q1,τ and Q2,τ be rectangles with properties (2.1) and (2.2). Suppose that Q1,τ := (x0 − τ, x0 + τ ) × (a, b) and Q2,τ := (x0 − τ, x0 + τ ) × (c, d), where a < b < c < d. Observe now that the following embeddings hold:

Lq(·) (Q2,τ ) → Lq2 (Q2,τ ), Lp (·) (Q1,τ ) → L(p1 ) (Q1,τ ), where q2 = inf q = Q2 ∩F2

p2 1−αp2 ,

(pQ1 ) =

p1 p1 −1 .

(2.3)

Recall that (see, e.g., Theorem

2.8 in [15]) the norm of embedding operators in (2.3) does not exceed 2τ (d − c) + 1 and 2τ (b − a) + 1 respectively. Further, by using (2.1) and (2.3) we have for the rectangle Qτ := (x0 − τ, x0 + τ ) × (a, d), 1

χQτ Lq(·) χQτ Lp (·) |Qτ |1−α 1 ≥

χQ2,τ ∩F2 Lq(·) χQ1,τ ∩F1 Lp (·) [2τ (d − a)]1−α 1 C 1− 1 ≥ [2τ (d − c)] q2 [2τ (b − a)] p1 1−α [2τ (d − a)]

sup AR ≥ R

= Cτ

α−1+ q1 +1− p1 2

1

= Cτ

α−

1 p1

− q1

2

.

The last expression tends to 0 as τ → 0 because α− p11 + q12 = α− p11 + p12 −α < 0 and the constant C does not depend on τ and ε for small τ and ε (recall also that a, b, c and d are fixed). This contradicts the condition supR AR < ∞.

p(·)

3. Double Hardy transform in Lw spaces Let

x y (H2 f )(x, y) =

f (t, τ )dtdτ, 0

(x, y) ∈ R2+ ,

0

R2+

where := [0, ∞) × [0, ∞). First we prove the following lemma: Lemma 3.1. Let p be a constant satisfying the condition 1 < p < ∞. Suppose that 0 < b ≤ ∞. Let ρ be an almost everywhere positive function on [0, b).


111

Then there is a positive constant c such that for all f ∈ Lpρ ([0, b)), f ≥ 0, the inequality b 0

p

x

1 λ([0, x])

b λ(x)dx ≤ C

f (t)dt 0

(f (x)ρ(x))p dx 0

holds, where λ(x) = ρ−p (x) and λ([0, x]) :=

x

λ(t)dt.

0

Proof. It is enough to show that (see e.g., [17], Ch. 1) the condition t p−1 b λ([0, x])−p λ(x)dx λ(x)dx 0

esssupΦ(y, s) 0

y∈[t,s]

1/q w(s)ds

V −1/p (t).

(3.7)

On a Supremum Operator

237

Proof. Let t > 0 and ϕt (x) := χ[0,t] (x). Then Rϕt (x) = sup Φ∞ (y, x)χ[0,t] (y) = χ[0,t] (x)Φ∞ (t, x). y≥x

We have

J≥

1/q

t

0

Φ∞ (t, s)w(s)ds

V −1/p (t), t > 0.

Hence, J ≥ A. As ϕq ∈ M ↓ if and only if ϕ ∈ M ↓ we can change ϕq by ϕ on the right hand side of (3.5). Then, using (3.4), we find 6 ∞5 supy≥t Φq∞ (y, t)ϕ(y)dy w(t)dt 0 q J = sup . ∞ p/q v q/p ϕ∈M↓ ϕ 0 Applying representation (2.3) α = q, we obtain q/p p α/p ∞ ∞ q −1 p q q sup Φ∞ (y, t) h h(s)ds sup Φ∞ (y, t)ϕ(y) = q y≥t y≥t y s q/p p q/p ∞ ∞ q −1 p ≤ sup h Φq∞ (s, t)h(s)ds . q y≥t y s Now, applying Minkowskii’s inequality, we get 4 ∞3 q sup Φ∞ (y, t)ϕ(y) w(t)dt 0

y≥t

q/p q/p ∞ ∞ ∞ pq −1 p ≤ h Φp∞ (s, t)h(s)ds q 0 t s q/p q/p ∞ ∞ ∞ pq −1 p q ≤A h h(s)V (s)ds q 0 t s q/p ∞ = Aq ϕp/q v . 0

The remaining part of the proof of the upper bound J ≤ A follows by applying Lemma 2.1 and the Monotone Convergence Theorem . Remark 3.3. In the case Φ(y, t) = esssup u(s), s∈[t,y]

where u(s) ≥ 0 is a measurable function, we obtain Rϕ(t) = esssup u(s)ϕ(s), s∈[t,∞]

and Theorem 3.2 extends ([1], Theorem 3.2 (i)), where the function u(s) was supposed to be continuous.

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4. The case q < p Definition 4.1. A measurable function Φ(x, y) ≥ 0 on {(x, y) : x ≥ y ≥ 0}, we name Oinarov kernel, Φ(x, y) ∈ O, if there exist a constant D ≥ 1, independent of x, y and z such that D−1 (Φ(x, z) + Φ(z, y)) ≤ Φ(x, y) ≤ D (Φ(x, z) + Φ(z, y))

(4.1)

for all x ≥ z ≥ y ≥ 0. Proposition 4.2. Let Φ(x, y) ∈ O. Then Φ∞ (x, y) ∈ O. Proof. For all x ≥ y ≥ 0 from (3.1) and (3.2) we have Φ∞ (x, y) = esssupΦ(s, y) = sup esssupΦ(t, y) = sup Φ∞ (s, y). s∈[y,x]

s∈[y,x] t∈[y,s]

(4.2)

s∈[y,x]

Then Φ∞ (x, y) is non-decreasing with respect to x for x ∈ [y, ∞). Similarly, using (4.1), we find Φ∞ (x, y) = esssupΦ(s, y) = sup esssupΦ(t, y) ≥ D−1 sup Φ(s, y) s∈[y,x]

≥D

−1

s∈[y,x] t∈[s,x]

s∈[y,x]

Φ(x, y).

Therefore, Φ(x, y) ≤ DΦ∞ (x, y)

(4.3)

for all x ≥ y ≥ 0. Let x ≥ z ≥ y ≥ 0. Then it follows from (4.2), that Φ∞ (x, y) ≥ Φ∞ (z, y).

(4.4)

Moreover, again using (3.2) and (4.1), we find Φ∞ (x, y) = sup esssupΦ(t, y) ≥ sup esssupΦ(t, y) s∈[y,x] t∈[s,x]

≥D

−1

s∈[z,x] t∈[s,x]

sup esssupΦ(t, s) = D

−1

s∈[z,x] t∈[s,x]

sup Φ∞ (x, s) ≥ D−1 Φ∞ (x, z).

(4.5)

s∈[z,x]

From this and (4.4) the left hand side of (4.1) follows for Φ∞ (x, y). Let x ≥ y ≥ 0 and z ∈ [y, x], s ∈ [y, x]. Then it follows from (4.1) that Φ(s, y) = χ[y,z] (s)Φ(s, y) + χ[z,x] (s)Φ(s, y) ≤ D χ[y,z] (s)Φ(z, y) + χ[z,x] (s) (Φ(s, z) + Φ(z, y)) = D Φ(z, y) + χ[z,x] (s)Φ(s, z) . From this and (4.3)

Φ∞ (x, y) ≤ D Φ(z, y) + esssupΦ(s, z)

≤ D2 (Φ∞ (z, y) + Φ∞ (x, z)) .

s∈[z,x]

Hence, Φ∞ (x, y) ∈ O with a constant D2 in (4.1).

Remark 4.3. If 0 < q < ∞ and Φ(x, y) ∈ O, that is (4.1) holds, then Φq (x, y) ∈ O, so that (4.1) holds with some constant Dq , dependent only on D and q.


239

Theorem 4.4. Let 0 < q < p < ∞, 1r := 1q − p1 and Φ(x, y) ∈ O. Suppose Φ(x, y) be continuous with respect to x for x ∈ [y, ∞) for all y ≥ 0 and t assume that the weight functions v and w such that 0 < V (t) := 0 v < t ∞, 0 < W (t) := 0 w < ∞ and V (∞) = W (∞) = ∞. If the supremum operator R and the functional J are defined by (3.3) and (3.5), respectively, and r1 rq

xk+1 − rp q Φ∞ (xk+1 , t)w(t)dt [V (xk+1 )] , (4.6) B := sup {xk } k

xk

where the sup is taken over all increasing sequences {xk } ⊂ R+ , then J ≈ B.

(4.7)

Proof. We start with the proof of the upper bound J # B. To this end we note, that because of (3.4) and (4.5) we have Rϕ(s) ≤ DRϕ(t) for all s ≥ t ≥ 0, if ϕ ∈ M↓ . Let a > 1 be a number, which we choose later and let {xk }, {yk } ⊂ R+ be such increasing sequences, that W (xk ) = V (yk ) = ak , k ∈ Z. We have ∞ 0

(Rϕ)q w ≤ Dq

= Dq (a − 1)

[Rϕ(xk )]

q

(4.8)

xk+1

w xk

k

q

ak sup [Φ∞ (s, xk )ϕ(s)] =: Dq (a − 1) s≥xk

k

ak Ik .

k

Since Φ(x, y) ∈ O, then Φ∞ (x, y) ∈ O by Proposition 4.2 and by Remark 4.3 we have Φq∞ (x, y) ∈ O. Let Dq ≥ 1 be a constant, for which (4.1) holds for Φq∞ (x, y). Then, applying (4.1) with Φq∞ (x, y) and Dq , we obtain Ik ≤ ≤

sup xk ≤s≤xk+1

sup xk ≤s≤xk+1

Φq∞ (s, xk )ϕ(s) + sup Φq∞ (s, xk )ϕ(s) s≥xk+1

Φq∞ (s, xk )ϕ(s) (

'

+ Dq

Φq∞ (xk+1 , xk )ϕ(xk+1 )

≤ (1 + Dq )

sup xk ≤s≤xk+1

+ sup s≥xk+1

Φq∞ (s, xk )ϕ(s)

Φq∞ (s, xk+1 )ϕ(s)

+ Dq sup Φq∞ (s, xk+1 )ϕ(s) s≥xk+1

=: (1 + Dq )Lk + Dq Ik+1 . We find from this

I := ak Ik ≤ (1 + Dq ) ak L k + D q ak Ik+1 k

= (1 + Dq )

k

k

k

Dq k Dq I. ak L k + a Ik = (1 + Dq ) ak L k + a a k

k

240

V.D. Stepanov

Now we choose a > 1 such, that a > 2Dq . Then I ≤ 2(1 + Dq ) Consequently, ' (q ∞

q k sup (Rϕ) w # a Φ∞ (s, xk )ϕ(s) . 0

k

ak L k .

xk ≤s≤xk+1

k

It follows from the continuity of Φ(x, y) that Φ∞ (x, y) is continuous. Moreover, taking into account Proposition 3.1 without a loss of generality we may and shall assume Φq∞ (s, xk )ϕ(s) to be continuous on [xk , xk+1 ]. while the upper bound is proving. Then there exist a point zk ∈ [xk , xk+1 ] such, that sup xk ≤s≤xk+1

Φ∞ (s, xk )ϕ(s) ≤ a1/q Φ∞ (zk , xk )ϕ(zk ).

Now using (4.1) and zk−2 ≤ xk−1 < xk ≤ zk , it follows, that ∞

xk q (Rϕ) w # w(t)dt Φq∞ (zk , xk )ϕq (zk ) 0

xk−1

k

#

k

≤

zk

w(t)Φq∞ (zk , t)dt

w(t)Φq∞ (z2k , t)dt

ϕq (z2k )

z2k+1

z2k−1

k

ϕq (zk )

z2k

ϕq (zk )

w(t)Φq∞ (zk , t)dt

z2k−2

k

+

xk−1

zk−2

k

=

xk

w(t)Φq∞ (z2k+1 , t)dt ϕq (z2k+1 )

=: Seven + Sodd . Further we estimate only Seven , the arguments for Sodd are similar. Put Yk := {l ∈ Z : yl ∈ [z2k−2 , z2k ]}, Y := {k ∈ Z : Yk = ∅}, where yl are taken from the definition (4.8). Denote θk := min{yl : l ∈ Yk }, k ∈ Y ; Θ := {θk }k∈Y ⊂ {yl }l∈Z and renumerate Θ so, that Θ =: {yn }n∈Z and yn < yn+1 . It is shown in the proof of ([1], Theorem 3.2 (ii)) that if n, k ∈ Z such, that yn < z2k ≤ yn+1 , then q/p yn+1 −q/p yn+1 q p [ϕ(z2k )] # v ϕ v . (4.9) 0

yn−1

Denote An := {k ∈ Z : yn < z2k ≤ yn+1 }, n ∈ Z. Then Seven =

n

An

z2k

z2k−2

Φq∞ (z2k , t)w(t)dt ϕq (z2k ).


241

It frollows from the properties of Φ∞ (x, y) ∈ O that Φ∞ (z2k , t) #

sup t≤s≤yn+1

Φ∞ (s, t) # Φ∞ (yn+1 , t).

(4.10)

Applying (4.9), (4.10) and Hölder’s inequality, we find q/p

yn+1 −q/p yn+1 p v ϕ v Seven # n

×

⎛

≤⎝

yn+1

yn−1

n

×

Φq∞ (yn+1 , t)w(t)dt

Φq∞ (yn+1 , t)w(t)dt

yn+1 yn−1

n

yn−1

z2k−2

An

≤

0 z2k

# Bq

∞

q/p

yn+1

v

p

ϕ v

0

yn−1

r/q Φq∞ (yn+1 , t)w(t)dt

−r/p

yn+1

v

⎞q/r ⎠

0

q/p

yn+1

p

ϕ v

yn−1

n

−q/p

yn+1

q/p

ϕp v

.

0

From this and analogous bound for Sodd the upper bound J # B follows. Let {xk } ⊂ R+ be an arbitrary increasing sequence, N - any positive integer. The proof of the lower bound J $ B is proceeding with the help of the test function N N 1/p 1/p N

ϕN (t) := χ(0,x−N ) (t) αi + χ[xk ,xk+1 ] (t) αi , i=−N

where

xi+1

αi := xi

k=−N

r/q Φq∞ (xi+1 , t)w(t)dt

i=k

V −r/q (xi+1 ).

We have 0

∞

q

[RϕN ] w ≥

N

k=−N

≥

k=−N

=

sup xk+1

sup t≤s≤xk+1

xk

N

k=−N

t≤s≤xk+1

xk

N

xk+1

xk+1 xk

Φq∞ (s, t)w(t)dt

N

q/p αi

i=k q/p

Φq∞ (xk+1 , t)w(t)dtαk r/q

Φq∞ (xk+1 , t)w(t)dt

r V −r/p (xk+1 ) =: BN .

242

V.D. Stepanov

On the other hand ∞ N

p ϕN v = αk 0

=

αi

i=−N

=

N

i=−N

v +

0

k=−N N

x−N

v + 0

αi

k=−N i=k

x−N

xi+1

v 0

N N

N

i=−N

αi

xk+1

αi

v xk

i

k=−N

xk+1

v

xk

r = BN .

Consequently, J $ BN and the lower bound J $ B follows.

References [1] Gogatishvili A., Opic B. and Pick L. Weighted inequalities for Hardy-type operators involving suprema. Collect. Math., 57 (2006), 227–255. [2] Gogatishvili A. and Pick L. A reduction theorem for supremum operators. J. Comp. Appl. Math., 208 (2007), 270–279. [3] Cwikel M. and Pustylnik E. Weak type interpolation near ”endpoint” spaces. J. Funct. Anal., 171 (2000), 235–277. [4] Evans W.D. and Opic B. Real interpolation with logarithmic functions and reiteration. Canad. J. Math., 52 (2000), 920–960. [5] Pick L. Optimal Sobolev Embeddings. Rudolph-Lipshitz-Vorlesungsreihe no. 43, Rheinische Friedrich-Wilhelms-Universit¨ at Bonn, 2002. [6] Prokhorov D. V. Inequalities for Riemann-Liouville operator involving suprema. Collect. Math., 61 (2010), 263–276. [7] Stepanov V.D. Integral operators on the cone of monotone functions. J. London Math. Soc. 48 (1993), 465–487. [8] Sinnamon G. Transferring monotonicity in weighted norm inequalities. Collect. Math. 54 (2003), 181–216. Vladimir D. Stepanov Department of Mathematical Analysis and Function Theory Peoples Friendship University 117198 Moscow Russia e-mail: [email protected]


Entropy Numbers of Quadratic Forms and Their Applications to Spectral Theory Hans Triebel Dedicated to my dear friends David Edmunds and Des Evans

Abstract. The paper deals with positive definite quadratic forms n ∂f (x) ∂g(x) Eb (f, g) = b(x) dx ∂xj ∂xj Ω j=1 in L2 (Ω) in limiting situations. We estimate the entropy numbers of related compact embeddings and apply the outcome to say something about the distribution of eigenvalues of the generated degenerate positive definite self-adjoint elliptic operators n ∂f ∂ b(x) Ab f = − ∂xj ∂xj j=1 with pure point spectrum. Mathematics Subject Classification (2010). 46E35, 41A46, 35P15. Keywords. Quadratic forms, degenerate elliptic operators, entropy numbers, distribution of eigenvalues.

1. Introduction This paper may be considered as the continuation of [14], [15, Section 3.4], and also of [36, 37]. Let Ω be a bounded domain in Rn , n ≥ 2, and let −1 ∈ Ln/2 (Ω). Then b ∈ Lloc 1 (Ω) be a real function with b(x) > 0 a.e. and b the closure Eb (Ω) of the closable quadratic form n

∂f ∂¯ g b(x) (x) (x) dx, f, g ∈ D(Ω), Eb (f, g) = ∂xj ∂xj Ω j=1 is continuously embedded in L2 (Ω) (positive definite quadratic form). This is a limiting situation. If b−1 belongs to the Zygmund spaces Ln/2 (log L)d (Ω),

244

H. Triebel

n ≥ 3, with d > 0, then id :

Eb (Ω) → L2 (Ω)

is compact.

For d > 4/n and a suitable constant c > 0 one has for the corresponding entropy numbers ek (id), ek (id) ≤ c b−1 |Ln/2 (log L)d (Ω) 1/2 k −1/n ,

k ∈ N.

For 0 < d ≤ 4/n, ε > 0, and suitable constants cε > 0 one has ek (id) ≤ cε b−1 |Ln/2 (log L)d (Ω) 1/2 k − 4 +ε , d

k ∈ N.

This will be complemented by corresponding assertions for bounded domains Ω in the plane R2 , hence for n = 2. We rely on extrapolation techniques which go back (in this context) to [33, 14, 15]. Section 2 deals with the indicated assertions about entropy numbers. Our main results here are the Theorems 2.3, 2.4. In Section 3 we apply these results to spectral assertions for generated positive definite self-adjoint elliptic operators in L2 (Ω) with pure point spectrum of type Ab f = −

n

∂ ∂f b(x) ∂x ∂x j j j=1

with Theorem 3.1 as our related main result. This will be complemented in Section 4 by some discussions, examples and further references.

2. Entropy numbers of quadratic forms 2.1. Preliminaries Let Ω be a bounded domain in the Euclidean n-space Rn , n ≥ 2. Domain means open set. We use standard notation. In particular, Lp (Ω) with 1 ≤ p ≤ ∞ are the usual complex Lebesgue spaces with respect to the Lebesgue measure indicated by dx. The space of all locally Lebesgueintegrable functions in Ω is denoted by Lloc 1 (Ω). Let D(Ω) be the collection of all complex-valued C ∞ functions on Ω with compact support and let D (Ω) be the related dual space of all distributions. We are interested in symmetric quadratic forms n

∂¯ g ∂f b(x) (x) (x) dx, f, g ∈ D(Ω), (2.1) Eb (f, g) = ∂x ∂x j j Ω j=1 in L2 (Ω) with b ∈ Lloc 1 (Ω) and b(x) > 0 a.e. (almost everywhere) in Ω. We recall some notation in an abstract setting. Let H be a separable complex Hilbert space with scalar product (u, v)H 1/2 and norm u |H = (u, u)H . Let D be a dense linear subset of H and E : D × D → C (complex numbers) be a bilinear symmetric map, hence E(λ1 u1 + λ2 u2 , v) = λ1 E(u1 , v) + λ2 E(u2 , v),

E(u, v) = E(v, u),

Entropy Numbers of Quadratic Forms

245

where u1 , u2 , u, v ∈ D and λ1 , λ2 ∈ C. A positive definite quadratic form is a densely defined bilinear symmetric map such that E(u, u) ≥ c u |H 2

for some c > 0 and all u ∈ D.

(2.2)

Recall that a positive definite quadratic form E is called closable on H when E(uk , uk ) → 0 for k → ∞ for all sequences {uk } ⊂ D with uk → 0 in H and for any ε > 0, E uk − ul , uk − ul ≤ ε if k ≥ l ≥ l(ε) (Cauchy E-sequence). Then the abstract completion of D with respect to the E-norm (the space of all E-Cauchy sequences) can be identified in a one-toone way with a related linear subset in H, the domain of definition dom E of the closure of E. One has (2.2) now with u ∈ dom E. Let dom A = u ∈ dom E : E(u, v) = (u , v)H for some u ∈ H and all v ∈ dom E . (2.3) Then u is uniquely determined and Au = u generates a positive definite self-adjoint operator in H. As usual, dom E = dom A1/2 is called the energy space, and A1/2 dom E = H, A−1/2 H = dom E, (2.4) are isomorphic maps. Details about closable and closed (positive definite) forms may be found in [28]. The construction (2.3) coincides essentially with Friedrichs’ extension of positive definite symmetric operators in complex Hilbert spaces. We refer for details to [10, Section IV,2, pp. 172–180] and [32, Sections 4.1.9, 4.4.3, pp. 213–215, 253]. A description may also be found in [34, pp. 190/191] with a reference to [9, Section 4.4, pp. 81–84] for a short direct proof. We return to the quadratic form Eb (f, g) in a bounded domain Ω in Rn , n ≥ 2, according to (2.1). Let Lp (Ω) with 1 ≤ p ≤ ∞ be the usual complex ◦ ∂f n Lebesgue spaces and let ∇f (x) = ∂x . Then W 1p (Ω) with 1 ≤ p < ∞ j=1 j is the completion of D(Ω) in the norm n

◦

f |W 1p (Ω) = |∇f | Lp (Ω) ∼ j=1

∂f |Lp (Ω) . ∂xj

(2.5)

Recall that for some c > 0, ◦

f |L2 (Ω) ≤ c f |W 12n (Ω) , n+2

◦

f ∈ W 12n (Ω).

(2.6)

n+2

If n ≥ 3, then (2.6) is the well-known Sobolev embedding. In case of n = 2 we refer for a short elegant proof of

f |L2 (Ω) ≤ c

∂f ∂f |L1 (Ω) + c |L1 (Ω) , ∂x1 ∂x2

to [39, Theorem 2.4.1, p. 56].

f ∈ D(Ω),

246

H. Triebel

If Eb (f, g) is closable, then the domain of its closure dom Eb will be denoted by Eb (Ω). Proposition 2.1. Let Ω be a bounded domain in Rn , n ≥ 2. Let b ∈ Lloc 1 (Ω), b(x) > 0 a.e. in Ω and b−1 ∈ Ln/2 (Ω). Then Eb (f, g) according to (2.1) is a closable positive definite quadratic form in L2 (Ω), and 1/2

f |L2 (Ω) ≤ c b−1 |Ln/2 (Ω) 1/2 b(x) |∇f (x)|2 dx , (2.7) Ω

for some c > 0 and all f ∈ Eb (Ω). Proof. Step 1. Let f ∈ D(Ω). Then (2.7) follows from (2.6) and Hölder’s inequality, n+2 n n 2n b− n+2 (x) b(x) |∇f (x)|2 n+2 dx

f |L2 (Ω) ≤ c Ω (2.8) 1/n 2 1/2 −n/2 ≤c b (x) dx b(x) ∇f (x) dx . Ω

Ω

Step 2. We prove that Eb is closable. Let {fk }∞ k=1 ⊂ D(Ω) be a Cauchy sequence in the Eb -norm. Let in addition fk → 0 in L2 (Ω). In particular ∂f ∂ϕ k → 0 if k → ∞, ϕ ∈ D(Ω), , ϕ = − fk , ∂xl ∂xl where l = 1, . . . , n. By (2.8) it follows that {fk }∞ k=1 is also a Cauchy sequence ◦

in W 12n (Ω). Then n+2

∂f k 2n (Ω), (x) → 0 if k → ∞ in L n+2 ∂xl

(2.9)

l = 1, . . . , n. It remains to prove that Eb (fk , fk ) → 0

if k → ∞.

(2.10)

By Eb (fk , fk ) = Eb (fk − fm , fk − fm ) + Eb (fk − fm , fm ) + Eb (fm , fk ) ≤ Eb (fk − fm , fk − fm ) + Eb (fk − fm , fk − fm )1/2 Eb (fm , fm )1/2 + Eb (fm , fk ) (2.11) one can reduce (2.10) to Eb (fk , ϕ) → 0 if k → ∞ for any ϕ ∈ D(Ω).

(2.12)

BK = {x ∈ Ω : b(x) < K}.

(2.13)

Let Then it follows from (2.9) that n

∂fk ∂ ϕ¯ b(x) · dx → 0 if k → ∞. ∂xl ∂xl BK l=1

(2.14)


247

Let Eb (f, g)K be given by (2.1) where the integration over Ω is replaced by the integration over Ω \ BK . One has by the triangle inequality that 1/2

1/2

Eb (fk , fk )K ≤ Eb (fk − fm , fk − fm )1/2 + Eb (fm , fm )K . Now (2.10) follows from (2.11)–(2.15) by standard arguments.

(2.15)

We need a second preparation. Let again Ω be a bounded domain (= open set) in Rn , n ≥ 2. Let f be a complex-valued a.e. finite Lebesgue measurable function on Ω. The distribution function μf and the non-increasing rearrangement f ∗ of f have the usual meaning, λ ≥ 0, μf (λ) = {x ∈ Ω : |f (x)| > λ}, and

f ∗ (t) = inf λ : μf (λ) ≤ t , t ≥ 0. Let 0 < p < ∞ and a ∈ R. Then the Zygmund space Lp (log L)a (Ω) consists of all complex-valued a.e. finite Lebesgue measurable functions on Ω for which |Ω| ap 1/p f ∗ (t)p 1 + | log t| dt < ∞. (2.16)

f |Lp (log L)a (Ω) = 0

Detailed information about rearrangement of functions and Zygmund spaces may be found in the standard references [2, 11]. We refer in particular to [11, Sections 3.2–3.4] and [2, Chapter 2, Section 4.6]. If a = 0, then Lp (log L)0 (Ω) = Lp (Ω) quasi-normed as usual by 1/p |f (x)|p dx .

f |Lp (Ω) = Ω

We need the following extrapolation characterisation of Lp (log L)a (Ω) with a = 0. Let 0 0, j ∈ N, j ≥ j0 (p). (2.17) j p(j) p n·2 p[j] p n · 2j Here N is the collection of all natural numbers. (i) Let a < 0. Then Lp (log L)a (Ω) is the collection of all Lebesgue measurable a.e. finite functions f on Ω such that ∞ 1/p 2jap f |Lp(j) (Ω) p < ∞, (2.18) j=j0

equivalent quasi-norm. (ii) Let a > 0. Then Lp (log L)a (Ω) is the collection of all Lebesgue measurable a.e. finite functions f on Ω which can be represented as ∞

f= fj , fj ∈ Lp[j] (Ω), (2.19) j=j0

such that

∞ j=j0

2jap fj |Lp[j] (Ω) p

1/p

< ∞.

(2.20)

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H. Triebel

Furthermore, the infimum of all expressions (2.20) taken over all representations (2.19), (2.20) is an equivalent quasi-norm on Lp (log L)a (Ω). This is one of the main results in [8, Corollary 3.1, p. 74]. Assertions of this type go back to [14] and [15, Section 2.6] with p ≥ 1 in case of a > 0. As for further information, generalisations and abstract versions we refer to [16, 8]. Let 0 < ε < p < ∞ and −∞ < a2 < a1 < ∞. Then Lp+ε (Ω) → Lp (log L)a1 (Ω) → Lp (log L)a2 (Ω) → Lp−ε (Ω)

(2.21)

and Lp (log L)ε (Ω) → Lp (Ω) → Lp (log L)−ε (Ω).

(2.22)

Here → means continuous embedding. Hence the Zygmund spaces Lp (log L)a (Ω) are refinements of the Lebesgue spaces Lp (Ω). For our purpose it is reasonable to introduce the auxiliary modifications Gp,a (Ω) of Lp (log L)a (Ω). If a < 0, then one replaces (2.18) in part (i) by

f |Gp,a (Ω) = sup 2ja f |Lp(j) (Ω) . j≥j0

If a > 0, then Gp,a (Ω) is the collection of all f which can be represented by (2.19) with sup 2ja fj |Lp[j] (Ω) < ∞

(2.23)

j≥j0

quasi-normed by f |Gp,a (Ω) which is the infimum over (2.23) for f represented by (2.19). We need the following simple observation. Proposition 2.2. Let Ω be a bounded domain in Rn , n ≥ 2. Let 0 < p, r < ∞ and 1t = 1p + 1r . Let a = 0 and ε > 0. Then Lp (log L)a (Ω) → Gp,a (Ω) → Lp (log L)a−ε (Ω)

(2.24)

gf |Gt,a (Ω) ≤ g |Lp (log L)a (Ω) · f |Lr (Ω) ,

(2.25)

and where g ∈ Lp (log L)a (Ω) and f ∈ Lr (Ω). Proof. The embedding (2.24) is obvious, whereas (2.25) follows from H¨ older’s inequality for Lebesgue spaces. Finally we recall the definition of the (fractional) Sobolev spaces Hps with 0 < p < ∞ and s ∈ R on Rn and on domains. We use standard notation. In particular, the Schwartz space S(Rn ), the space S (Rn ) of tempered distributions, the Fourier transform fB and its inverse f ∨ of S (Rn ) have the usual meaning. Let ϕ0 ∈ S(Rn ) with ϕ0 (x) = 1 if |x| ≤ 1 and let

and ϕ0 (y) = 0 if |y| ≥ 3/2,

ϕk (x) = ϕ0 2−k x − ϕ0 2−k+1 x ,

x ∈ Rn ,

k ∈ N.


249

n Then ϕ = {ϕj }∞ j=0 is a dyadic resolution of unity in R . Let 0 < p < ∞ and s n s ∈ R. Then the (fractional) Sobolev space Hp (R ) is the collection of all f ∈ S (Rn ) such that ∞ 2 1/2 (2.26)

f |Hps (Rn ) ϕ = 22js (ϕj fB)∨ (·) |Lp (Rn ) j=0

is finite (equivalent quasi-norms for different choices of ϕ). Of interest for us are the classical Sobolev spaces Wp1 (Rn ) with 1 ≤ p < ∞, normed by

f

|Wp1 (Rn )

n

∂f = f |Lp (R ) + |Lp (Rn ) . ∂xl n

l=1

Wp1 (Rn )

Hp1 (Rn ).

If 1 0, c2 > 0 such that for all p with 1 < p < ∞ and all f ∈ Lp (Rn ), c1 (p + p )−1 f |Hp0 (Rn ) ≤ f |Lp (Rn ) ≤ c2 (p + p ) f |Hp0 (Rn ) ,

(2.27)

1 1 p + p n

where = 1. This is the well-known Littlewood–Paley characterisation of Lp (R ) with 1 < p < ∞. It is essentially covered by [17, p. 339]. There is a similar inequality with Hp1 (Rn ) in place of Hp0 (Rn ) and Wp1 (Rn ) in place of Lp (Rn ). 2.2. Quadratic forms in higher dimensions Higher dimensions means n ≥ 3. But first we assume n ≥ 2 and indicate where the cases n ≥ 3 and n = 2 differ. Let Ω be a bounded domain in Rn , n ≥ 2, and let n

∂f ∂¯ g Eb (f, g) = b(x) (x) (x) dx, f, g ∈ D(Ω), (2.28) ∂xj ∂xj Ω j=1 be the above quadratic form where b ∈ Lloc 1 (Ω), b(x) > 0 a.e. in Ω and b−1 ∈ Ln/2 (Ω). By Proposition 2.1 this quadratic form Eb (f, g) is closable in L2 (Ω) and positive definite and one has (2.7) on the closure Eb (Ω). In particular the embedding id :

Eb (Ω) → L2 (Ω)

(2.29)

is continuous. But we do not know whether it is always compact. The situation improves if one strengthens b−1 ∈ Ln/2 (Ω) by b−1 ∈ Ln/2 (log L)d (Ω) with d > 0, (2.22). We measure compactness in terms of entropy numbers and recall briefly the standard definition of entropy numbers in an abstract setting. Let A and B be two (complex) quasi-Banach spaces with the respective unit balls UA and UB . Let T ∈ L(A, B) be a linear and bounded operator from A into B. Then for k ∈ N the k-th entropy number ek (T ) of T is defined as the infimum of all ε > 0 such that T (UA ) ⊂

k−1 2

j=1

bj + ε UB

for some b1 , . . . , b2k−1 ∈ B.

250

H. Triebel

We assume that the reader is familiar with basic assertions for entropy numbers of compact embeddings between function spaces. In [15] and also in [24, 36] one finds historical comments. A few more recent and more specific papers will be mentioned later on. Theorem 2.3. Let Ω be a bounded domain in Rn , n ≥ 3. Let b be a real Lebesgue-measurable function on Ω with b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

b−1 ∈ Ln/2 (log L)d (Ω)

with d > 0. Then the embedding id in (2.29) is compact. If d > 4/n, then ek (id) ≤ c b−1 |Ln/2 (log L)d (Ω) 1/2 k −1/n ,

k ∈ N,

(2.30)

for some c > 0 which is independent of b and k. If 0 < d ≤ 4/n, then for any ε > 0 there is a constant cε > 0 such that for all b and k, ek (id) ≤ cε b−1 |Ln/2 (log L)d (Ω) 1/2 k − 4 +ε , d

k ∈ N.

(2.31)

2n be the Sobolev index Proof. Step 1. First we assume that n ≥ 2. Let t = n+2 according to (2.6). Let Gt,a (Ω) be the spaces introduced in (2.23) with

a>0

and

1 1 1 = − > 0, t[j] t n · 2j

j ∈ N, j ≥ j0 (t),

according to (2.17). With f = b−1/2 b1/2 f ∈ D(Ω) and from (2.25) that

1 t

=

1 2

+

f |Gt,a (Ω) ≤ b−1/2 |Ln (log L)a (Ω) · b1/2 f |L2 (Ω) .

(2.32) 1 n

it follows (2.33)

If n ≥ 3, then t > 1. But if n = 2, then we have t = 1. This requires some extra care. In order to cover both cases, n ≥ 3 and n = 2, we introduce the Littlewood–Paley counterpart of Gt,a (Ω) with a > 0 according to (2.23). We use the norms in (2.26) (without the subscript ϕ). Let t be as above and a, 0 (Ω) is the collection of all f ∈ L1 (Ω) which can be t[j] as in (2.32). Then Ht,a represented as ∞

f= fj , fj ∈ Lt[j] (Ω), (2.34) j=j0

such that 0 sup 2ja fj |Ht[j] (Rn ) < ∞

(2.35)

j≥j0

with fj extended by zero outside of Ω. Furthermore, 0 0

f |Ht,a (Ω) = inf sup 2ja fj |Ht[j] (Rn )

(2.36)

j≥j0

where the infimum is taken over all representations (2.34) (since t[j] > 1 one 0 has Lt[j] (Rn ) = Ht[j] (Rn ) with the Littlewood–Paley assertions in (2.27)). 0 would not We assume now n ≥ 3. Then t > 1 and the step from Gt,a to Ht,a 0 be necessary. But we stick at formulations in terms of Ht,a -spaces in order


251

to prepare our later considerations of n = 2. Let f ∈ D(Ω). Then it follows from (2.33) that n 1/2

∂f 0 −1/2 Ht,a (Ω) ≤ c b |Ln (log L)a (Ω) b(x) |∇f (x)|2 dx . ∂xl Ω l=1 (2.37) In [15, Section 2.6.3] we introduced logarithmic Sobolev spaces Hps (log H)a (B) in smooth domains B, say a ball B with Ω ⊂ B. In Theorem 2.6.3(iv), p. 79, based on (29), (30), p. 80, in [15] we justified that optimal decompositions of f ∈ Hps (log H)a (B) with 1 0, can be reduced to corresponding optimal decompositions of Dα f ∈ Lp (log L)a (B), 1 (B) based 0 ≤ |α| ≤ s. This applies also to obviously defined spaces Hp,a 0 on Gp,a (B). If one replaces Gp,a (B) by Hp,a (B), then one can extend these assertions also to p = 1, which will be needed later on in case of n = 2. We return to n ≥ 3. Let 1/2 b(x) |∇f (x)|2 dx ≤1

b−1/2 |Ln (log L)a (Ω) Ω

in (2.37). Then it follows from the above considerations that f can be decomposed by ∞

f=

fj ,

1 fj ∈ Ht[j] (B),

1

fj |Ht[j] (B) ≤ c 2−ja

(2.38)

j=j0

where c > 0 is independent of j. Step 2. We apply [15, Proposition 1, p. 139] to the embedding idj : where p1j = such that

1 Ht[j] (B) → L2 (Ω) 1 2

with

n+2 1 1 1 1 = − = + j t[j] 2n n2 pj n

− n21 j . One obtains that for any ε > 0 there is a constant cε > 0 2

ek (idj ) ≤ cε 2j(ε+ n ) k −1/n ,

k ∈ N.

(2.39)

Here cε is independent of j and k. Hence the image of a ball of radius c 2−ja 2 −1/n 1 in Ht[j] (B) in L2 (Ω) can be covered by 2kj balls of radius c 2j(ε+ n −a) kj centred at glj ∈ L2 (Ω) where l = 1, . . . , 2kj . Let J > j0 . Then f in (2.38) can be approximated by f−

J

j=j0

gljj =

J

fj − gljj + f J ,

where

j=j0

fJ =

fj .

j>J

Choosing gljj optimally one obtains that f−

J

j=j0

gljj |L2 (Ω) ≤ cε

J

j=j0

2

−1/n

2j(ε+ n −a) kj

+ c 2−Ja .

(2.40)

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H. Triebel

Step 3. Let a > 2/n and 0 < δ < a. We choose kj > 0 such that −1/n

2

2j(ε+ n −a) kj

= 2−Ja+(J−j)δ ,

j = j0 , . . . , J.

(2.41)

To avoid clumsy formulations we identify kj with its nearest natural number, hence assuming kj ∈ N. Choosing ε > 0, δ > 0 small one obtains that k=

J

J

2

kj = 2Jn( n +ε)

j=j0

2

2(J−j)n(a− n −ε−δ) ∼ 2Jn(a−δ) .

(2.42)

j=j0

By (2.40)–(2.42) we have f−

J

gljj |L2 (Ω) ≤ c 2−J(a−δ) ∼ c k −1/n

(2.43)

j=j0

for one of the the ∼ 2k elements of Jj=j0 gljj . Then one obtains that ek id : Eb (Ω) → L2 (Ω) ≤ c b−1/2 |Ln (log L)a (Ω) k −1/n , k ∈ N, (2.44) 62 5 where a > 2/n. Recall that (b−1 )∗ (t) = (b−1/2 )∗ (t), [2, p. 41, (1.20)], hence by (2.16)

b−1/2 |Ln (log L)a (Ω) ∼ b−1 |Ln/2 (log L)2a (Ω) 1/2 .

(2.45)

Then (2.30) follows from (2.44) with d = 2a > 4/n. Step 4. Let 0 < a ≤ 2/n. Instead of (2.41) we choose 1/n

kj

2

= 2j(ε+ n −a) 2Ja 2jκ ,

j = j0 , . . . , J,

(2.46)

where κ > 0. Then the counterpart of (2.42) is given by k=

J

kj = 2nJa

j=j0

J

2j(nε+2−na+nκ) ≤ cσ 22J(1+σ)

(2.47)

j=j0

where σ > 0 can be chosen arbitrarily small. We insert (2.46) in (2.40). Using (2.47) one obtains that

f −

J

gljj |L2 (Ω) ≤ c 2−Ja ≤ c k − 2(1+σ) . a

(2.48)

j=j0

This is the counterpart of (2.43). The rest is now the same as in Step 3. This proves (2.31). 2.3. Quadratic forms in two dimensions Let Ω be a bounded domain (= open set) in the plane R2 and let 2

∂f ∂¯ g b(x) dx, f, g ∈ D(Ω), Eb (f, g) = ∂x ∂x j j Ω j=1


253

be the two-dimensional version of (2.28) with b ∈ Lloc 1 (Ω), b(x) > 0 a.e. in Ω and b−1 ∈ L1 (Ω). By Proposition 2.1 the quadratic form Eb (f, g) is closable and positive definite. One has 1/2 −1 1/2

f |L2 (Ω) ≤ c b |L1 (Ω) b(x) |∇f (x)|2 dx Ω

on its closure Eb (Ω). In particular, id :

Eb (Ω) → L2 (Ω)

(2.49)

is continuous. Again we strengthen b−1 ∈ L1 (Ω) by b−1 ∈ L1 (log L)d (Ω) with d > 0, (2.22), and ask for compactness. Let ek (id) be the corresponding entropy numbers. Then one has the following counterpart of Theorem 2.3. Theorem 2.4. Let Ω be a bounded domain in the plane R2 . Let b be a real Lebesgue-measurable function on Ω with b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

b−1 ∈ L1 (log L)d (Ω)

where d > 2. Then the embedding (2.49) is compact. If d > 4, then ek (id) ≤ c b−1 |L1 (log L)d (Ω) 1/2 k −1/2 ,

k ∈ N,

for some c > 0 which is independent of b and k. If 2 < d ≤ 4, then for any ε > 0 there is a constant cε > 0 such that for all b and k, 1

ek (id) ≤ cε b−1 |L1 (log L)d (Ω) 1/2 k − 4 + 2 +ε , d

k ∈ N.

Proof. We follow the proof of Theorem 2.3 and indicate the necessary modifications. Now one has t = 1 and the counterparts a > 0,

1 1 = 1 − j+1 , t[j] 2

j ∈ N,

and

f |G1,a (Ω) ≤ b−1/2 |L2 (log L)a (Ω) · b1/2 f |L2 (Ω) 0 of (2.32), (2.33). Let again H1,a (Ω) be the Littlewood–Paley version of G1,a (Ω) according to (2.34)–(2.36). In contrast to n ≥ 3 one must now apply in addition (2.27) with p = t[j]. This produces an extra factor 2j and the counterpart of (2.37) is now given by 2 1/2

∂f 0 −1/2 |H1,a (Ω) ≤ c b |L2 (log L)a+1 (Ω) b(x) |∇f (x)|2 dx . ∂xl Ω l=1

Otherwise one can follow the arguments from the proof of Theorem 2.3 with a > 1 in Step 3 and 0 < a ≤ 1 in Step 4. The counterpart of (2.45) is now given by b−1/2 |L2 (log L)a+1 (Ω) ∼ b−1 |L1 (log L)d (Ω) ,

d = 2 + 2a.

Hence d > 4 if a > 1 and 2 < d ≤ 4 if 0 < a ≤ 1. The rest is the same as in the proof of Theorem 2.3 based on (2.44), (2.48).

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H. Triebel

3. Spectral theory 3.1. Preliminaries We complement the abstract preliminaries from Section 2.1. Let again H be a separable complex Hilbert space and let E be a positive definite closed quadratic form with E(u, u) ≥ c u |H 2

for some c > 0 and all u ∈ dom E.

Let A be the related positive definite self-adjoint operator with dom A according to (2.3). One has the isomorphic maps (2.4). Recall that a positive definite self-adjoint operator A is said to be an operator with pure point spectrum if its spectrum consists solely of eigenvalues {λk }∞ k=1 of finite (geometric) multiplicity, repeated according to their geometric multiplicity, and ordered by if k → ∞. (3.1) 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · → ∞ The necessary information may be found in, for example, in [32, Section 4.5]. In particular, A is an operator with pure point spectrum if, and only if, id :

dom E → H

is compact. Furthermore, −1/2

λk

≤ c ek (id),

k ∈ N,

(3.2)

which is essentially an easy consequence of Carl’s inequality. This observation underlies the related spectral assertions in [14, 15]. We refer to [24, Chapter 6]. There one finds also the following assertions about approximation numbers. Recall that the k-th approximation number ak (T ) of a compact linear operator T ∈ L(B1 , B2 ) between the complex quasi-Banach spaces B1 and B2 is given by (3.3) ak (T ) = inf T − S : S ∈ L(B1 , B2 ), rank S < k , k ∈ N, where rank S = dim(image S). With B1 = dom E, B2 = H and the above self-adjoint positive definite operator A with pure point spectrum and the eigenvalues λk ordered by (3.1) one has −1/2 k ∈ N. (3.4) = ak A−1/2 : H → H ∼ ak id : dom E → H , λk This equality between eigenvalues and approximation numbers is well known since a long time. Proofs may be found in [10, p. 91] and [24, Theorem 6.21, p. 195]. 3.2. Distributions of eigenvalues Let Ω be a bounded domain in Rn with n ≥ 2 and let Eb (f, g) be the closed quadratic form on dom Eb = Eb (Ω) according to Proposition 2.1. Let Ab be the generated positive definite self-adjoint operator in L2 (Ω). If ϕ ∈ D(Ω) ∩ dom Ab , then Ab ϕ ∈ D (Ω) can be written as n

∂ϕ ∂ b(x) ∈ L2 (Ω). Ab ϕ = − ∂xj ∂xj j=1


255

This follows from the definition of Ab according to (2.3). If id :

Eb (Ω) → L2 (Ω)

is compact, then Ab is an operator with pure point spectrum and its eigenvalues λk (Ab ) can be ordered as in (3.1) by 0 < λ1 (Ab ) ≤ λ2 (Ab ) ≤ · · · ≤ λk (Ab ) ≤ · · · → ∞

if k → ∞.

Theorem 3.1. (i) Let Ω be a bounded domain in Rn with n ≥ 3. Let b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

b−1 ∈ Ln/2 (log L)d (Ω)

(3.5)

with d > 0. Then Ab is a positive definite self-adjoint operator in L2 (Ω) with pure point spectrum {λk (Ab )}∞ k=1 . If, in addition, d > 4/n, then λk (Ab ) ≥ c b−1 |Ln/2 (log L)d (Ω) −1 k 2/n ,

k ∈ N,

(3.6)

where c > 0 is independent of b and k. If, in addition, 0 < d ≤ 4/n, then for any ε > 0 there is a constant cε > 0 such that for all b and k, λk (Ab ) ≥ cε b−1 |Ln/2 (log L)d (Ω) −1 k 2 −ε , d

k ∈ N.

(3.7)

b−1 ∈ L1 (log L)d (Ω)

(3.8)

(ii) Let Ω be a bounded domain in the plane R2 . Let b ∈ Lloc 1 (Ω),

b(x) > 0 a.e. in Ω,

with d > 2. Then Ab is a positive definite self-adjoint operator in L2 (Ω) with pure point spectrum {λk (Ab )}∞ k=1 . If, in addition, d > 4, then λk (Ab ) ≥ c b−1 |L1 (log L)d (Ω) −1 k,

k ∈ N,

(3.9)

where c > 0 is independent of b and k. If, in addition, 2 < d ≤ 4, then for any ε > 0 there is a constant cε > 0 such that for all b and k, λk (Ab ) ≥ cε b−1 |L1 (log L)d (Ω) −1 k 2 −1−ε , d

k ∈ N.

(3.10)

(iii) Let Ω be a bounded domain in Rn with n ≥ 2. Let b be as in part (i) if n ≥ 3 and as in part (ii) if n = 2. Then there is a constant c > 0 such that 2 λk (Ab ) ≤ c inf |B|−1− n b(x) dx · k 2/n , k∈N (3.11) B

for all admitted b and k, where the infimum is taken over all balls B with B ⊂ Ω. Proof. Step 1. Part (i) follows from Theorem 2.3 and (3.2). Similarly one obtains part (ii) from Theorem 2.4. Step 2. We prove part (iii) and begin with a preparation. Let r > 0, Br = x ∈ Rn : |x| < r , ◦

and let C 1 (Br ) be the completion of D(Br ) in the norm ◦

f |C 1 (Br ) =

n

∂f (x) . sup ∂x x∈B j r j=1

256

H. Triebel

Let ak (idr ), k ∈ N, be the approximation numbers of the compact embedding idr :

◦

C 1 (Br ) → L2 (Br ),

r > 0.

(3.12)

We wish to show that ak (idr ) = r1+ 2 ak (id1 ) ∼ r1+ 2 k −1/n , n

n

k ∈ N.

(3.13)

The equivalence in (3.13) is covered by [15, Theorem, Section 3.3.4, p. 119]. The first equality in (3.13) follows from the isometries ◦

◦

f (r·) |C 1 (B1 ) = r f |C 1 (Br ) ,

g(r−1 ·) |L2 (Br ) = rn/2 g |L2 (B1 ) , and the well-known properties of approximation numbers. If B is a ball in Rn with B ⊂ Ω, then n 1/2

◦ ∂f (x) 2 1/2 b(x) dx ≤c b(x) dx

f |C 1 (B) , f ∈ D(B). ∂x j Ω B j=1 One has for the corresponding approximation numbers 1/2 ◦ ak id : C 1 (B) → L2 (Ω) ≤ c b(x) dx ak id : Eb (Ω) → L2 (Ω) , Ω

and by (3.13) that −1/2 1 1 b(x) dx k −1/n . ak id : Eb (Ω) → L2 (Ω) ≥ c |B| 2 + n Ω

Using (3.4) one obtains (3.11). Remark 3.2. If d > 4/n in case of n ≥ 3 and d > 4 in case of n = 2, then λk (Ab ) ∼ k 2/n ,

k ∈ N,

as in the classical case. This applies to any admitted b how rough it might be. One may think about ∞

cj b(x) = , x ∈ Ω, |x − xj | j=1 where {xj } ⊂ Ω collects all points with rational components and cj > 0 appropriately chosen. The breaking points d = 4/n if n ≥ 3 in part (i) and d = 4 if n = 2 in part (ii) can surely be improved. There are good reasons to believe that the alternative decomposition techniques used in the remarkable paper [13] by D.E. Edmunds and Yu. Netrusov can also be employed in the above context supporting the following expectation. Conjecture 3.3. (i) Let n ≥ 3. Then (3.6) remains valid for all d > 2/n. If 0 < d < 2/n, then (3.7) can be improved by λk (Ab ) ≥ c b−1 |Ln/2 (log L)d (Ω)

−1

for some c > 0 which is independent of b and k.

kd ,

k ∈ N,


257

(ii) Let n = 2. Then (3.9) remains valid for all d > 3. If 2 < d < 3, then (3.10) can be improved by λk (Ab ) ≥ c b−1 |L1 (log L)d (Ω)

−1

k d−2 ,

k ∈ N,

for some c > 0 which is independent of b and k.

4. Complements 4.1. An example One may ask to which extent the conditions for b in Theorems 2.3, 2.4, 3.1 and in Conjecture 3.3 are natural. But it seems to be a sophisticated interplay between local and global smoothness and singularity properties of b and b−1 . We illuminate the situation by a simple example. Let Ω = B = {x ∈ R2 : |x| < 1} be the unit circle in the plane and let λ b(x) = |x|2 1 + log |x| , x ∈ B, λ ≥ 0. (4.1) Let Eb (f, g) =

b(x) B

2

∂f ∂¯ g (x) (x) dx, ∂xj ∂xj j=1

f, g ∈ D(B),

(4.2)

be as in (2.1). Proposition 4.1. Eb (f, g) according to (4.1), (4.2) is a closable positive definite quadratic form in L2 (B) and 1/2 λ

f |L2 (B) ≤ |x|2 1 + log |x| |∇f (x)|2 dx (4.3) B

on its closure Eb (B). The embedding id :

Eb (B) → L2 (B)

(4.4)

is compact if, and only if, λ > 0. Proof. Step 1. Let f ∈ D(B) be real. Then (4.3) with λ = 0 (and hence also with λ ≥ 0) follows from 0≤

2 xj B j=1

= B

=

B

|x|

f (x) + |x|

f 2 (x) +

2

j=1

xj

∂f 2 dx ∂xj

∂f 2 (x) + |x|2 |∇f (x)|2 dx ∂xj

2 |x| |∇f (x)|2 − f 2 (x) dx.

(4.5)

258

H. Triebel

We used integration by parts. We prove that Eb (f, g) is closable. Let ∞ {fk }∞ k=1 ⊂ D(B), with fk → 0 in L2 (B) and let {fk }k=1 be a Cauchy sequence in the Eb -norm. Let ϕ ∈ D(B). Then ∂f √ √ ∂f k k b ,ϕ = b ϕ¯ dx ∂xj ∂x 1 B ∂ √b √ ∂ ϕ¯ dx → 0 if k → ∞. =− fk ϕ¯ + b ∂xj ∂xj B √ k Then b ∂f ∂xj → 0 in L2 (B). This proves that Eb (f, g) is closable and by (4.3) positive definite on its closure Eb (B). Step 2. Let ϕj (x) = 2j/2 ϕ(2j x) with ϕ ∈ D(B). One can choose ϕ such that the functions ϕj have disjoint supports, 2 and |x|2 ∇ϕj (x) dx ∼ 1.

ϕj |L2 (R) = 1 B

This shows that the embedding id in (4.4) with λ = 0 is not compact. Let λ > 0 and let ψ ∈ D(B)

with

ψ(x) = 1 if

|x| ≤ 1/2.

Let with idj = ψ(2j ·) id, j ∈ N. id = idj + idj Then idj : f → ( 1 − ψ(2j ·) f is compact and idj ≤ c j −λ/2 → 0 if j → ∞ as will be justified below. Hence id is compact. Remark 4.2. First we prove that idj ≤ c j −λ/2 . Let g ∈ C ∞ (B) with supp g ⊂ {y : |y| ≤ δ}

where δ < 1.

Then it follows from (4.3) that 2

2 ∂ log |x|λ |g(x)|2 dx ≤ | log |x| |λ/2 g(x) dx |x|2 ∂xj B B j=1 2 λ log |x| λ−2 |g(x)|2 dx. |x|2 log |x| ∇g(x) dx + c ≤c B

B

If δ > 0 is small, then one obtains λ 2 log |x|λ |g(x)|2 dx ≤ c |x|2 log |x| ∇g(x) dx. B

B

Inserting g = id f one obtains id ≤ c j −λ/2 . We add a few further discussions. With b as in (4.1) one has b−1 ∈ L1 (B) if, and only if, λ > 1. Then one can rely on Proposition 2.1 with n = 2. One can calculate for which λ > 1 Theorems 2.4 and 3.1(ii) can be applied. But it seems to be more effective to deal directly with problems of this type for all λ > 0. This will be done in [38] in a more general context. We add a comment about the inequality (4.5) and (4.3) with λ = 0. The proof is very easy but the outcome is nevertheless remarkable and (of course) well known. see, e.g., [29, p. 97]. Let κ(t) with j

j


259

0 < t ≤ 1 be a positive monotonically decreasing function. Then there is a constant c > 0 with κ(|x|) |f (x)|2 dx ≤ c |∇f (x)|2 dx for all f ∈ D(B) 2 |x|2 B 1 + | log |x| | B if, and only if, κ is bounded. This is a special case of [35, Theorem 16.2(i), p. 237]. It shows that the weight |x|2 in (4.3) with λ = 0 can not be shifted from the right-hand side to the left-hand side as it is quite often the case in non-limiting situations (as a rule of thumb). 4.2. Extrapolations, decompositions, weights We add a few (partly historical) comments. Remark 4.3. Let again Ω be a bounded domain Rn , n ≥ 2. If a < 0, then the definition of Lp (log L)a (Ω) can be extended to p = ∞, where (2.16) must be modified by

f |L∞ (log L)a (Ω) =

sup (1 + | log t|)a f ∗ (t) < ∞.

0 2/n d d and one can improve k − 4 +ε in (2.31) by k − 2 +ε if 0 < d ≤ 2/n. If n = 2, then the situation might be a little bit different. Let again n ≥ 3. Then one obtains corresponding improvements in Theorem 3.1(i). The breaking point is shifted from 4/n to 2/n and one has (3.6) for all d with d > 2/n. If 0 < d ≤ 2/n, d then one can replace k 2 −ε in (3.7) by k d−ε . This supports Conjecture 3.3. But as said details remain to be checked.

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Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends

Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends

Spectral Theory, Function Spaces and Inequalities: New Techniques and Recent Trends