THEORY OF HP SPACES
This is Volume 3 8 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAUL A. SMITH AND SAMUEL ElLENBERG A complete list of titles in this series appears at the end of this volume
THEORY OF HP SPACES
Peter L. Duren Department of Mathematics University of Michigan Ann Arbor, Michigan
Academic Press New York and London 1970
COPYRIGHT© 1970, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
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LmRARY OF CoNGRESs CATALOG CARD NUMBER:
PRINTED IN THE UNITED STATES OF AMERICA
74-117092
TO MY FATHER William L. Duren
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CONTENTS
Preface, xi 1. HARMONIC AND SUBHARMONIC FUNCTIONS
1.1. 1.2. 1.3. 1.4. 1.5. 1.6.
Harmonic Functions, 1 Boundary Behavior of Poisson-Stieltjes Integrals, 4 Subharmonic Functions, 7 Hardy's Convexity Theorem, 8 Subordination, 1 0 Maximal Theorems, 11 Exercises, 13
2. BASIC STRUCTURE OF HP FUNCTIONS
2.1. 2.2. 2.3. 2.4. 2.5. 2.6.
Boundary Values, 15 Zeros, 18 Mean Convergence to Boundary Values, 20 Canonical Factorization, 23 The Class N+. 25 Harmonic Majorants, 28 Exercises, 29
3. APPLICATIONS
3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.
Poisson Integrals and H'. 33 Description of the Boundary Functions, 35 Cauchy and Cauchy-Stieltjes Integrals, 39 Analytic Functions Continuous in [z[ ~ 1, 42 Applications to Conformal Mapping, 43 Inequalities of Fejer-Riesz, Hilbert, and Hardy, 46 Schlicht Functions, 49 Exercises, 51
4. CONJUGATE FUNCTIONS
4.1. Theorem of M. Riesz, 53 4.2. Kolmogorov's Theorem, 56 vii
viii
CONTENTS
4.3. 4.4. 4.5. 4.6.
Zygmund's Theorem, 58 Trigonometric Series, 61 The Conjugate of an h' Function, 63 The Case p < 1 : A Counterexample. 65
Exercises, 67 5. MEAN GROWTH AND SMOOTHNESS
5.1. 5.2. 5.3. 5.4. 5.5. 5.6.
Smoothness Classes, 71 Smoothness of the Boundary Function, 74 Growth of a Function and its Derivative, 79 More on Conjugate Functions, 82 Comparative Growth of Means, 84 Functions with HP Derivative, 88
Exercises. 90 6. TAYLOR COEFFICIENTS
6.1. 6.2. 6.3. 6.4.
Hausdorff-Young Inequalities, 93 Theorem of Hardy and Littlewood, 95 The Case p:::::: 1. 98 Multipliers, 99
Exercises. 106 7. HP AS A LINEAR SPACE
7.1. 7.2. 7.3. 7 .4. 7.5. 7.6.
Quotient Spaces and Annihilators, 110 Representation of Linear Functionals, 112 Beurling's Approximation Theorem, 113 Linear Functionals on HP, 0 < p < 1. 11 5 Failure of the Hahn-Banach Theorem, 118 Extreme Points, 123
Exercises. 126 8. EXTREMAL PROBLEMS
8.1. 8.2. 8.3. 8.4. 8.5.
The Extremal Probl.em and its Dual, 129 Uniqueness of Solutions, 132 Counterexamples in the Case p = 1, 134 Rational Kernels, 136 Examples, 139
Exercises, 143 9. INTERPOLATION THEORY
9.1. Universal Interpolation Sequences, 147 9.2. Proof of the Main Theorem, 149
CONTENTS
9.3. The Proof for p < 1, 153 9.4. Uniformly Separated Sequences, 154 9.5. A Theorem of Carleson, 156
Exercises, 164 10, HP SPACES OVER GENERAL DOMAINS
1 0.1. 1 0.2. 10.3. 10.4. 10.5.
Simply Connected Domains, 167 Jordan Domains with Rectifiable Boundary, 169 Smirnov Domains, 173 Domains not of Smirnov Type, 176 Multiply Connected Domains, 179
Exercises. 183 11. HP SPACES OVER A HALF-PLANE
11.1. 11.2. 11.3. 11.4. 11.5.
Subharmonic Functions, 188 Boundary Behavior, 189 Canonical Factorization, 192 Cauchy Integrals, 194 Fourier Transforms, 195
Exercises. 197 12. THE CORONA THEOREM
12.1. 12.2. 12.3. 12.4. 12.5.
Maximal Ideals, 201 Interpolation and the Corona Theorem, 203 Harmonic Measures, 207 Construction of the Contour r, 211 Arclength of 215
r.
Exercises. 218 Appendix A. Rademacher Functions, 221 Appendix B. Maximal Theorems, 231 References, 237 Author Index, 253 Subject Index. 256
lx
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PREFACE
The theory of HP spaces has its origins in discoveries made forty or fifty years ago by such mathematicians as G. H. Hardy, J. E. Littlewood, I. I. Privalov, F. and M. Riesz, V. Smirnov, and G. Szego. Most of this early work is concerned with the properties of individual functions of class HP, and is classical in spirit. In recent years, the development of functional analysis has stimulated new interest in the HP classes as linear spaces. This point of view has suggested a variety of natural problems and has provided new methods of attack, leading to important advances in the theory. This book is an account of both aspects of the subject, the classical and the modern. It is intended to provide a convenient source for the older parts of the theory (the work of Hardy and Littlewood, for example), as well as to give a self-contained exposition of more recent developments such as Beurling's theorem on invariant subspaces, the Macintyre-RogosinskiShapiro-Havinson theory of extremal problems, interpolation theory, the dual space structure of HP with p < 1, HP spaces over general domains, and Carleson's proof of the corona theorem. Some of the older results are proved by modern methods. In fact, the dominant theme of the book is the interplay of" hard" and "soft" analysis, the blending of classical and modern techniques and viewpoints. The book should prove useful both to the research worker and to the graduate student or mathematician who is approaching the subject for the first time. The only prerequisites are an elementary working knowledge of real and complex analysis, including Lebesgue integration and the elements offunctional analysis. For example, the books (cited in the bibliography) of Ahlfors or Titchmarsh, Natanson or Royden, and Goffman and Pedrick are more than adequate background. Occasionally, particularly in the last few chapters, some more advanced results enter into the discussion, and appropriate references are given. But the book is essentially self-contained, and it can serve as a textbook for a course at the second- or third-year graduate level. In fact, the book has evolved from lectures which I gave in such a course at the University of Michigan in 1964 and again in 1966. With the student in mind, I have tried to keep things at an elementary level wherever possible. xi
PREFACE
xii
On the other hand, some sections of the book (for example, parts of Chapters 4, 6, 7, 9, 10, and 12) are rather specialized and are directed primarily to research workers. Many of these topics appear for the first time in book form. In particular, the last chapter, which gives a complete proof of the corona theorem, is "for adults only." Each chapter contains a list of exercises. Some of them are straightforward, others are more challenging, and a few are quite difficult. Those in the last category are usually accompanied by references to the literature. Many of the exercises point out directions in which the theory can be extended and applied. Further indications of this type, as well as historical remarks and references, appear in the Notes at the end of each chapter. Two appendices are included to develop background material which the average mathematician cannot be expected to know. The chapters need not be read in sequence. For example, Chapters 8 and 9 depend only upon the first three chapters (with some deletions possible) and upon the first two sections of Chapter 7. Chapter 12 can be read immediately after Chapters 8 and 9. The coverage is reasonably complete, but some topics which might have been included are mentioned only in the Notes, or not at all. Inevitably, my own interests have influenced the selection of material. I wish to express my sincere appreciation to the many friends, students, and colleagues who offered valuable advice or criticized earlier versions of the manuscript. I am especially indebted to J. Caughran, W. L. Duren, F. W. Gehring, W. K. Hayman, J. Hesse, H. J. Landau, A. Macdonald; B. Muckenhoupt, P. Rosenthal, W. Rudin, J. V. Ryff, D. Sarason, H. S. Shapiro, A. L. Shields, B. A. Taylor, G. D. Taylor, G. Weiss and A. Zygmund. I am very thankful to my wife Gay, who accurately prepared the bibliography and proofread the entire book. Renate McLaughlin's help with the proofreading was also ihost valuable. In addition, I am grateful to the Alfred P. Sloan Foundation for support during the academic year 1964-1965, when I wrote the first coherent draft of the book. I had the good fortune to spend this year at Imperial College, University of London and at the Centre d'Orsay, Universite de Paris. The scope of the book was broadened as a result of my mathematical experiences at both of these institutions. In 1968-1969, while at the Institute for Advanced Study on sabbatical leave from the University of Michigan, I added major sections and made final revisions. I am grateful to the National Science Foundation for partial support during this period.
Peter L. Duren
THEORY OF HP SPACES
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HARMONIC AND SUBHARMONIC FUNCTIONS
CHAPTER 1
This chapter begins with the classical representation theorems for certain classes of harmonic functions in the unit disk, together with some basic results on boundary behavior. After this comes a brief discussion of subharmonic functions. Both topics are fundamental to the theory of HP spaces. In particular, subharmonic functions provide a strikingly simple approach to Hardy's convexity theorem and to Littlewood's subordination theorem, as shown in Sections 1.4 and 1.5. Finally, the Hardy-Littlewood maximal theorem (proved in Appendix B) is applied to establish an important maximal theorem for analytic functions. 1.1. HARMONIC FUNCTIONS
Many problems of analysis center upon analytic functions with restricted growth near the boundary. For functions analytic in a disk, the integral means 1 Mp(r,J) = {21t lf(rei 8 )1P d() , O 2n 2n
2n
0
00
1
=limn-> co 211:
0
J P(r, (}- t)u(rne' 2n
0
. 1)
dt =lim u(rnz) = u(z). n-> oo
(Here, as always, z = rei 8 .) As a corollary to the proof, we see that every positive harmonic function in the unit disk can be represented as a Poisson-Stieltjes integral with respect to a nondecreasing function JL(t). This is usually called the H erglotz representation. The function JL(t) of bounded variation corresponding to a given u E h 1 is
4
1 HARMONIC AND SUBHARMDNIC FUNCTIONS
essentially unique. Indeed, if JP(r, () - t) dJ1(t)
=0, analytic completion gives
+z d}l(t) =iy, fo2"ei'-;,e - z where y is a real constant. Since
we conclude that 2lt
Jo
eint
d}l(t) = 0,
n = 0,
± 1, ±2, ...
Since the characteristic function of any interval can be approximated in L 1 by a continuous periodic function, hence by a trigonometric polynomial, this shows that the measure of each interval is zero. Thus dJl is the zero measure. 1.2. BOUNDARY BEHAVIOR OF POISSON-STIELTJES INTEGRALS
If u(z) is the Poisson integral of an integrable function cp(t), then for any point t = () 0 where cp is continuous, u(z)-+ cp(0 0 ) as z -+e;80 • This can be generalized to Poisson-Stieltjes integrals: u(z)-+ J1'(0 0 ) wherever Jl is continuously differentiable. Actually, it is enough that Jl be differentiable; or, slightly more generally, that the symmetric derivative
DJ1(0o) =lim Jl(Oo + t)- Jl(()o- t) t--+0
2t
exist, as the following theorem shows. THEOREM 1.2. Let u(z) be a Poisson-Stieltjes integral of the form (2), where Jl is of bounded variation. If the symmetric derivative DJ1(0 0 ) exists at a point 00 , then the radial limit lim,__. 1 u(rei8 o) exists and has the value DJ1(0 0 ). PROOF. We may assume
00 1
=
0. Set A = DJ1(0), and write
"
u(r)- A=- J P(r, t)[dJl(t)- A dt] 2n _, 1 2n
= - [P(r, t)lfl(t)- At]]':,
[a
]
- - 1 J " [Jl(t)- At] - P(r, t) dt. 2n _, ot
1.2 BOUNDARY BEHAVIOR OF POISSON-STIELTJES INTEGRALS
The integrated term tends to zero as r-+ I. For 0 < c5 ~ It I~ - r IiJPiJt I-< [1 - 2r(1 2r cos c5 + r 2
-+
)
Hence for each fixed c5 > 0, u(r) - A - I,
[a
J
0
as
2] 2
-+
5
TC,
r-+ 1.
0, where
]
1 " [.u(t)- At] - P(r, t) dt I,= - -
2TC
=~ 7'C
iJt
-IJ
J" [p.(t)-2tp.( -t)- A] t[- ~iJt P(r, t)] dt. 0
Given e >0, choose c5 >0 so small that
Ip.(t) -2~( -t)- A I~
1:
for 0 < t
~
c5.
Then
II"I
e ~ -2 7'C
J" t(-iJP) - dt < 2£ iJt _,
for r sufficiently near I, as an integration by parts shows. Thus u(r)-+ A as r -+I, and the proof is complete. Since a function of bounded variation is differentiable almost everywhere, we obtain two important corollaries. COROLLARY 1.
Each function u E h 1 has a radial limit almost everywhere.
COROLLARY 2. If u is the Poisson integral of a function cp u(re; 8)-+ cp(O) almost everywhere.
E
I!, then
By a refinement of the proof it is even possible to show that u(z) tends to D/1(00 ) along any path not tangent to the unit circle. However, we shall arrive at this result (almost everywhere) by an indirect route. For the present, we content ourselves with showing that a bounded analytic function has such a nontangential limit almost everywhere. For 0 < oc < TC/2, construct the sector with vertex ei 8 , of angle 2oc, symmetric with respect to the ray from the origin through e; 8 • Draw the two segments from the origin perpendicular to the boundaries of this sector, and let S,.((}) denote the "kite-shaped" region so constructed (see Fig. I).
6 1 HARMONIC AND SUBHARMONIC FUNCTIONS
Figure 1
THEOREM 1 .3. If f e Hrr;,, the radial limit lim, .... J(rei 8 ) exists almost everywhere. Furthermore, if (} 0 is a value for which the radial limit exists, then f(z) tends to the same limit as z -+e;8 " inside any region S~~.(0 0 ), ex < 1r.j2. PROOF. The existence almost everywhere of a radial limit follows from Corollary 1 to Theorem 1.2, since hrr:J c: h1 • To discuss the angular limit, it is convenient to deal instead with a bounded analytic function f(z) in the disk lz- 11 < 1, having a limit Las z-+ 0 along the positive real axis. Letf..(z) = f(zjn), n = l, 2, .... The functions f..(z) are uniformly bounded, so they form a normal family (see Ahlfors [2], Chap. 5). This implies that a subsequence tends to an analytic function F(z) uniformly in each closed subdomain of the disk, hence in the region
larg zl :::;; ex < 7r./2;
(cos ex)/2 :::;; lzl :::;; cos ex.
(3)
(The ray arg z = ex has a segment of length 2 cos ex in common with the disk lz- II < 1.) But for all real z in the interval 0 < z < 2, f..(z)-+ L. It follows that F(z) = L, and that f..(z)-+ L uniformly in the region (3). This implies that f(z)-+ Las z-+ 0 inside the sector jarg zl :::;; ex, which proves the theorem. The function f is said to have a nontangential limit L at e; 8 " if f(z) -+ Las z-+ eiBo inside each region Si0 0 ), ex < 1r.j2. Thus each f E Hrr:J has a nontangential limit almost everywhere.
1.3 SUBHARMONIC FUNCTIONS
7
1.3. SUBHARMONIC FUNCTIONS
A domain is an open connected set in the complex plane. Let D be a bounded domain. The boundary an of D is defined to be the closure l5 minus D. A real-valued function g(z) continuous in D is said to be subharmonic if it has the following property. For each domain B with B c D, and for each function U(z) harmonic in B, continuous in B, such that g(z) ::;; U(z) on aB, the inequality g(z) ::;; U(z) holds throughout B. In particular, if there is a function U(z) harmonic in B with boundary values g(z), then g(z) ::;; U(z) in B. Subharmonic functions are also characterized by the" local sub-mean-value property," which is often easier to work with. THEOREM 1.4. Necessary and sufficient that a continuous function g(z) be subharmonic in D is that for each z0 E D there exist Po > 0 such that the disk lz - z 0 1< Po is in D and
g(zo) ::;; 2171:
J2"g(z 0
0
+ pe'8 ) d(}
l4)
for every p < Po . PROOF. The necessity is easy. Let lz- z0 1< p be in D, and let U(z) be the function harmonic in this disk and equal to g(z) on lz- z0 1= p. Then
J2"
g(z 0 )::;; U(z 0 ) = 1 U(z 0 2n o
1 + pe'8 ) d(} = -
J2"g(z
2n o
0
+ pei8 ) d(}.
To prove the sufficiency, suppose there exists a domain B with B c D and a harmonic function U(z) such that g(z)::;; U(z) on aB, yet g(z) > U(z) somewhere in B. Let E be the set of points in B at which h(z) = g(z) - U(z) attains its maximum m > 0. Then E c B, because h(z) ::;; 0 on aB. Since E is a closed set, some point z 0 E E has no circular neighborhood entirely contained in E. Hence there exists a sequence {Pn} tending to zero such that the disk lz- z0 1 1, then this inequality persists in some neighborhood lz- z 0 1::;p. Thus log+ 1/(z)l coincides with the harmonic function loglf(z)l in lz- z 0 1::; p. 1.4. HARDY'S CONVEXITY THEOREM
The class HP was introduced in Section 1.1 as the set of all functions f(z) analytic in lzl < 1 for which the means MP(r, f) are bounded. It is natural to ask how these means may behave as functions of r, for an arbitrary analytic function/ The case p = 2 is especially simple. If f(z) = L an zn is analytic in lzl < 1, then by Parseval's relation co
M 2 2 (r, f)
=
L lanl 2 r 2 n. n=O
This shows that Mir,f) increases with r, and that /E H 2 if and only if L lanl 2 < oo. Likewise, it follows from the maximum modulus principle that M 00 (r,f) increases with r. The situation is more complicated for other values of p, but Mp(r,f) is always a nondecreasing function. In fact, much more is true.
1.4 HARDY'S CONVEXITY THEOREM
THEOREM 1.5 (Hardy's convexity theorem).
9
Let f(z) be analytic in
Jzl < 1, and let 0 < p::::;; oo. Then (i) MP(r,f) is a non decreasing function of r; (ii) log Mp(r,f) is a convex function of log r. To say that log Mp(r,f) is a convex function of log r means that if log r = a log r 1
+ (1
- a) log r 2
(0 < r 1 < r 2 < 1; 0 0, A is any real number, and f(z) is analytic. Thus, by the remark following Theorem 1.6, r;.M/(r,f) is a convex function of log r. Given 0 < r 1 < r 2 < 1, let A< 0 be chosen so that
r/M/(r 1 ,f) = r/M/(r 2 ,f)
= K,
say. Let r = r 1'" ri-'" (0 <ex< 1). Then r;.M/(r,f) :-:::; K
= K'"K 1 -'"
=
{r/M/(r 1 ,f)}'"{r/M/(r2 ,/)} 1 -
=
r;.{M/(r 1 ,f)}'"{M/(r2 ,/)} 1 -'",
..
which completes the proof. 1.5. SUBORDINATION
Let F(z) be analytic and univalent in lzl < 1, with F(O) = 0. Letf(z) be analytic in lzl < 1, withf(O) = 0, and suppose the range off is contained in that of F. Then w(z) = F- 1 (/(z)) is well-defined and analytic in lzl < 1, w(O) = 0, and lw(z)l :-:::; 1. By Schwarz's lemma, then, lw(z)l :-:::; lzl. This implies, in particular, that the image under f(z) = F(w(z)) of each disk lzl :-:::; r < 1 is contained in the image of the same disk under F(z). More generally, a functionf(z) analytic in lzl < 1 is said to be subordinate to an analytic function F(z) (written/-< F) ifJ(z) =F(w(z)) for some function w(z) analytic in lzl < 1, satisfying lw(z)l :-:::; lzl. F(z) need not be univalent. The following result has many applications. THEOREM 1.7 (Littlewood's subordination theorem). Let f(z) and F(z) be analytic in lzl < 1, and suppose/-< F. Then MP(r,J) :-:::; Mp(r, F), 0 < p :-:::; oo.
1.6 MAXIMAL THEOREMS
11
PROOF. As in the proof of Hardy's theorem, we shall deduce this from a more general result concerning subharmonic functions. Let G(z) be subharmonic in lzl < 1, and let g(z) = G(w(z)), where w(z) is analytic in lzl < 1 and lw(z)l : 1,
to show that Theorem 1.8 is false for p = 1. 7. Show that if f(z) and g(z) are subharmonic, then so is max{f(z), g(z)}. NOTES
In the language of functional analysis, the Belly selection theorem simply asserts the weak-star sequential compactness of the unit sphere of the dual space of the space C of continuous functions over the interval [a, b]. That is, ifcpnEC* and supllcfJnll 1
while 11/llco = sup IJ(z)l = ess sup lf(e19)1. 1~1 0, analytic completion gives
{ J -ee'',-+, -zz dJl(t)}.
S(z) = exp -
2x
o
Putting everything together, we have the factorizationf(z)
(5) = e'YB(z)S(z)F(z).
We now introduce some terminology. An outer function for the class HP is a function of the form } 1 2" ei' + z F(z) = e1Y exp {-ilog t{!(t) dt , (6) 1 2n o e - z
J
where ')'is a real number, t{!(t) ;;:.: 0, log t{!(t) E I!, and t{!(t) E IJ'. Thus (4) is an outer function. An inner function is anyfunctionf(z) analytic in lzl < 1, having the properties lf(z)l ::;; l and lf(e 16)1 = l a.e. We have shown that every inner function has a factorization eiy B(z)S(z), where B(z) is a Blaschke product and S(z) is a function of the the form (5), Jl(t) being a boundea nondecreasing singular function (!l'(t) = 0 a.e.). Such a function S(z) is called a singular inner function. THEOREM 2.8 (Canonical factorization theorem). Every functionf(z) ¢. 0 of class HP (p > 0) has a unique factorization of the form.f(z) = B(z)S(z)F(z), where B(z) is a Blaschke product, S(z) is a singular inner function, and F(z)
2.5 THE ClASS N+
25
is an outer function for the class HP (with 1/J(t) = !.f(ei 1)!). Conversely, every such product B(z)S(z)F(z) belongs to HP. PROOF. We have already shown that everyf E HP can be factored as claimed, and the uniqueness is obvious. To prove the converse, it suffices to show that an outer function (6) must belong to HP. Applying the arithmetic-geometric mean inequality (see Exercise 2), we find
jF(z)jP
1 :$;-
J2"P(r, 8 -
2n o
t)[l/J(t)JP dt.
Thus 27t
J
27t
!F(rei6 )!P de ::-::;
0
J
[1/J(t)]P dt.
0
There is a similar factorization for/EN. A function F(z) of the form (6), where 1/J(t) ;;::: 0 and log 1/J(t) E L1 , will be called an outer function for the class N. (Note that the condition 1/1 E I! has been dropped.) THEOREM 2.9.
Every functionf(z) oj. 0 of class N can be expressed in the
form
f(z) = B(z)[S1 (z)/S2 (z)]F(z),
(7)
where B(z) is a Blaschke product, S 1 (z) and S 2 (z) are singular inner functions, and F(z) is an outer function for the class N (with 1/J(t) = !f(ei1)1). Conversely, every function of the form (7) belongs to N. PROOF. Letf(z) =eiYB(z)g(z), whereg E N,g(z) of. Oin lzl < 1, andg(O) > 0. Since loglg(z)l E h 1 , it has a representation as a Poisson-Stieltjes integral
J2"
1 loglg(z)l = P(r, 8- t) dv(t) 2n v
with respect to a function v(t) of bounded variation. Analytic completion and separation of v(t) into its absolutely continuous and singular components gives the desired representation. The converse follows directly from the fact that every Poisson-Stieltjes integral is of class h1 • 2.5. THE CLASS N+
The two preceding theorems point out the sharp structural difference between functions in the classes HP and N. In factoring functions of class N, it is recessary not only to enlarge the class of admissible outer functions, but also to replace the singular factor by a quotient of two singular inner functions.
26
2 BASIC STRUCTURE OF H" FUNCTIONS
This allows, for instance, our "pathological" example exp{(l + z)/(1 - z)}, which we now recognize as the reciprocal of a singular inner function. It is useful to distinguish the class N+ of all functions fEN for which S 2 (z) l. That is,.f EN+ if it has the form/= BSF, where B is a Blaschke product, Sis a singular inner function, and F is an outer function for the class N. In a sense, N+ is the natural limit of HP asp-.. 0. The proper inclusions HP c N+ c N are obvious.
=
THEOREM 2.10. A functionfE Nbelongs to the class N+ if and only if 2x
lim
J
2x
log+lf(rei 6)1
d()
=
0
r~1
J
log+lf(ei 6)1
d(),
(8)
0
PROOF. Suppose first that/EN+, so thatf=BSF. Then, in view of (4),
log+lf(rei 6)1
~ log+IF(rei 6)1 ~ 2~ S:xP(r, ()- t) log+lf(e;')l dt.
Hence 2x
lim r~l
J
log+l/(re16)1 d() ~
0
2x
J
log+lf(e;')l dt.
0
Fatou's lemma gives the reverse inequality. It is more difficult to prove the sufficiency of the condition (8). We first observe that for an arbitrary Blaschke product B(z), 2x
lim r--+
1
J
logiB(re; 6)1 d() = 0.
(9)
0
This is obvious if B(z) has only a finite number of factors, so we may assume
By Jensen's theorem, l 2n
f2x logiB(rei6)1 d() = L
r
log-+ logiB(O)I. \an\ laNI 1 2-11:
2x
J 0
logiB(rei6)1
d()
~
N
L
r
oo
log - 1+ L loglanl n=l lan n=l 00
= N log r +
L loglanl· n=N+l
2.5
THE CLASS N+
27
Consequently, co
2n
L
2n
J
loglanl :5: lim
n=N+1
r-+1
loglB(rei 6 )l d8 ::;;; 0,
0
and (9) follows by letting N ~ oo. Continuing the proof of the theorem, let the given function/ E N be expressed in the formf(z) = B(z)g(z), where g EN and g(z) -:1 0 in izl < 1. Since logJB(z)l
+ log+/g(z)l::;;; log+lf(z)l::;;; log+lg(z)J,
combination of (8) and (9) gives
J
2n
lim r-+1
Since logJg(z)l
E
J
2"
log+Jg(rei 6)l d8 =
0
1og+Jg(e16)l d8.
(10)
0
h 1, it has a representation 1 27t logJg(z)l = P(r, 8 -t) dv(t) 2n o
J
(11)
with respect to a function v(t) of bounded variation. Recalling the proof of Theorem 1.1, we see that v can be chosen to have the form 6
v(8) = lim n-+oo
J logJg(rn ei')l dt,
0::;;; 8::;;; 2n,
0
where {r"} is an appropriate sequence increasing to 1. On the other hand, by Fatou's lemma, 6
6
f log+Jg(~')l dt :s; lim inf J log+Jg(rn~')l dt = v+(8), 0
n-+co
(12)
0
say. If there were strict inequality for some 8, then a similar application of Fatou's lemma in [8, 2n] and addition of the two results would give a contradiction to (10). Equality therefore holds in (12) for all 8, which shows that v+(8) is absolutely continuous. On the other hand, 6
v _(8) = v +(8) - v(8) =lim inf n-+co
J log-Jg(rn ~')I dt 0
is nondecreasing. In view of(ll), this shows thatg = SG, where Sis a singular inner function and G is an outer function. Hence fEN+, which was to be shown. The following useful result is an easy consequence of the factorization theorems.
28
2 BASIC STRUCTURE OF HP FUNCTIONS
THEOREM 2.11.
If/EN+ andf(ei 6)EI!forsomep>0,thenjEHP.
The a priori assumption that fEN+ cannot be relaxed. The reciprocal of any (nontrivial) singular inner function is bounded on lzl = 1, but is not of classN+. 2.6. HARMONIC MAJORANTS
We noted in Section 1.3 that if f(z) is analytic in a domain D, then lf(z)IP is subharmonic in D. This means that in each disk contained in D, lf(z)IP is dominated by a harmonic function, the Poisson integral of its boundary function. However, there may not be a single harmonic function which dominates lf(z)IP throughout D. In general, a function g(z) is said to have a harmonic majorant in D if there is a function U(z) harmonic in D such that g(z) :$; U(z) for all z in D. If g is continuous and has a harmonic majorant, it is obviously subharmonic; but the converse is false. THEOR.EM 2.12. lf/(z) is analytic in lzl 0) has a radial limit almost everywhere, and that loglf(e; 6)1 E L1 unless f(z) 0. He also proved the mean convergence of f(rei') to f(ei') (Theorem 2.6). Lemma 1, which simplifies the proof of this theorem, is also due to F. Riesz [6], at least for p ~ 1. The proof given in the text is in Littlewood's book [5]. A somewhat different approach to the structure of HP functions is in the paper of M. and G. Weiss [1]. For further information on the boundary behavior of HP functions, see Tanaka [1]. Theorem 2.1 is due to F. and R. Nevanlinna [1]; it unifies the presentation of the HP theory. A paper of Ostrowski [1] is also relevant here. Blaschke [1] introduced "Blaschke products" and proved Theorem 2.4. Regarding matters of priority, however, see Landau [3]. The canonical factorization theorems (Theorems 2.8 and 2.9) are due to Smirnov [2], who also noted a weaker form of Theorem 2.11 : iff E HP and f(e; 6) E Lq for some q > p, then
=
=
=
NOTES
31
fe Hq. Beurling [1] coined the terms "inner function" and" outer function." Smimov [3] introduced the class N+ (called "D" in his paper and in subsequent Soviet literature) and cited Theorem 2.10, which he attributed to "Madame P. Kotchine". More about the class N+ may be found in Privalov's book [4]. The condition (9) actually characterizes Blaschke products among all analytic functions with 1/(z)l :::;;; l (see Exercise 6). This theorem is due to M. Riesz, but was first published in Frostman [1]. (See also Zygmund [4], Vol. I, p. 281.) Theorem 2.12 was mentioned in passing by Smirnov ([3], p. 341). There is a large literature on the boundary behavior of Blaschke products; see, for example, Frostman [2], Cargo [1, 2], and Somadasa [1]. Paley and Zygmund [2] have shown that if the hypothesis f e N is "slightly relaxed" (which they interpret in two different ways), thenf(z) need not have a radial limit on any set of positive measure.
This Page Intentionally Left Blank
APPLICATIONS
CHAPTER 3
We now turn to some applications of the HP theory to such diverse topics as measure theory (the F. and M. Riesz theorem), Cauchy and Poisson integrals, and conformal mapping. Some of the results obtained will be useful in the later development of the HP theory. Further applications will appear in subsequent chapters.
3.1. POISSON INTEGRALS AND H 1
In Chapter 1, the harmonic functions u E h1 were characterized as PoissonStieltjes integrals. We shall now show that if the harmonic conjugate of u also belongs to h 1 , then the representing function 11-(t) is absolutely continuous. This result will have a number of interesting consequences. NOTE.
u(z)
The harmonic conjugate of u is any function v such that f(z) =
+ it(z) is analytic in the disk. We shall speak of" the" harmonic conjugate
even though it is determined only up to an additive constant. Later it will be convenient to normalize v by the requirement v(O) = 0.
34
3 APPLICATIONS
THEOREM 3.1.
A functionf(z) analytic in lzl < 1 is representable in the
form 1
27t
f(z) = 2,.
Jo
P(r, 8 - t)rp(t) dt
(1)
as the Poisson integral of a function rp E I! if and only iff E H 1 • In this case, rp(t) = f(ei') a.e. PROOF. If an analytic function f(z) has the form (1 ), then 27t
J
27t
lf(rei 6)1 d8 :::;:;
0
J
Irp(t)l dt,
0
so thatjE H 1 • Conversely, supposefEH 1 , and write
f 2n
1 (z) = -
2n
o
P(r, 8- t)f(e;') dt.
For any fixed p, 0 < p < 1, f(pz)
=1 -
J2"P(r, 8 -
2n o
t)f(pe 1') dt.
But, by Theorem 2.6, Jlf(pe 11 ) - f(e 11)1 dt ~ 0 as p ~ 1, so f(pz) ~ (z). Hence (z) = f(z), and the theorem is proved. Let u E h 1 , so that it is the Poisson-Stieltjes integral of a function 11-(t) of bounded variation. If the conjugate harmonic function v E h 1 , then 11-(t) is absolutely continuous. COROLLARY 1.
COROLLARY 2. A functionf(z) analytic in lzl < 1 is the Poisson integral of a function rp E IJ' (1 :::;:; p :::;; oo) if and only iff E HP. COROLLARY
then S 1 (z)/S2 (z)
3. Letf(z) =B(z)[S1 (z)/S2 (z)]F(z) EN. Iflog[f(z)/B(z)] EH\
=1.
Corollary 2 follows from Theorem 2.11. We shall prove in Chapter 4 that the harmonic conjugate of any function u E hP (1 < p < oo) is also of class hP. This cannot be true for p = 1, as Corollary 1 shows, even if u is a positive harmonic function. In fact, the Poisson kernel P(r, 8) itself is a counterexample. The following weaker theorem, however, remains true. THEOREM 3.2. Every analytic functionf(z) with positive real part in lzl < 1 is of class HP for all p < 1.
3.2 DESCRIPTION OF THE BOUNDARY FUNCTIONS
35
PROOF. Without loss of generality, suppose f(O) =I. The range off is contained in the right half-plane, sofis subordinate to
l+z 1 _ z = P(r, lJ) + iQ(r, lJ), where P(r, lJ) is the Poisson kernel and
2r sin lJ Q (} (r, ) = 1 - 2r cos(}+ r2 is the conjugate Poisson kernel. It follows from Littlewood's subordination theorem (Theorem I. 7) that
f2"1f(reie)IP d(J ::::; J2" 111 +- rere:: iP dlJ ~ J2"1Q(1, lJ)IP d(J < oo o
o
o
for any p < I, since (I + z)/(1 - z) is in HP and P(l, lJ) = 0 for(} # 0. COROLLARY. If u
e h 1 , then its harmonic conjugate belongs to h11 for all
p 0, b > 0,
3.2 DESCRIPTION OF THE BOUNDARY FUNCTIONS
37
To complete the proof, we have to show that :TfP is closed. But suppose that a sequence Un(e; 8)} of :TfP functions converges in I! mean to qJ(8) E I!. Then by the lemma, Un(z)} is uniformly bounded in each disk lzl ::;; R < 1. Thus {j~} is a normal family, so a diagonalization argument produces a subsequence {J,,(z)} which converges uniformly in each disk lzl ::;; R < 1 to an analytic function f(z). It is clear that fE HP. We wish to show that qJ(8) = f(ei 8 ) a.e. But given e > 0, choose N such that 11/n-fmiiP < e for n, m;;::: N. Then for m;;::: Nand r < 1, Mp(r,f- fm) =lim Mp(r,fn,- fm)::;; lim supll/n,- fmllp <E. k--+oo
k--+oo
Letting r--. 1, we find II/- fmiiP < e for all m ;;::: N; thus II/- /,liP--. 0, and rp(B) = f(eiB) a.e. The same argument, in simpler form, shows that :Tt 00 is closed. COROLLARY 1.
If 1 ::;; p::;; oo, HP is a Banach space.
If p < 1, then II I!P is not a true norm and in fact the space HP is not normable. However, the expression d(f, g)= 11/- gil/
defines a metric on HP if p < 1. This can be verified using the inequality
a> 0, b > 0, which is valid for 0 < p < 1. (See Section 4.2.) Furthermore, the theorem shows that HP is complete under the topology induced by this metric. COROLLARY 2.
If 0 < p < 1, HP is a complete metric space.
Theorem 3.3 is false for p = oo, since :Tt 00 contains functions which do not coincide almost everywhere with continuous functions. One example is the singular inner function exp{(z + 1)/(z -1)}, whose boundary function is exp{ - i cot(0/2)}, e =F 0. There is another approach, however, which leads to a description of :TfP for 1 ::;; p ::;; oo. The Fourier coefficients of a function qJ E I! are the numbers n
= 0, ± 1, ± 2, ....
It is important to note that the Taylor coefficients of a function /E HP (1 ::;; p::;; oo) coincide with the Fourier coefficients of its boundary function. The following theorem expresses this more precisely.
38
3 APPLICATIONS
THEOREM 3.4. Let f(z) = L,""=o anzn belong to H 1 , and let {en} be the Fourier coefficients of its boundary function f(e; 1). Then en= an for n ~ 0, and en = 0 for n < 0. Furthermore, JfP (I ::5; p ::5; oo) is exactly the class of I! functions whose Fourier coefficients vanish for all n < 0.
PROOF. The Taylor coefficients off can be expressed in the form
O1
a.e.
(4)
PROOF. If f(z) is analytic in lzl < 1 and continuous in lzl::;;; 1, it is the Poisson integral of its boundary function:
1
f(z) = 2n
2~
Ia
P(r, 8 - t)f(ei') dt.
3.5 APPLICATIONS TO CONFORMAL MAPPING
43
Differentiate with respect to 8:
J
" aP izf'(z) = 1 - (r, 2n o 2
ae
e- t)f(ei') dt;
and integrate by parts [using the absolute continuity of f(ei')] to obtain
J2"
1 izf'(z) = P(r, 8- t)iei'f'(ei') dt. 2n o
(5)
Thus izf'(z) E H 1, which implies f' E H 1 • Conversely, iff' E H 1, then izf'(z) can be represented in the form (5) as the Poisson integral of its boundary function. Here it is understood that f'(ei') denotes lim, ..., 1 f'(rei'). Since the function 8
g(8) =
J iei'f'(ei') dt 0
is absolutely continuous in [0, 2n], and g(O) = g(2n) = 0, integration by parts in (5) gives
aea {f(reiB)} = aea{12n fo27tP(r, e- t)g(t) dt } . Thus
J2"
f(rei 8 ) = 1 P(r, 2n o
e- t)g(t) dt + C(r).
Since C(r) is the difference of two (complex-valued) harmonic functions, it is itself harmonic: C"(r)
+ (1/r)C'(r) = 0.
Thus C(r) =a log r + b, where a and b are constants. To ensure continuity at the origin, a must be zero. Hencef(z) is the Poisson integral of the continuous function [g(t) + b], and is therefore continuous in lzl ::;:; 1 and has boundary valuesf(e;8) = g(8) +b. The relation (4) now follows from the definition of g(8). 3.5. APPLICATIONS TO CONFORMAL MAPPING
The theorems just given, together with a few general facts about HP functions, can be applied to obtain some rather deep results in the theory of conformal mapping. A Jordan curve (or a simple closed curve) C is the image of a continuous complex-valued function w = w(t) (0::;:; t::;:; 2n) such that w(O) = w(2n) and
44
3 APPLICATIONS
w(td "# w(t2 ) for 0::;:; t 1 < t2 < 2n. The curve C is said to be rectifiable if w(t) is of bounded variation. Its length Lis then defined as the total variation ofw(t): n
L =sup
L lw(tk)- w(tk-1)1, k=1
where the supremum is taken over all finite partitions 0 = t0 < t 1 < · · · < tn = 2n of [0, 2n]. It is easily seen that Ldepends only on the curve C, and is invariant under a change of parameter. If w(t) is absolutely continuous, the length of the arc of C corresponding to an arbitrary interval a ::;:; t ::;:; b is given by b
J lw'(t)l dt. a
(A proof may be found in Natanson [1 ], Vol. II, p. 227.) It follows that any Lebesgue measurable set E c [0, 2n] has an image on C of measure
tlw'(t)l dt. Indeed, the formula is obviously valid for open sets E, hence for G6 sets (countable intersections of open sets). Therefore, since an arbitrary measurable set E is contained in a G6 set with the same measure, the formula holds generally. A famous theorem of Caratheodory asserts that every conformal mapping w = f(z) of lzl < 1 onto the interior of a Jordan curve C has a one-one continuous extension to lzl ::;:; 1. In particular, w = f(ei 8 ) is a parametrization of C. Applying Theorems 3.10 and 3.11, we therefore obtain the following important result. THEOREM 3.12. Let f(z) map lzl < 1 conformally onto the interior of a Jordan curve C. Then C is rectifiable if and only iff' E H 1.
This theorem is highly plausible, perhaps even "obvious," when viewed geometrically. It says that the lengths
of the images of the circles lzl = r are bounded if and only if the boundary has finite length. In the presence of a rectifiable boundary, this result puts the general HP theory at our disposal. The derivative of the mapping function has an angular
3.5 APPLICATIONS TO CONFORMAL MAPPING
45
Iimitf'(ei 8 ) almost everywhere on the boundary, and loglf'(e; 8)1 is integrable. The function f'(ei 8 ) is related as in (4) to the derivative of the absolutely continuous boundary function f(e; 8 ). A measurable set Eon the unit circle is carried onto a subset of C with measure
Consequently, a subset of lzl = I has measure zero if and only if its image on C has measure zero. In other words, the boundary sets of measure zero are
preserved under the conformal mapping. This remains true (in view of the Riemann mapping theorem) for a conformal mapping between any two Jordan domains with rectifiable boundaries. More can be said. If f(z) is a conformal mapping of lzl < 1 onto the interior of a rectifiable Jordan curve C, then its continuous extension to lzl ::::;; I is conformal at almost every boundary point. To be precise, let y be a continuous curve in lzl < 1 which terminates at a point z 0 = eiBa in a well-defined direction not tangent to the unit circle; i.e., the limit of arg{z- z 0 } is to exist as z-+ z 0 along y, and is not to equal e0 ± nf2. The image of y is then a curve r inside C, terminating atf(z 0 ). Since (djde){f(e; 8 )} exists a.e., C has a tangent direction at almost every point. We assert that for almost every e0 , the angle between rand the tangent to C atf(z 0 ) exists and is equal to the angle between y and the tangent to the unit circle at z 0 . In other words, the mapping preserves angles at almost every boundary point. We have to show that } arg{ [.!!.._ f(ei 8 )] e~ea de
-
lim arg{f(z)- f(z 0 )}
z->zo Z E y
=eo+ nj2- Jim arg{z -
Zo}•
z--+zo zEy
In view of relation (4), it is eno'.lgh to prove lim f(z)- J(zo) = f'(zo) z-za
(6)
Z - Zo
zEy
wherever the angular limitf'(z 0 ) off'(z) exists; hence almost everywhere. But
f(z)- J(zo) z- z 0
=
_1_ z- z 0
Jz f'(O d(,
(7)
za
the integration being performed along the segment joining z 0 and z. If f'(z 0 ) exists, the right-hand side of(7) approachesf'(z 0 ) as z-+ z 0 along y, which was assumed to be a nontangential path. This proves (6).
46
3 APPLICATIONS
3.6. INEQUALITIES OF FEJ~R-RIESZ, HILBERT, AND HARDY
We shall now discuss some interesting inequalities which will have applications in later chapters. THEOREM 3.13 (Fejer-Riesz inequality). If /E HP (0 < p < oo), then the integral of lf(x)IP along the segment -1 ::;; x::;; 1 converges, and
(8)
t
The constant
is best possible.
PROOF. Consider first the case p = 2, and suppose for the moment that f(z) is real on the real axis. By Cauchy's theorem with a semi-circular path,
r
+i
[f(x)] 2 dx
r
[f(rei 8 )] 2 ei 8 de
=0
0
-r
for each r < 1. It follows that
Adding a similar inequality for the lower semicircle, we find 2n
r
2
J
[f(x)] 2 dx::;;
J
2n
lf(re;8)1 2 de::;;
0
-r
J
IJ(e;8 )1 2 de.
0
The desired inequality is now obtained by letting r tend to 1. More generally, we may expreso; an arbitrary H 2 functionf(z) in the form co
f(z)
=
co
n~o
n~o
where g and h are in proved,
1
co
L (cxn + i{Jn)zn = L CXnZn + in=O L f1nzn = g(z) + ih(z),
2~
: ; -2 J 0
H2
and are real on the real axis. By what we have just
1
lg(e;8)1 2 de+2
2~
J
lh(e;8)1 2 de
0
1 J2~ lf(e;a)l2 de--i J2~[h(e;a)g(e-ia)- g(e;a)h(e-;a)] de. 2 0 2 0
=-
But the last integral must vanish, since the integrand is an odd function of e. Hence inequality (8) is established in the case p = 2. To deduce the result for general p, we use a familiar device. lf/(z) belongs
3.6 FEJtR-RIESZ. HILBERT. AND HARDY INEQUALITIES
47
to HP and vanishes nowhere in lzl < 1, then [f(z)]Pf 2 E H 2 , and (8) follows from the special case p = 2 already proved. If f(z) has zeros, we factor out the Blaschke product B(z) to obtain a nonvanishing function g(z) = f(z)/B(z) which is again of class HP. Thus 1
f_
1
1/(x)IP dx 1
:$;
f_
27t
lg(x)IP dx 1
:$;
t
Jo
27t
lg(e; 8)IP de =
t
Jo
lf(ei8)IP de.
In view of Theorem 3.12, there is an interesting geometric application. COROLLARY. If the unit disk lzl < 1 is mapped conformally onto the interior of a rectifiable Jordan curve C, the image of any diameter has length at most half the length of C.
We may now finish the proof of the theorem by showing that tis the best possible constant. Let w = qJ(z) map lzl < 1 conformally onto the interior of the rectangle with vertices ± 1 ± ie, the diameter - 1 :$; z :$; 1 corresponding to the real segment - 1 :$; w :$; 1. The ratio of the length of this segment to the perimeter of the rectangle is 2[4(1 + e)r\ which tends to } as e __. 0. The functionf(z) = [qJ'(z)] 1 fP E HP therefore shows that the constant cannot be improved. Two further inequalities, named after Hilbert and Hardy, lie in the same circle of ideas. We shall prove them in generalized form, with a view to later applications. For a complex vector x = (x 0 , x 1 , ..• , xN), let N
llxll 2 THEOREM 3.14.
Let 1/J
E
L lxnl
=
2•
n~o
L"' and n = 0, 1, 2, ....
Let N
AN(x, y) =
L
A.n+m
xn Ym.
n,m=O
Then
IAN(x, x)l
= 1 2171: Jo27t[P(t)] 2 1/J(t) dt :$;
1 111/JIIoo 2
I
27t
nfo
IP(t)l 2 dt = 11!/JIIoollxll 2 •
(9)
48
3 APPLICATIONS
To deduce the more general result, observe that AN(x, y)
= -!-AN(x + y, x + y) - -!-AN(x - y, x - y).
Thus
Y)l::;; Hl/llloo{llx + Yll 2+ llx- Yll 2} = tlll/111 {11xll 2+ llyll 2 }. This shows that IAN(x,y)l::;; 111/ll oo if llxll = IIYII = 1, which is enough to IAN(x,
00
prove the theorem. COROLLARY (Hilbert's inequality).
In.t~o n :n~m+ 1 1::;; nllxiiiiYII. PROOF. Choose 1/l(t) = ie- ir(n
Let
THEOREM 3.15.
f(z) =
L anzn
E
- t), so that .A.n = (n + 1) - 1and 111/111 = n. 00
.A.n ~ 0 be given by (9) for some
1/1
E
L00 • Then if
H\ CXl
L .A.nlanl::;; 111/JIIooll/111· n=O PROOF. Every /E H 1 has a factorization f = gh, where g and h are H 2 functions with llgll/ = llhll/ = 11/11 1. Indeed,/= BqJ, where B is a Blaschke product and qJ is a nonvanishing H 1 function (Theorem 2.5). Let g = BqJ 1f 2 and h = qJ 1'2. Now let g(z) = bn~ and h(z) = cnzn; then
L
L
n
an=Lbkcn-k· k~O
Hence N
N
n
L An Iani ::;; n=O L An L lbkllcn-kl n=O k=O
N
::;; L
k,m=O
.A.k+mlbkllcml::;; lll/JIIoollgll2llhll2 = 111/JIIooll/111•
by Theorem 3.14. COROLLARY (Hardy's inequality).
Iff(z) =
L anzn
Ia I L ~1 ::;; nllfll1· n=O n + CXl
E
H\ then
3.7 SCHLICHT FUNCTIONS
49
Hardy's inequality is proved with the same choice of 1/J that gave Hilbert's inequality. One interesting consequence should be mentioned. Suppose f(z) = a"z" is analytic in lzl < 1. If lanl < oo, then f has a continuous extension to lzl::;:; 1, but the converse is false (see Exercise 7). Hardy's inequality shows, however, that iff' E H 1 (or equivalently, in light of Theorem 3.11, ifjis continuous in lzl::;:; 1 and absolutely continuous on lzl = 1), then la.l < oo. In particular, lanl < oo iff is a conformal mapping of the unit disk onto a Jordan domain with rectifiable boundary.
L
L
L
L
3.7. SCHLICHT FUNCTIONS
A function analytic in a domain is said to be sch/icht (or univalent) if it does not take any value twice; that is, if/(z 1 ) "# f(z 2 ) whenever z1 "# z2 . Our main aim in this section is to prove that every schlicht function in the unit disk is of class HP for all p
(r > 0)
and apply the Cauchy-Riemann equation
r
a( log R) a ar = ae
to obtain
-d J
2"
dr o
lf(re; 8)IP de =
Jo -;ora;-- (RP) de 2"
PJ2"RPa de = Pf -
=-
r
0
ae
r r,
RP d,
where r, is the image underfofthe circle lzl = r. Thus if M(r) = the maximum of lf(z)l on lzl = r, then for each e > 0 the circle lwl surrounds r,' and it follows from Lemma 2 that d p -d {M/(r,f)}::;:;- [M(r) r
r
+ e]P.
M~(r,f)
is
= M(r) + e
EXERCISES
51
Now let e ~ 0, integrate from 0 to r, and apply Lemma 1 :
M/(r,f) ~ p [ [M(r)]P dr o r
~p
JrP- (1- r)1
1
2P
dr < oo
0
if 0 < p < ! . This concludes the proof. As a function of class HP(p < -t), each schlicht function factorization f(z) = B(z)S(z)F(z)
f has a canonical
(see Theorem 2.8). The Blaschke product B(z) obviously has at most one factor. Less obvious is the fact thatfcan have no singular part. THEOREM 3.17. lf/(z) is analytic and schlicht in lzl
factor S(z)
=1.
< 1, then its singular
PROOF. Iff does not vanish in the open disk, then 1/fis analytic and schlicht, so 1/f E HP for all p < -t. Hence by the uniqueness of the canonical factorization of a function of class N (Theorem 2.9), S(z) = 1. If /(0 = 0 for some (, lei < 1, then the function
g(z) = (1 -
t(t: c~)
1'1 2 )- 1 [/'
!·
From this it follows that Blf E H 00 , where
z-C
B(z) = 1 _ (z. In particular, f cannot have a (nontrivial) singular factor. EXERCISES
1. Prove: If f(z) is analytic and Re{f(z)} > 0 in lzl < 1, thenf is an outer function. (Hint: Show thatf.(z) = e + f(z) is outer and let e ~ 0.) 2. Showthat
f(z) = - 1-log(-1- )
1-z
1-z
belongs to HP for all p < 1, but has unbounded Taylor coefficients. In particular, show thatf(z) is not a Cauchy-Stieltjes integral, so that the converse to Theorem 3.4 is false.
52
3 APPLICATIONS
3. Prove Theorem 3.9. 4. Prove the integral analogue of Hilbert's inequality:
IJooJoo 0
0
f(x)g(y) dx dy
X+ y
I~ nii!II2IIYII2
iff and g are in L2 (0, oo ). 5. It is a natural conjecture that iff EN andf(ei 8)
27t
J
ein 8f(e; 8 ) de = 0,
E
L1 , then
n = 1, 2, ....
0
Show this is false. 6. Construct an analytic functionf(z) =
f' ¢ Hl.
L anzn such that L lanl < oo, but L
7. Give an example of a function f(z) = anzn analytic in lzl < 1 and continuous in lzl ~ 1, such that lanl = oo. (Suggestion: Let f map the unit disk conformally onto a Jordan domain constructed in such a way that the radius 0 ~ z < 1 corresponds to a curve of infinite length.)
L
NOTES
Theorems 3.1 and 3.8 are essentially due to F. and M. Riesz [1]. They also showed that the coefficients of an H 1 function tend to zero. Theorem 3.4 and the falsity of its converse (Exercise 2) may be found in Smirnov [2]. Theorem 3.9 is due to Fichtenholz [1 ]. Privalov's book [4] discusses these results. Helson [1] has given a" soft" proof of the F. and M. Riesz theorem, making no use of complex function theory. Rudin [6] obtained a generalization of this theorem in which Jeinr dp. = 0 only outside a certain "thin" set of positive integers. Theorem 3.11 is due to Privalov [1, 2], as are most of the results in Section 3.5. Theorems 3.10 and 3.12 appear in a paper of Smirnov [4]. The theorem of Caratheodory mentioned before Theorem 3.12 may be found in Goluzin [3], Chap II, Sec. 3, or in Zygmund [4], Chap. VII, Sec. 10. Golubev [1] gave the first example of a conformal mapping of the unit disk onto a Jordan domain which carries a boundary set of measure zero onto a set of positive measure on the (nonrectifiable) boundary of the image domain. Theorem 3.13 is in a paper of Fejer and Riesz [1]. Hardy [1] had essentially proved it for p = 2. Many proofs of Hilbert's inequality have been given; see Hardy, Littlewood, and P6lya [1]. "Hardy's inequality" seems to have appeared first in a paper of Hardy and Littlewood [1]. Theorem 3.16 is essentially in a paper of Prawitz [1]; see also Goluzin [3], Chap. IV, Sec. 6. Another proof is in Littlewood [5]. Theorem 3.17 is due to Lohwater and Ryan [1].
CONJUGATE FUNCTIONS
CHAPTER 4
If a harmonic function has a certain property, must the same be true of its conjugate? Questions of this kind have been widely investigated, both for their own interest and for their importance in applications. In this chapter and the next, we consider growth and smoothness properties of functions harmonic in the unit disk. Generally speaking, a harmonic function and its conjugate behave alike, but there are some rather surprising exceptions.
4.1. THEOREM OF M. RIESZ
Given a real-valued function u(z) harmonic in lzl < 1, let v(z) be its harmonic conjugate, normalized so that v(O) = 0. Thusf(z) = u(z) + iv(z) is analytic in lzl < 1, andf(O) is real. If f(z) = Len zn and en= an- ibn, then co
u(z) = ao
+ L rn(an cos ne + bn sin ne), n=l
co
v(z) =
L rn 0. For general u E hP, fix r < l and set
Then
u(re 18)
= U1 (0)- U2 (0);
lu(re; )1P = IU1(0)IP 8
+ IUiO)IP.
(4)
56
4 CONJUGATE FUNCTIONS
For 0 :s; p < 1, define
f 2n 1
ui(pei 8 )
=-
27t
P(p,
o
e- t)Uit) dt,
(5)
j = 1, 2.
Let vi be the harmonic conjugate of ui. Then u(pre;o)
= ut(pe;o)-
uipe;o);
v(preio) = vt (pe;o) - vipe;o). Since u1 and u2 are positive harmonic functions, it follows from what has been proved that ("1v(pre;8 )IP de :s; 2P{("Iv 1 (pe; 8)1P de+ ("1v2(pei 8)IP de} 0
0
0
2Pp { :s; - -
J
p- 1
27t
lut(pei8 )IP de+
0
J
27t
lu2(pe;oW de
}
.
0
Letting p--. 1 and using relations (4) and (5), one now obtains (3) with
- (-p)lfp.
AP- 2
p-1
The factor 2 can be removed by a more refined argument (see Exercise 5). As pointed out in Section 3.1, Riesz's theorem breaks down for p = 1. The Poisson kernel P(r, e) E h 1 , but its analytic completion (1 + z)/(1 - z) ¢ H 1 . The next two sections will be concerned with weaker theorems which may be said to replace Riesz's theorem in the case p = 1. Let us note that Riesz's theorem also fails in the case p = oo. Consider, for example, the function f(z) = u(z)
1+z + iv(z) = i log 1 _ z,
which maps lzl < 1 conformally onto the vertical strip -n/2 < u < n/2. Here u is bounded, but v is not. On the other hand, Riesz's theorem guarantees that f E H P for all p < oo, a fact not easy to verify by direct calculation. 4.2. KOLMOGOROV'S THEOREM
Although the harmonic conjugate of an h 1 function u(z) need not be in h 1 , it does belong to hP for all p < 1. We have already proved this (Corollary to Theorem 3.2) by appeal to Littlewood's subordination theorem. We now give an independent proof of a slightly stronger result. The following lemma will be needed.
4.2 KOLMOGOROV'S THEOREM
57
LEMMA. For arbitrary positive numbers a and b,
O 1, increasing if 0 < p < I. Since g(l) = 2p-l and g(x)-> I as x-> oo, the lemma is proved. For p > I, the result also follows easily from the fact that xP is a convex function. THEOREM 4.2 (Kolmogorov). If u E h 1, then its conjugate v E hP for all
p < I. Furthermore, there is a constant BP, depending only on p, such that O~r
0, and set
f(z) = u(z)
+ iv(z) =
Re; 41 ,
I\
is analytic in lzl < I. By the mean value theorem,
-2n1 Jo F(rei Z7t
8)
dO= F(O)
= [u(O)]P =
[M 1(r, u)]P.
Hence
In view of the inequalities lv(z)l < R and 0 < cos pn/2 < cos p, it follows that Mp(r, v) < (secpn/2) 11PM1 (r, u)
for positive u. The constant is in fact best possible for u(z) > 0 (see Exercise 3).
58
4 CONJUGATE FUNCTIONS
The extension to general u E h 1 is similar to that in the proof of Riesz's theorem. Using the same notation, we find by the lemma that
::;; sec pnf2{[M 1(p, u 1 )]P
+ [M 1(p, u2)]P}.
The other part of the lemma now gives (since 1/p > 1)
Mp(pr, v)::;; BP[M1 (p, u 1)
+ M 1(p, u2 )],
where
BP = 2°fpl- 1 (sec pn/2) 1 1P. The proof is completed by letting p --. 1. 4.3. ZYGMUND'S THEOREM
Since u E h1 is not enough to ensure v E h1, but the stronger hypothesis u E hP (for some p > 1) is sufficient, it is natural to ask for the "minimal" growth restriction on u which will imply v E h1• Such a condition is the boundedness of 27t
J
lu(re; 6)llog+ lu(rei 6)1
de,
r < 1.
0
We shall denote by h log+ h the class of harmonic functions u(z) for which these integrals are bounded. Clearly, hP c h log+ h for all p > 1. THEOREM 4.3 (Zygmund).
If u E h log+ h, then its conjugate vis of class
h 1 , and 27t
J
27t
lv(rei6)1
0
de::;;
J
lu(re; 6)llog+ lu(rei6)1
de+ 6ne,
O::;;r C for some constant C, then u e h log+ h. PROOF. We may assume u(z) > I, since addition of a constant to u does not affect v. Set
f(z) = u(z)
+ iv(z) =
Rei41 ,
11
< 7t/2.
The function f(z) logf(z) is analytic in lzl < 1. Applying the mean value theorem to its real part, we find
f
2n
R cos log R dO =
0
f
2n
R sin dO
+ 21ru(O) log u(O).
0
Thus
f
J u log R dO 7t J2" lv(rei )1 dO+ 21tu(O) log u(O). (t), and (I 0) is the Fourier series of q>(t). PROOF. Let A a.= an+l- a. and A2 a"= A(A a"). The convexity of {a"} means that A 2 a.~ 0; thus {A a"} is an increasing sequence. Since {a.} converges, A a.-+ 0; hence A a. :s;; 0. Since a.-+ 0, it follows that a.~ 0. Let SN(t) denote the Nth partial sum of (10). After two summations by parts, we find N
SN(t) =
L (A2 a.)(n + 1)K.(t)
n=O
(11)
where 1 N sin(N + -!)t DN(t) = -2 + L cos nt = 2 . 12 n= 1 sm t
and 1
KN(t) = N
N
+ 1 n~O
D.(t)
sin 2 (N + l)t/2 1) sin 2 (t/2)
= 2(N +
are the Dirichlet and the Fejer kernel, respectively. For each t # 0, the sequences {DN(t)} and {NKN_ 1(t)} are bounded; hence the last two terms in (11) tend to zero. Consequently, 00
SN(t)-+ q>(t) =
L (A 2 a.)(n + 1)K.(t).
(12)
n=O
Because (n + l)K.(t) is uniformly bounded in each set 0 < {J :S iti ~ n, the series (12) converges uniformly there, and so represents a function q>(t) continuous for t # 0. Since the terms of the series are all nonnegative, so is q>(t). By the Lebesgue monotone convergence theorem,
f,
q>(t) cos mt dt =
Jo
(A 2 a.)(n
+ 1) f,K"(t) cos mt dt
00
= n L (A 2 a.)(n- m + 1), n=m
m
= 0, 1, 2, ....
4.6 THE CASE p < 1 : A COUNTEREXAMPLE
65
This last series may be evaluated through summation by parts: k
L (A
k
LA an+ (k- m + 1) A ak+
an)(n- m + 1) = -
2
n=m
1
+ (k - m + 1) A ak+ 1 • the right-hand side approaches a,., and so =am- ak+ 1
Ask.-. oo,
J" rp(t) cos mt dt = am'
-1 1T.
m = 0, 1, 2, ....
-n
[In particular, rp(t) is integrable.] Thus (l 0) is a Fourier series, which was to be proved. The fact that k A ak-+ 0 follows from a classical theorem of Abel: If {bd is a monotonically decreasing sequence and L bk converges, then kbk-+ 0. For a proof, observe that 2"
2nb2" =::; 2 k=
4.6. THE CASE p
< 1:
L2"-
bk · 1
A COUNTEREXAMPLE
For 1 < p < oo, the class hP is preserved under conjugation. This is false for h1, but it is "almost true" in the sense that the conjugates of h 1 functions belong to hP for all p < 1. If the hypothesis is further weakened by requiring only that u E hP for some p < 1, hardly a trace of the theorem remains. The conjugate function v may not belong to hq for any positive q. In fact, we are about to construct an analytic functionf(z) = u(z) + iv(z) such that u E hP for allp < 1, yet v rt hP for any p > 0. The example is co
J(z) = u(z)
+ iv(z) =:Len n=1
z2"
1-z
(13)
zn+l,
We are going to show that for every choice of the signs en, u E hP for all p < 1; while for" almost every" sequence of signs,f(z) has a radial limit on no set of positive measure. In particular, some choice of the e. gives a function f(z) which is not even of class N, but whose real part belongs to hP for all p < 1. LEMMA. For each p
f" (1 -n
> t,
dO
2r cos
e + r2 )P =
( 1 ) 0 (1 - r) 2 P 1
as
r-+ I.
66
4 CONJUGATE FUNCTIONS
PROOF.
~
Since sin x
(2/n)x for 0 :-:::; x :-:::; n/2,
I - 2r cos()+ r 2 =(I - rf + 4r sin 2 ()j2 ~(I - r) 2 + (4rfn 2)()2. Hence, for r
~
t, the integral does
f-x" [(1- r)2 d()+ 2n
not exceed 1
The last integral is convergent because p >
dt
JCX)
2()2]p < (1- r)2P
1
-oo [1
+ 2n-2t2]p"
f.
In showing u E hP for all p < I, we may suppose p > begin by computing
t.
With z = rei 6, we
e{ z 2 " } _ Rn(l - Rn 2 ) cos 2n() R 1 - z2 "+' - 1 - 2Rn 2 cos 2n+ 1 () + Rn 4 ' From this we find, using the lemma, 2>r
f
CX)
iu(re;6 )1P d() :-:::;
0
d()
2>
0. (Hint: Try the Poisson kernel.) 4. Use Green's theorem to prove Kolmogorov's theorem with 2lfp-1(1- p)-lfp.
5. Letf(z) = u(z) + iv(z) be analytic in lzl < l, and suppose u p, 1 < p ::;:; 2. For fixed e > 0, set G(z) = {[u(z)] 2
Show that
+ e}Pf2 ;
H(z) = {lf(z)l 2
E
+ e}P 12 •
BP =
hP for some
68
4 CONJUGATE FUNCTIONS
Apply Green's theorem, integrate, and let
Mp(r,
f)~
E-+
0 to obtain
p ) 1/p (p _ Mp(r, u), 1
thus proving the theorem of M. Riesz with a smaller constant. (This idea is due toW. K. Hayman. It is an open problem to find the best possible constant, for p = 2, while the best even for u(z) > 0. The above constant reduces to possible constant in this case is I.)
J2
6. For the example of Section 4.6, show that there is a choice of signs such that
NOTES
Theorem 4.1, or an equivalent form of it, is in the paper of M. Riesz [1 ]. The proof based on Green's theorem, as presented in the text, is due to P. Stein [1]. This approach has the advantage of leading to a relatively good value of the constant AP. Calderon [1] has given still another proof; see also Zygmund [4], Chap. VII. Shortly after Kolmogorov [1] proved Theorem 4.2, Littlewood [2] suggested the proof using his subordination theorem. Hardy [3} then discovered the elementary argument given in the text as a second proof. Theorem 4.3 and the converse results (Theorem 4.4 and its corollary) are due to Zygmund [1, 4]. Another proof is in a paper of Littlewood [3]. Theorem 4.5 and similar results may be found in Zygmund [4], Chap. V. Phenomena of the type illustrated in Sec. 4.6 are discussed in the paper of Hardy and Littlewood [6]. Previously, Littlewood [1] had constructed a harmonic function which belongs to hP for all p < 1, yet has a radial limit almost nowhere. The example (13) given in Sec. 4.6 is essentially due to Hardy and Littlewood [6], who showed by a highly nonelementary argument that a function similar to this (but with En== I) fails to have a radial limit on some set of positive measure. Paley and Zygmund [2] introduced Rademacher functions and "constructed" an example of the type (13) for which/¢ HP for all p < 1. The argument given in the text is considerably simpler than theirs. Hardy and Littlewood [6] also proved that if u E hP for some p ~ 1, then its conjugate v satisfies r-+ 1.
For p = 1, }, !, ... , they showed by an elementary example that this estimate is best possible. Whether or not it can be improved for other values of p
NOTES
69
remains an open question, although Swinnerton-Dyer [I] has shown that it cannot be improved to 1-). Mp(r, v)= o(log1- r (Unfortunately, Hardy and Littlewood stated the positive result incorrectly in the introduction to their paper, and Swinnerton-Dyer reproduced this error.) Gwilliam [1] simplified some of the work of Hardy and Littlewood in this area.
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MEAN GROWTH AND SMOOTHNESS
CHAPTER 5
If f(z) is analytic in the unit disk, there is a very close relation between the mean growth of the derivative f'(z) and the smoothness of the boundary functionf(e; 6). This principle takes several precise forms, as we shall see in the present chapter. Other topics to be discussed are the relations between the growth of Mp(r,f) and Mp(r,f'), between the growth of Mir,f) and Mq(r,f), and between the growth and smoothness of a harmonic function and its conjugate. Some of these results will be applied repeatedly in later chapters. We shall begin by discussing several ways to measure smoothness and exploring the connections between them. 5.1. SMOOTHNESS CLASSES
Let rp(x) be a complex-valued function defined on - oo < x < oo and periodic with period 2n. The modulus of continuity of rp is the function w(t) = w(t; rp) =
sup !rp(x)- rp(y)!. !x- Yl :51
Thus rp is continuous if and only if w(t)-+ 0 as t-+ 0. We say that rp is of class
72
5 MEAN GROWTH AND SMOOTHNESS
A,. (0 < oe::;; I) if w(t) = O(tj as t-+ 0. Alternatively, A,. is the class of functions which satisfy a Lipschitz condition of order oe: iq>(x)- q>(y)l ::;; A l(x- YW·
The definition is of no interest for oe > I, since A,. would then contain only the constant functions. It is clear that Ap c A,. if oe < p. A continuous function q>(x) is said to be of class A* if there is a constant A such that (I) Iq>(x + h) - 2q>(x) + q>(x - h)l ::;; Ah for all x and for all h > 0. The condition (I) alone does not imply continuity;
indeed, it is well known that there exist nonmeasurable functions such that q>(x
+ y) = q>(x) + q>(y).
For any oe < I, the proper inclusions AI cA. c A,.
hold. In fact, every q> E A* has modulus of continuity ro(t) = O(t log 1/t), as we shall see later. (See also Exercise 14.) It often turns out that A., not A1, is the "natural limit" of A,. as oe-+ I. Another class, apparently larger than A,., could be defined by requiring that the left-hand side of (I) be O(h"). However, this class is actually the same as A,. if oe < I. For q> E £P = LP(O, 2n), I ::;; p < oo, the function wp(t) = wp(t; q>) = sup {t"lq>(x +h)- q>(x)l Pdx}l/p O, wp(t)-+ 0 as t-+ 0, by a theorem of F. Riesz (see, for example, Titchmarsh [I], p. 397). If wp(t) = O(t"), 0 < oe::;; I, we say that q> belongs to the class A/. Because the LP means increase with p, it is evident that A,.q c A/ if p < q. If q> is continuous, wP(t)-+ w(t) asp-+ oo. The classes A/ may therefore be viewed as generalizations of A,.. Obviously, A,. c A/. In the proof of Theorem 5.4 we shall need the following result, which is of some interest in itself. LEMMA 1 (Hardy-Littlewood). If q> is of bounded variation over [0, 2n], then q> E A1 1 • Conversely, every function q> E A 11 is equal almost everywhere to a function of bounded variation. PROOF. Suppose first that q>(x) is of bounded variation, and let V(x) be the total variation of q> on [0, x]. Then for small h > 0,
5.1 SMOOTHNESS CLASSES
2x
73
2x
J if'(pei(6+r>) _ ei(e-r>f'(pei(6-t))} dt. h
0
Analyzing the integrand as in (9), we see that it is dominated by et{log
k+ ~}.
where Cis independent of e. Hence 11h2 f(pe 16 ) is uniformly O(h), and the proof is complete. The next theorem may be viewed as the LP analogue of Theorem 5.1. It is included here for the sake of completeness; we shall make no use of it in this book. THEOREM 5.4 (Hardy-Littlewood). Let f(z) be analytic in /zl < I. Then /E HP andf(e 16) E A/ (l ::;p < oo; 0 0.
Then
PROOF. For 1 < p < oo, we could apply the theorem of M. Riesz, but we shall give a proof which makes no appeal to this deeper result. Let p = 1{1 +r), and express f(z) by the Poisson formula:
1 2" peir + z . f(z) = ir u(pe'1) dt 2n o pe - z
J
+ iy.
Then 1 2" iu(peHB+rl)l dt lf'(re; 6)1 ~2 n o p - 2pr cos t + r 2
J
.
By Minkowski's inequality, M ( P
r'
f') < ~ - n
f21t 0
p2
-
Mp(p, u) dt 2pr cos t + r 2
_ 2Mp(p, u) _ - p2 _ r2 -
a(
1 ) ( 1 _ r)/1+1 ·
Theorem 5.5 now gives the desired conclusion. Generally speaking, then, a harmonic function and its conjugate have the same rate of growth. We are now in a position to show that they also have the same degree of smoothness on the boundary. THEOREM 5.8. Let f(z) = u(z) + iv(z) be analytic in lzl < 1, and suppose u(z) is continuous in lzl ~ 1. If u(ei6) E A~ (ex < 1), then v(z) is continuous in lzl ~ 1 and v(ei 6) E A~. If u(e;6) E A*, then v(ei'1 E A*.
PROOF. If we represent f(z) as a Poisson integral of u(e; 1) and follow the proof of Theorem 5.1, we see that the weaker condition u(e 16 ) E A~ still implies f'(z) = 0((1 - rY- 1J- Thus f(ew) E A~, by Theorem 5.1. To show that A* is
also preserved under conjugation, we may express u(z) as the Poisson integral ofu(ei 1) and follow the proof of Theorem 5.3 to see that u 6 o(z) = 0((1 - r)- 1).
84
5 MEAN GROWTH AND SMOOTHNESS
Theorem 5.7 then implies v99(z) = 0((1 - r)- 1), so thatf"(z) = 0((1 - r)- 1). Hence f(e; 9) E A*, by Theorem 5.3. COROLLARY. Every function q>(x) of class A* has modulus of continuity w{t) = O(t log lft). PROOF. Let u(z) be the Poisson integral of the given function q> E A*, and let f(z) = u(z) + iv(z) be its analytic completion. Then f(z) is continuous in lzl :::;; I, and f(ei 9) E A*. The result therefore follows from the special case already noted as a corollary to Theorem 5.3.
5.5. COMPARATIVE GROWTH OF MEANS
If f(z) E H P (0 < p < oo ), it is possible to give a sharp estimate on the growth of Mq(r,f) for any q > p. This theorem has a number of interesting applications. The proof will make use of the following lemma, which can be established by essentially the same argument used to prove the lemma in Section
4.6. LEMMA 3. If a> I and p
t
= !(! + r), then
2"
lpeit- ri-a dt
= 0((1
- r) 1 -a),
r-+ 1.
THEOREM 5.9 (Hardy-Littlewood). Let f(z) be analytic in lzl
< I, and
suppose 0 1. 12. Letf(z) = u(z) + iv(z) be analytic in Jzl < 1. Suppose u E hP (1 < p < oo) and u(e;'1 E A/ (0 < ex< 1). Show that v(ei 6 ) E A/. In other words, show that the class A/ is self-conjugate. 13. Prove that iff E H P (1 < p < oo) and f(ei'1 E A/ (1/p < ex :$; 1), then f(z) is continuous in lzl :$; 1 and f(ei 6 ) E A11 , {3 = ex - lfp. 14. Show that the function
rp(x)
= x log 1/x,
X
>0,
belongs to A* but not to A1 .
15. Prove Theorem 5.2. 16. Using the example f(z) = (1 - z)- 1 , show that the exponent ({3 + 1/p - lfq) in Theorem 5.9 is best possible. NOTES
Almost everything in this chapter is due to Hardy and Littlewood. Lemma 1 is in their paper [2). Theorems 5.1, 5.4-5.6, 5.9, and 5.11 occur in their paper [5]. The proofs of Theorems 5.5 and 5.7 for the case p < 1 may be found in their paper [6]. The simple but enormously useful remark that every HP function is the sum of two nonvanishing ones (Lemma 2) can be traced to their
92
5 MEAN GROWTH AND SMOOTHNESS
paper [1]. Further results and generalizations are in their papers [5] and [8]. The "o" growth condition given in Theorem 5.9 is best possible; see G. D. Taylor [1] and Duren and Taylor [1]. Flett [2] has recently based a proof of Theorem 5.11 on the Marcinkiewicz interpolation theorem. Zygmund [2, 4] introduced the class A* and proved Theorem 5.3 in somewhat different form. The self-conjugacy of A« (Theorem 5.8) goes back to Privalov [1]. Zygmund [2] showed that A* is self-conjugate. For a direct proof that every A* function has modulus of continuity which is O(t log 1/t), see Zygmund [4], Vol. I, p. 44. The converse to Theorem 5.12 is totally false. The "Bloch-Nevanlinna conjecture" asserted that if/EN, thenf' EN, but this has been disproved in many ways. In fact, there exist functions /E H 00 continuous in the closed disk, such that f'(z) has a radial limit almost nowhere. See, for example, Lohwater, Piranian, and Rudin [1] and Duren [4]. Hayman [3] has recently shown that f' E N does not imply f E N, an unexpected result in view of Theorem 5.12. Caughran [2] has obtained results comparing the canonical factorizations off and f' in case f' E H 1 •
TAYLOR COEFFICIENTS
CHAPTER 6
L
If a functionf(z) = anzn belongs to a certain HP space, what can be said about its Taylor coefficients an? Clearly, one can hope to describe only the "eventual behavior" of {an} as n-> oo, since any finite number of coefficients can be changed arbitrarily without upsetting the fact thatfis in HP. It is also interesting to ask how an HP function can be recognized by the behavior of its Taylor coefficients. Ideally, one would like to find a condition on the an which is both necessary and sufficient for fto be in HP. For p = 2, of course, the problem is completely solved: /E H 2 if and only if lanl 2 < oo. But the general situation is much more complicated, and no complete answer is available. If 1 < p < oo, the problem is equivalent to that of describing the Fourier coefficients of LP functions, as the M. Riesz theorem shows. This chapter contains some scattered information about the coefficients of HP functions. Curiously, the results are most complete in the case 0 < p < 1.
L
6.1. HAUSDORFF-YOUNG INEQUALITIES
Any available information about the Fourier coefficients of LP functions can be applied, in particular, to the Taylor coefficients of HP functions. In fact, the
94
6 TAYLOR COEFFICIENTS
two sets of coefficient sequences are essentially the same if 1 < p < oo. The Hausdorff-Young theorem states that if rp(x) e U = U[O, 2n], 1 < p ~ 2, then its sequence {c,} of Fourier coefficients is in tq (1/p + lfq = 1) and
Conversely, every tP sequence {c,} of complex numbers (1 < p ~ 2) is the sequence of Fourier coefficients of some rp e Lq (1/p + lfq = 1), and
This result may also be expressed in HP language, as follows. THEOREM 6.1. If 00
f(z) = then {a,} e
tq (lfp + 1/q =
L a,z" e HP n=O
(1
~
p ~ 2),
1) and
I {a,}llq ~ 11/llp. Conversely, if {a,} is any tP sequence of complex numbers (1 = a,z" is in Hq (lfp + 1/q = 1) and
f(z)
L
(1) ~
p ~ 2), then
11/llq ~ II {a,} liP.
(2)
PROOF. The first statement follows easily from the Hausdorff-Young theorem. Indeed, if fe HP, then the radiallimitf(ei9) is in U and {a,} is its sequence of Fourier coefficients. On the other hand, if {a,} e tP (1 ~ p ~ 2), then fe H 2 , and the numbers a, are the Fourier coefficients of f(ei 9 ). The Hausdorff-Young theorem then tells us thatf(e 19 ) e Lq, which implies f e Hq, and (2) follows.
Neither part of the theorem remains valid if p > 2. In fact, the hypothesis thatfe H P for some p > 2 implies nothing more about the Ia, I than {a,} e t 2 • And if the condition {a,} e t 2 is weakened to {a,} e tP for some p > 2, nothing reasonable can be said aboutf(z). These claims are justified by Theorem A.5 of Appendix A. According to this theorem, the mere assumption that {a,} et 2 implies that, for almost every choice of signs {e,}, the function e, a,z" is in HP for allp < oo. On the other hand, if {a,}¢ t 2 , almost every choice of signs produces a function e, a, z" having a radial limit almost nowhere.
L
L
6.2 THEOREM OF HARDY AND LITTLEWOOD
95
6.2. THEOREM OF HARDY AND LITTLEWOOD
The following theorem provides further information about the coefficients of HP functions. THEOREM 6.2 (Hardy-Littlewood).
If
co
f(z) = I, anzn
E
HP,
0< p::;:; 2,
n=O
{J (n + I)p- 1an1pf'p::;:; cpllfllp. 2
0
where
cp
(3)
depends only on p.
PROOF. We suppose 1 ::;:; p ::;:; 2, postponing the case 0 < p < 1 to Section 6.4. If p = 1, the theorem reduces to Hardy's inequality (Corollary to Theorem 3.15). If p = 2, it follows from the Parseval relation. A proof for intermediate values of p can be based on the Marcinkiewicz interpolation theorem, but the argument given here will be self-contained. Let p. be the measure defined on the set of integers by
p.(n) = (lnl + 1)- 2 ,
n
= 0, ± 1, ± 2, ....
If gEL and 2
co
g(t)""'
I.
bn elnr,
n=- oo
let g(n) =(In I + l)bn,
n
= 0, ± 1, ±2, ....
For s > 0, let £~
= {n: l§(n)l > s}.
Then ... co
s 2p.(E,)::;:;
co
I, lii(nWp.(n) = n=- oo
I, lbnl 2
= IIYII/.
(4)
n= -oo
Furthermore, since lhnl ::;:; 11911 1,
E, c F, = {n: (lnl + l)llgll1 > s}; thus p.(E,)::;:;p.(F,)= I,(lnl+l)- 2 ::;;cs- 1 ilgil 1. neF5
(5)
96
6 TAYLOR COEFFICIENTS
Now write
g(t) = q>,(t)
+ t/l,(t),
where
ig(t)i??:s lg(t)i < s. Then 00
00
I s}).
Then in view of (4) and (5),
f
n=-oo
llP.(n)IP.u(n) = - Joo sPdcx(s) = p Joo sr 1cx(s) ds
~
0
pC
Jooo
pC =-
oo J sp- J lcp.(t)l dt ds 2n o o pC
sP- 2 IIcp.ll 1
f2"
2n o
0
ds
2n
2
=-
2n
lg(t)l
lg(t)l
Jo
Jo
sr 2 ds dt = AP
lg(t)IP dt,
where AP is a constant depending only onp. Similarly, by (4),
f sP- p(s) ds ~PJ sp- llt/1.1i/ds=-J sp- J lt/l.(tWdtds o 2n o o
n~ooltfr.(n)IP.u(n) = - IaoosP dp(s) = p
00
p
2n
1
27t
00
3
p = -2
00
0
3
oo
J lg(tW J n o
sp- 3 ds dt ~ BP
lg(tJI
2n
Jo lg(t)IP dt.
Combining these two estimates with (6), we have (7)
6.2 THEOREM OF HARDY AND LITTLEWOOD
97
lf/E H P (1 < p < 2), the desired estimate (3) now follows after approximating the boundary functionf(ei') by 9M(t)
1/(e'')l ::;; M, 1/(ei')l > M,
= {f(ei'), 0,
applying (7) to gM, and Jetting M--. oo. As with the Hausdorff-Young theorem, the converse to Theorem 6.2 is false if p < 2. However, the converse is true for indices larger than 2, and can be deduced from Theorem 6.2 by a duality argument. The exact statement is as follows. THEOREM 6.3. Let {an} be a sequence of complex numbers such that
L nq-
1anlq < oo for some q, 2::;; q < oo. Then the functionf(z) is in Hq, and 2
llfllq::;; where
cq{~o(n + l)q-
2
= L:'=o anzn
lanlq} lfq'
cq depends only on q.
PROOF. Let p = qf(q- 1), and let
G(eio)
n
L ck e'ko
=
k=-n
be an arbitrary trigonometric polynomial with II GIIP::;; 1. By the M. Riesz theorem, its " analytic projection " n
g(eio) =
L ck eiko k=O
n
sn(z)= :Lak~k=O
Then
=
n
L lckl(k + l)(pk=O
2
J!Piakl(k
+ l)(q- 2 Jfq
98
6 TAYLOR COEFFICIENTS
Taking the supremum over all G with IIGIIP:::;; 1, we have
Mq(r, s,.):::;; CPAP{X0(k
+ 1)q- 2 lakiqf'q,
and the result follows by letting n--+ oo. 6.3. THE CASE p,:::;; 1
As pointed out in the corollary to Theorem 3.4, the Taylor coefficients of an H 1 function must tend to zero, by the Riemann-Lebesgue lemma. The simple example (1 - z)- 1 shows this is false for HP functions withp < l, but a sharp asymptotic estimate for the coefficients can be given as follows. THEOREM 6.4. If co
f(z) =
L a,.z" E HP,
0< p:::;; 1,
n=O
then (8)
and (9)
Furthermore, the estimate (8) is best possible for each p: given any positive sequence {~,.}tending to zero, there existsfe HP such that
a,. =1-
O(~,.ntfrt).
We have already observed that a,.= o(l) iffe H 1 . This statement cannot be improved, even iffe H"". Indeed, given a sequence{~,.} tending to zero, choose integers 0 < n1 < n2 < · · · such that ~"k < 2-k. Define PROOF.
- {k
a,.- 0 ~11k
if n = nk, otherwise.
Then L la,.i < oo, sof(z) = L a,.z" is continuous in lzl:::;; 1, yet a,. =1- 0(~,.). Now suppose f E HP, p < 1. The coefficient a,. has the representation
a,.= -21 . f 1tl
Jl•! =r
~~~~ dz,
O n 2 , ••• be a lacunary sequence of integers in the sense that nk+1/nk ~
Q > l.
Let
A. n
If nk
~
=
{l0
if n=nk otherwise.
N < nk+ 1, N
k
L n21A.nl2 = L n/ ~ nk2{l + Q-2 + ... + Q20-kl}:::;; CN2.
n= 1
j~
1
We thus obtain PALEY'S THEOREM.
If 00
f(z)= LanznEH 1, n=O then for every lacunary sequence {nd,
6.4 MULTIPLIERS
105
As a final corollary of Theorem 6.7, we now show that, more generally, {A.,} is a multiplier of H 1 into tq (2::;:; q < oo) if(and only if) it satisfies (12). Let li-n = IA.nlqf 2 , and observe that (12) gives a condition equivalent to N
L n2p./ = O(N2).
n=1
Thus {p.n} multiplies into t 2 , which implies (since the coefficients of an H 1 function are bounded) that {A.n} multiplies H 1 into tq. The multipliers from H 1 to t 1 are more difficult to describe. Hardy's inequality shows that the sequence {(n + 1)- 1 } is one example. It is possible to characterize all the multipliers, but only by a condition difficult to verify in most situations. Thus the following theorem, interesting though it may be, is really more a translation of the problem than a solution. H1
THEOREM 6.8. The sequence {A.n} is a multiplier of H 1 into t 1 if and only if there is a function 1/J E L oo such that
n
= 0,
1, 2, ....
(18)
PROOF. The sufficiency of the condition (18) was established in Theorem 3.15. Conversely, suppose {A.n} is a multiplier from H 1 to t 1. Then by the closed graph theorem, the mapping
L anzn-+ {A.n an} is a bounded operator from H 1 to t 1 : CXl
L IA.nanl::;:; Cll/1!1, n=O In particular, CXl
r/J(f)
= n=O L IA.nlan
is a bounded linear functional on H 1 • By the Hahn-Banach theorem,¢ can be extended to a bounded linear functional on L 1 • Thus by the Riesz representation theorem, there exists 1/J E L oo with 1 (f) = -
27t
-
J f(e'')!/J(t) dt, 2n o
106
6 TAYLOR COEFFICIENTS
Choosing the H 1 function f(z)
=
zn (n
=
0, 1, 2, ... ), we have
IA.nl = ¢(!) = (f)(f) = 21n
J2" einrt/l(t) - dt, 0
which becomes (18) after conjugation. EXERCISES
1. Show that Theorem 6.2 is best possible if 0 < p < I: For each positive sequence {kn} increasing to infinity, there exists L anzn E HP with
L knnp-21aniP = 00.
•
(This is also true if I :$ p :$ 2; see Duren and Taylor [I].) 2. Show that Theorem 6.2 is false for p > 2. 3. Show that the converse to Theorem 6.2 is false (0 < p :$ 2).
4. Show that Paley's theorem does not generalize to Fourier series: There existsfe L 1 with Fourier series L cne"' 9 , for which L lc 2 ,.1 2 = oo. (Suggestion: Try L (log n)-a cos ne, a> 0. See Theorem 4.5.)
5. Show thatf(z) may be analytic in lzl < I and continuous in izl :$ I, yet f'(z) have a radial limit almost nowhere. (This disproves the" Bloch-Nevanlinna conjecture" thatfe N implies/' eN. See Duren [4].) 6. A function/(z) analytic in izl < I is said to have finite Dirichlet integral (feD) if
~ lzlJJ
11 for cJ>
every extension cJ>; and, by the Hahn-Banach theorem, there is at least one extension for which 111/111 = l c/>11. In other words, for the coset of extensions of 1/1, the infimum defining the norm is attained and is equal to 111/111. PROOF OF THEOREM 7.2.
a linear functional c/> in
sl.
In terms of a given C1> E(X/S)*, one can define unambiguously by cj>(x)
= (J)(x + S).
Conversely, given c/> E sl., this relation defines a linear functional cp on X/S. The correspondence is clearly an isomorphism. We assert that the boundedness of either functional implies that of the other, and in fact l c/>11 = IIC1>11This follows from the relations IC1>(x
lc/>(x)l + S)l
= IC1>(x
+ S)l
~
= lc/>(x)l = lc/>(x
I C1>11 llx + Sli ~ I C1>11 llxll; + Y)l ~ l c/>11 llx + yll, YES.
Hence 51. and (X/S)* are isometrically isomorphic. To prove the rest of the theorem, observe first that for any c/> e 51. with l c/>11 ~ 1, and for any x EX andy e S, lc/>(x)l = lc/>(x
+ Y)l
~ llx
+ Yll·
Thus sup
,Pes~.II(x)l ~ inf llx
+ Yll·
(1)
yeS
On the other hand, given x eX, a corollary of the Hahn-Banach theorem (see Dunford and Schwartz [1], p. 65) shows the existence of C1> e (X/S)* such that
IC1>(x + S)l
= llx
+ Sil
and
I C1>11
= 1.
Now let c/> e 51. correspond to C1> as above, so that cj>(x) = C1>(x + S) and l c/>11 = I C1>11 = 1. Equality therefore holds in (1), and the supremum is attained.
112
7 HP AS A LINEAR SPACE
7.2. REPRESENTATION OF LINEAR FUNCTIONALS
As we saw in Chapter 3, HP is a Banach space if 1 :::;;, p:::;;, oo, with norm 11/11 = Mp(l,f). The polynomials are dense inHP ifO
1, the representation becomes unique if we distinguish in each coset that function 9 for which
7.3 BEURLING'S APPROXIMATION THEOREM
t
21t
e-!n9g(ei9) d() = 0,
113
n = 1, 2, ....
Equivalently, there is a unique function g E Hq for which
¢(/) =
2_ J2"f(e;e)g(e;e) d() 2n o
(3)
for all f E HP, I < p < oo. Since I < q < oo, the M. Riesz theorem (Theorem 4.1) guarantees that the" analytic projection" g of the original Lq function is in Hq (see Chapter 4, Exercise 1). In summary: THEOREM 7.3. For 1 ~ p < oo, the space (HP)* is isometrically isomorphic to IJfHq, where 1/p + lfq = 1. Furthermore, if 1 < p < oo, each ¢ E (HP)* is representable in the form (3) by a unique function g E Hq, while each¢ E (H 1)* can be represented in the form (3) by some g E L00 •
We might have chosen to put g(e- ;9) instead of g(ei9) in (3). This would have the advantage of setting up an isomorphism between (HP)* and Hq. But in either case the correspondence need not be an isometry. In fact, 11¢11 is equal to the norm of the coset determined by g(ei 9) (or by g(e-; 9)), so that only the inequality
I
0 (see Theorem 7.5). This implies that g, regarded as an element of H 2 , is orthogonal to the subspace SH 2 • Therefore, by Theorem 7.6, 00
L nYibnl 2 = n=1 unless g
00
for each y > 0,
= 0. But this contradicts Lemma 3, since g e A".
We remark that SHP is a proper subspace unless S
= l.
COROLLARY 1. If Sis a singular inner function as described in the theorem, the quotient space HPf(SHP) has no continuous linear functionals except the zero functional, for each p < l.
7.6 EXTREME POINTS
123
COROLLARY 2. If p < 1, there is a subspace M of HP and a bounded linear functional on M which cannot be extended to a bounded linear functional onHP.
The proofs are left as exercises.
7.6. EXTREME POINTS
A setS in a linear space X is said to be convex if whenever x 1 and x 2 are inS, every proper convex combination 0 0 there exists a fJ > 0 such that llxll = IIYII = 1 and llx- Yll > E imply ll(x + Y)/211 < 1- fJ. If 1
1 and A. 2 > 1 such that
II/+ A.1gll = II/- A.2gll =
1.
This is possible, by continuity, since 11/11 < 1. Then (f + A.1 g) and(/- A. 2 g) are extreme points, as shown in (i). Iff is outer, the functions ±.1711/11 are extreme points. The next theorem identifies the extreme points of the unit sphere in H 00 • By way of motivation, let us first note that f E H 00 is an extreme point if 11/11 = 1 and lf(ei 6)1 = 1 on some set E of positive measure. Indeed, if g E Hoo and II/+ gil oo = II/- gil oo = I, theng(ei 6) = 0 a.e. onE, which implies g=O. A function f E H 00 with unit sphere in H"" if and only if THEOREM 7.9.
I f I = 1 is an extreme point of the
27t
Jo PROOF. Letg
E
log(1 - lf(ei 6)1) de= - oo.
H 00 such that
II/+ gil= II/ -gil= 1.
(17)
126
7 HP AS A LINEAR SPACE
Then /g(z)l 2
:::;;
1 - lf(zW :::;; 2(1 - lf(z)l),
and it follows from (17) that
t
21t
loglg(e;~l
d() = -
oo,
which implies g = 0. Hence fis an extreme point. Conversely, if the integral (17) converges, let
J
. } 1 2 " eir + z g(z) = exp {-2 -it-log(l - lf(e'')l) dt . 1t o e -z
Then lg(z)l :::;; 1 and lg(ei 6)1
:::;; 1 -
lf(e; 6)1
a.e.
Thus II!+ ull:::;; 1 and II/- ull:::;; 1, sofis not an extreme point. EXERCISES
1'1 < 1, and for each positive integer n, = Jco is a bounded linear functional on HP, p < 1.
1. Show that for each fixed (,
¢(!)
2. Prove Corollary 1 to Theorem 7.7. 3. Prove Corollary 2 to Theorem 7.7. 4. Show that every Hilbert space is uniformly convex. 5. Show that in a uniformly convex space, every boundary point of the unit sphere is an extreme point. 6. Show thatf E Leo with 11/11 = 1 is an extreme point of the unit sphere in Leo if and only if lf(ei 6)1 = 1 a.e. 7. Show that the unit sphere in I! has no extreme points. NOTES
A. E. Taylor [2, 3] was among the first to study HP (1 :::;; p:::;; oo) as a Banach space. He represented the linear functionals (Theorem 7.3) in his paper [3]. Beurling proved Theorem 7.4 for H 2 in his fundamental paper [1]. The proof given in the text is essentially his, suitably extended to 1 :::;; p < oo. Actually, Beurling showed that every subspace of H 2 invariant under multiplication by z has the form gl'[/0 ] for some (unique) inner function fo ; and he described
NOTES
127
the lattice structure of these invariant subspaces. Equivalently, he described the invariant subspaces of the shift operator on t 2 • Subsequently, a large literature has evolved on invariant subspaces. References up to 1964 may be found in Belson [3]. Gamelin [1) has extended Beurling's theory to HP with O;e::>;O
o < e < n,
and
have identical Fourier coefficients. This implies I/J 1 (e) which is obviously impossible. (ii)
= I/J 2 (e)
a.e., so F(z) = 0,
The elementary example 1
2~
J 2n
¢(!) = -
f(e's)e-is
de,
0
shows that a normalized extremal function need not be unique. Here it is obvious that ll¢ll = l, and thatf(z) = z is an extremal function. But a simple calculation shows that for every complex constant ex, ( ) _ (z
f z -
+ ex)(1 + ciz) 1 + lexl 2
is also extremal. (iii) A more elaborate example is needed to demonstrate the nonuniqueness of an extremal kernel. In view of the main theorem, this will automatically be another case in which no extremal function exists. Let
I, k(e 18 ) =
-1,
0,
1t
O::>;e::>;2 1t
2 <e::>;n
-n < e
; I. To show that II¢ I = l, consider the function g(z) which maps lzl < I conformally onto the rectangle with vertices ± I ± iE, in such a way that g(l) = -1 - iE, g(i) = l, and g( -I)= -I + iE. Let eia. (0 <ex< n/2) be the
136
8 EXTREMAL PROBLEMS
point carried into (1 - iE). Then g(ei 0, with h(l) = -1, h(i) = 0, h( -1) = 1. Obviously, K"' k. But lh(ei 6)1 = 1, IK(e 16)1 = ( 1 + h(ew) ::;:; 1, 1-1 + h(ei6 )1::;:; 1, Hence IlK II oo
-n 0, smce R(z) ~ 0 on lzl = 1. Some of the a.; which are zeros ofF may possibly coincide with numbers {3;. However, K has a pole at each {3,, and in fact its principal part at each pole must coincide with that of k. Let the a.; be renumbered, if necessary, so that a. 1, ... , a., are the zeros of K and a.,+ 1, ... , a.a those ofF in lzl < 1. Then Ia.; I= 1 fori= u + 1, ... , n- 1. The functions -
K(z)
• 1-~z" z-{3, - - TI - = i=1z- a.; i=11-{3;z
= K(z) TI
and -
F(z)
= F(z)
a 1-ii;z TI i=s+1 Z - CY.;
are analytic and nonvanishing in tzl < 1. FE HP, while K(z) is continuous in tzl :o:; 1. Furthermore, F and K are outer functions. To prove this, it suffices
138
8 EXTREMAL PROBLEMS
to show that the product R(z) = F(z)K(z) has no singular factor. In view of the structure of R(z), however, R(z) is a rational function without zeros or poles in Jzl ~ I, except perhaps for zeros on Jzl = I. Thus the problem reduces to showing that (z-oe) is an outer function if Joel = I. But this follows from the fact that (z- oe)- 1 E HP for p < I. Consequently,
F(z)
= e'Y. exp {-1
J
2x
eir
+z
.
}
(9)
-i-r-logJF(e'r)l dt ,
2n o e - z
and similarly for K(z). The relations (8) between F and K are yet to be used. Suppose first that 1
k.
(6)
156
9 INTERPOlATION THEORY
Consequently, co,z-z., ~ Il co(l-cn)2 > 0, Il 1 - z i zk 1 1+ C 1
k
j= 1
--n
n=
j'¢k
which shows that {zk} is uniformly separated. Now suppose 0 ::;; z 1 < z 2 < · · · and k = 1, 2, ....
Then n
= 1, 2, ... ,
so that 1 - zn+ 1
::;;
~ + zn 1 - - - ::;; (1 1 + ~z,
~)(1
- zn).
Thus {zn} satisfies (5), as claimed. Sequences {zn} which satisfy the condition (5) are called exponential sequences. 9.5. A THEOREM OF CARLESON
Part of the proof of the main interpolation theorem consisted in showing that if {zn} is uniformly separated and/ E HP (0 < p < oo), then
In other words, if JL is the discrete measure on lzl < 1 defined by n = 1, 2, ... ,
then
JJzl
lf(z)IP dp(z) < oo s},
s>O.
For the proof, we define for each E > 0 the sets A,' ={z:
t lqJ(t)l
dt > s(E
+ llzl)}
and
Observe that the
B.' = {z: /z c sets B.' expand
for some
/w
as
E ____.
E, =
0 and
U B.'.
0
wE
A.'}.
9.5 A THEOREM OF CARLESON
161
Hence (13)
JJ.(E,) =lim Jl.(B:). ..... o
If zn e
A: and the arcs r.,. are disjoint, then I (e + II."I) < I J lt
log+lcp'(re1'11 d()
:: 0 such that exp{- aF(z)} is the derivative of a functionf(z) which maps lzl < l conformally onto a Jordan domain, if and only if 11 E A*. The boundary of this domain is rectifiable if and only if 11it) is nondecreasing and exp{ -2naw(t)} is integrable. REMARK. This theorem shows, in particular, that the construction of a Jordan domain with rectifiable boundary, the derivative of whose mapping function is a singular inner function alone (as in the Keldysh-Lavrentiev example), is equivalent to the problem of constructing a singular, nondecreasing, bounded function /l(t) of class A*. This latter construction can be carried out directly. (See Notes.) PROOF OF THEOREM.
Set J(z) =
r 0
exp{- aF(()} d(,
178
10 H• SPACES OVER GENERAL DOMAINS
so that - {f(z), z} = aF"(z)
a2
+2
[F'(z)] 2 •
Let us first observe that there exists a number a > 0 such thatf(z) maps lzl < 1 conformally onto a Jordan domain, if and only if F'(z)
=
o(-r
1 )· 1-
(15)
Indeed, if(15) holds, then F"(z) = 0((1 -r)- 2 ), and the inequality (13) can be achieved by a suitably small choice of a. Conversely, if f(z) is univalent in lzl < 1, it must satisfy the elementary inequality l
zf"(z) f'(z)
-~~ 0}.
It turns out that the spaces HP(D) and EP(D), as defined in Chapter 10, do not coincide in this case, and in fact EP(D) is properly contained in HP(D). It is natural also to consider the space f)P (0 < p < oo) offunctionsjanalytic in D, such that lf(x + iy)IP is integrable for each y > 0 and (
IJJlP(y,f) =
CXl
lf_oolf(x + iy)IP dx
}1/p
is bounded, 0 < y < oo. f>oo will denote the space of bounded analytic functions in D. Eventually it will turn out that f)P = EP(D), but the general theory of Chapter lO is not entirely applicable because the boundary of D is not rectifiable. It is possible to develop the theory of f>P by mapping the half-plane onto the unit disk, but this approach runs into difficulties because the lines y = y 0 are mapped onto circles tangent to the unit circle. Mainly to deal with this problem, or rather to avoid it, we begin with some lemmas on subharmonic
188
11 H• SPACES OVER A HALF- PLANE
functions in D. We then discuss boundary behavior, factorization, and integral representations of f>P functions, basing most of the proofs on known properties of HP functions in the disk. The chapter concludes with the Paley-Wiener theorem, in which a Fourier transform plays the role of the Taylor coefficients. 11.1. SUBHARMONIC FUNCTIONS
LEMMA 1. If g(z) ~ 0 is subharmonic in the upper half-planeD and
f'-oo
g(x
+ iy) dx :::;:;
M,
y > 0,
then g(z) :::;:; 4Mj3ny, PROOF.
Fix z 0 = x 0
+ iy 0 (y 0 >
Z =X+
iy.
0), and map D onto the unit disk by
z- z 0
w=--. z - z0
Then G(w) =
is subharmonic in
lwl < 1, so by the mean value theorem, G(O):::;:;
where w
= u + iv.
g(z 0 )
=
g(Zozo w) 1- w
~ JJ
np lwlD [(x- Xo)2 + (y + Yo)2]2 x 4y 0- 2 J'YJ Joo (y n o - oo
:::;:; -
d
y
+ y 0 )- 4 g(x + iy) dx dy:::;:; -34M -. nyo
LEMMA 2. If a subharmonic function g(z) satisfies the hypotheses of Lemma 1, then it has a harmonic majorant in D.
11.2 BOUNDARY BEHAVIOR
PROOF.
189
Map D onto lwl < 1 by z-i
z
w=l/J(z)=-.;
z+z
= qJ(w) = i(1 + w). 1- w
(1)
The line y = b then corresponds to the circle Cb with center b( I + b) - 1 and radius R =(I+ b)- 1 • By Lemma I, G(w) = g(qJ(w)) is bounded inside Cb, so it has a least harmonic majorant Ub(w) there. If a< b, it is clear that Ua(w) ~ Ub(w) for each w inside Cb. Thus by Harnack's principle (see Ahlfors [2], p. 236) and a diagonalization argument, lim Ua(w) = U(w),
lwl < 1;
=
and U(w) is a harmonic majorant of G(w) unless U(w) oo. Hence u(z) = U(l/l(z)) is a harmonic majorant of g(z) if U(w) ¥:- oo. To show that U(w) ¥:- oo, let rb be a circle concentric with Cb and having radius p < R. Let Vb(w) be the Poisson integral of G over rb. Then, in particular,
vbC: b)= 2~p
(2)
Jr.G(w)ldwl.
But asp --. R, Vb( w) --. Ub( w) inside Cb; so it follows from (2) and the bounded convergence theorem that
ub(1+b b) = -2nR1- fc.G(w)ldwl < 2(1 +b) -
J
11:
1+bf :::;;-11:
G(w) ldwl c. 11- wl 2
00
( "b)d x:::;; (1+b)M . gx+z
11:
-oo
As b--. 0, this shows U(O):::;; Mjn. Thus U(w) ¥:- oo, and the proof is complete. 11.2. BOUNDARY BEHAVIOR
The following theorem is an immediate consequence of Lemma 2. THEOREM 11.1. COROLLARY.
If 0 < p < oo and f
E ~P,
then f
E
Iff E ~P, then the boundary function f(x) = limf(x y->0
+ iy)
HP(D).
190
11 HP SPACES OVER A HALF-PLANE
exists almost everywhere, f
E
LP, and
d J-oooo loglf(x)l X>1 +X 2
(3)
00.
PROOF OF COROLLARY. As in the proof of Theorem 10.3, the existence (more generally) of a nontangentiallimitj(x) = limz_,xf(z) follows from the fact that F(w) = f(qJ(w)) is in HP, where qJ is the mapping (1). Fatou's lemma shows lf(x)IP is integrable over (- oo, oo). Finally, (3) follows from the fact (Theorem 2.2) that
Jlwl=
logiF(w)lldwl > - oo. 1
It is also true that f(x + iy) tends to f(x) in the U mean. Before showing this, it is convenient to prove a Poisson integral representation for f) 1 functions and a factorization theorem analogous to that of F. Riesz (Theorem 2.5). THEOREM 11.2.
f(z) =
If jE f)P, 1 ~ p
n1 J<Xl-
y
CXl
Conversely, if hE U (1
~
(x- t)2
p
~
~
+ y2f(t) dt,
1t
E
Z =X+ iy.
(4)
oo) and
1 J<Xl f(z) = (
is analytic in D, then f
oo, then
-oo
X-
t
)y2
+ y2
lz(t) dt
f)P and its boundary function f(x) = h(x) a.e.
PROOF. Since F(w) = f(qJ(w)) is in HP (1 ~ p ~ oo), it has a Poisson representation (Theorem 3.1)
F(w)
{eio w} F(e'~ de.
1 2" + Re - 16- 2n o e - w
=-
J
This gives (4) after a change of variable and a straightforward calculation. The converse is obtained from Jensen's inequality. COROLLARY.
If jE f)P, 1 ~ p < lim y--+0
00,
then
Joo lf(x + iy)IP dx = - 00
r -
lf(x)IP dx. 00
(5)
11.2 BOUNDARY BEHAVIOR
191
PROOF. Applying Jensen's inequality to (4), we have
f'
lf(x
+ iy)IP dx:::;:;
-oo
f'
(6)
y >0.
lf(x)IP dx,
-oo
This together with Fatou's lemma gives the result. THEOREM 11.3. If jE f;)P (0 < p:::;:; oo) and f(z) =f. 0, then f(z) = b(z)g(z), where g is a nonvanishing f;)P function with lg(x)l = lf(x)l a.e., and
~)m f] +I n
b(z) = (zZ
lz,.: Zn
+ 11 . z+1 Z-
Zn
(7)
Zn
is a Blaschke product for the upper half-plane. Here m is a nonnegative integer and z" are the zeros (z" -:f. i) off in D, finite or infinite in number. Furthermore, "'
L... n
Yn 1 + IZn 12 < oo,
(8)
PROOF. According to Theorem 2.5,
.f(qJ(w))
= B(w)G(w),
where B is a Blaschke product in the disk and G E HP has no zeros. If we define b(z) = B(l/l(z)), the expression (7) follows from the corresponding formula for B; and (8) is equivalent to L (1 - ll/l(z")l) < oo. It remains to show that the nonvanishing function g(z) = G(l/J(z)) is in f;)P. But by Theorem 11.2, or rather by its proof, [g(z)]P is the Poisson integral of its boundary function; hence gP E f;) 1 • COROLLARY 1.
Jf jE f;)P, 0 < p < oo, then (5) holds .
PROOF. Since gP
E
...
f'
lf(x
f;} 1,
+ iy)IP dx:::;:;
-oo
lg(x
+ iy)IPdx-+
-oo
COROLLARY 2.
half-plane y
f'
~ ~
f'
lg(x)IP dx
-oo
=
f'
1/(x)IP dx.
-oo
Iff E f:JP, 0 < p < oo, then f(z)-+ 0 as z-+ oo within each
> 0.
PROOF. Since lf(z)IP:::;:; lg(z)IP, it is enough to prove this for f;) 1 functions. But each f E f> 1 has a Poisson representation of the form (4). Given e > 0, choose T large enough so that -T
J
-oo
oo
lf(t)l dt
+J
T
lf(t)l dt <e.
192
11 HP SPACES OVER A HALF-PLANE
Then (4) gives 1
Jr
e
y
lf(z)l .:s;( )2 2 lf(t)l dt + ~ n -rX-t +y nu
=
o(_!_) +~ lzl nD
(lzl
~ oo).
It is now a short step to the theorem on mean convergence. THEOREM 11.4. If fe ~P (0
< p < oo), then
lim{' lf(x y->0
PROOF.
+ iy)- f(x)IP dx = 0.
-00
Apply Corollary 1 above and Lemma 1 in Section 2.3.
A further application of the factorization theorem shows that 9Jlp(y,f) is a nonincreasing function of y iff e ~P. This is expressed by the following theorem. THEOREM 11.5. If fe ~P (0
< p < oo) and 0 < y1 < y 2 , then
IDlp(y1,J);;::: IDlp(Y2 ,f). PROOF. Let fi(z) = f(z + iy 1). Then / 1 e ~P, so it has the factorization fi = b1g 1 as in Theorem 11.3. But since g 1P e ~ 1 , an application of (6) gives
ID1.p{y2- Y1 •. ft)
:5;
IDlp(y2- Y1, U1)
:5;
IDliO, U1)
= IDlP(O,f1),
which proves the theorem. The hypothesis f e
~P
is essential, as the example f(z) = e-iz(i + z)-2fp
shows. Here 'lR/(y,f) = neY(y
+ 1)- 1 ,
which increases to infinity withy. 11 .3. CANONICAL FACTORIZATION
The factorization/= bg given in Theorem 11.3 can be refined as it was in the case of the disk (Theorem 2.8) to produce a canonical factorization for ~P functions. The space HP(D) will be considered first.
11.3 CANONICAL FACTORIZATION
THEOREM 11.6.
Each function f
E
193
HP(D), 0 < p < oo, has a unique factor-
ization of the form
f(z) where oc
~
= eiazb(z)s(z)G(z),
(9)
0, b(z) is a Blaschke product of the form (7),
s(z)
+ tz dv(t) } {J --=-
= exp
oo
i
-00
1 t
(10)
z
for some nondecreasing function v(t) of bounded variation over (- oo, oo) with v'(t) = 0 a.e., and G(z) = eiy exp{_!_ ni
Joo -oo
(1 + tz) log w(t) dt} (t- z)(1 + t 2 )
for some real number y and some measurable function w(t)
J
oo
-00
log w(t)
----=----=,..:. > 1 + t2
and
00
J
oo
-oo
[w(t)]P -1--2
+t
(11) ~
dt
(1/J(z)) has the form (II), with w(t) = lf(t)l. The properties (12) of w follow from the properties logjF(eil)l E L 1 and F(eil) E £P. Finally, taking into account the possible jumps of Jl at 0 and at 2n, we have log S(I/J(z)) = i
J -1 +-tz dv(t) + ia.z, oo
-oo
t- Z
where v(t) = p(arg{l/l(t)}) and
a. = p(O+) - p(O) + p(2n) - p(2n-)
~
0.
Conversely, iff is an arbitrary function of the form (9), then lf(z)IP::;; IG(z)IP = exp{! Joo y log[w(t)]P dt} n - oo (x - t)2 + y2
1 Joo y[w(t)]P P (I:::;:; p < oo), then
f(z)
=~ 2nz
f'-oo
Im{z} = y > 0;
f(t) dt,
t- z
and the integral vanishes for all y < 0. Conversely, if hE U (1 :::;:; p < oo) and
~ J<Xl
== 0,
lz(t) dt
0 this integral represents a functionfE f>P whose boundary function f(x) = h(x) a.e. PROOF.
If fE f)P, the Cauchy integral F(z)
= ~ Joo
-oo
2nz
f(t) dt
t- z
is analytic in both of the half-planes y > 0 andy < 0. According to (14), it is related to the Poisson integral by the identity F(z)- F(z)
= -1 J<Xl ( 1t
-oo
y X-
t)
2
+ y2
/(t) dt.
Therefore, in view of Theorem 11. 2, F(z)
=
y >0.
F(z)- f(z),
In particular, F(z) is analytic for y > 0, so F(z) must be identically constant in the lower half-plane. But since F(z)--. 0 as z--. oo, the constant is zero. Thus F(z) = f(z) in y > 0 and F(z) = 0 in y < 0, which was to be shown. The converse follows immediately from Theorem 11.2. 11.5. FOURIER TRANSFORMS
We come now to the Paley-Wiener theorem, a half-plane analogue of the fact that H 2 is the class of power series with lanl 2 < oo. For functions analytic in the upper half-plane, the Fourier integral
La" z"
f(z) =
L
Joo tt• F(t) dt 1
(15)
0
plays the role of a power series. Before stating the Paley-Wiener theorem, we recall a few facts about Fourier transforms of L 2 functions. If fEL 2 , its Fourier transform is defined as ](x)
1 JR = l.i.m.-
R->oo
2n
-R
.
e-"'''i(t) dt,
196
11 H• SPACES OVER A HALF-PLANE
where" l.i.m." stands for" limit in mean" in the L 2 sense. It is a theorem of Plancherel that] exists, 11/11/ = 2nl!]ll/, and R
f(t) = l.i.m. R-+oo
J
ei"1(x) dx.
-R
If g is another L 2 function with Fourier transform g, the Planche rei formula is
f'
J(t)O(t) dt
=
-oo
f'
(16)
](t)g(t) dt.
-oo
THEOREM 11.9 (Paley-Wiener). Afunctionf(z)belongsto~ 2 ifand only
if it has the form (15) for some FE L 2 • PROOF. Iff has the form (15) with FE L 2 , it is analytic in the upper halfplane, as an application of Morera's theorem shows. For fixed y > 0, the functionfy(x) = f(x + iy) is the inverse Fourier transform of t~O
t < 0.
Hence
f'-oo
lf(x
+ iy)l 2 dx = 2n
f'e- 2 Y1/F(tW dt 0
t
,<X)
:::;:; 2n
IF(t)l 2 dt < oo,
showing thatjE ~ 2 • Conversely, each fin ~ 2 is the Cauchy integral of its boundary function, by Theorem 11.8: f(z)
= ~ Joo f(t) dt,
y >0.
2nz -oot-z
(17)
But
Joo e-it~eiz~ d~ = (j
1 = __!,_ 2ni(t- z) 2n o
where u(~) = eiz~ for ~ ~ 0 and u(~) formula combined with (17) gives f(z)
=
= 0 for
Joo ]Wu(~) d~ = -oo
which proves the Paley-Wiener theorem.
~
'
< 0. Thus the Plancherel
Jooeizq(~) d~, 0
EXERCISES
197
COROLLARY. If fE .f> 2 and J is the Fourier transform of its boundary function, then]W = 0 for almost all ~ < 0.
PROOF. By Theorem 11.8, the Cauchy integral (17) vanishes for all y < 0. But if y < 0,
1 2ni(z- t)
=
2_ 2n
Jo-oo e-it~eiz~ d~,
and we find as before that 0
J
-oo
eizq(~) d~
=0,
y
< 0.
Jn particular, 0
J
-oo
e 2 ~1](~)1 2 d~ = 0,
which proves Jc~) = 0 for almost all ~ < 0. The argument can be generalized to give a similar representation for .f>P functions, 1 ::;; p < 2. We shall content ourselves with a discussion of f) 1 • If JEL 1 , its Fourier transform 1 JCX) • f(x) = e-u:'i(t) dt 2n - 00 A
is continuous on - oo < x < oo, and](x)--. 0 as x--. ± oo. If also g E L 1 , the formula (16) is a simple consequence of Fubini's theorem. Thus the proof of the Pa1ey-Wiener theorem can be adapted to obtain the following result. THEOREM 11.10. If/E .f> 1 and] is the Fourier transform of the boundary function, thenJ(~) = 0 for all ~ ::;; 0 and
f(z) =
Joo eizq(~) d~,
y > 0.
0
EXERCISES
.f>P if and only if f(tp(w))[ql(w)] 11P E HP, 0 < p < oo. Hence show that .f>P = EP(D), where D is the upper half-plane. [Suggestion: Use the canonical factorization theorems (Theorems 11.6 and 11.7).]
1. Show that
.fE
2. Show that EP(D) is properly contained in HP(D) if D is the upper halfplane.
198
11 HP SPACES OVER A HALF-PLANE
3_ Give an example of a function f(z) which is analytic in a half-plane y > - ~ (~ > 0), with f(x) E I}, but which is not the Cauchy integral of f(x). 4. Let p(t) be a complex-valued function of bounded variation over (- oo, oo), such that
f'
eixt dp(t) = 0
for all
x > 0.
-oo
Show that dp is absolutely continuous with respect to Lebesgue measure. 5. Prove the half-plane analogue of Hardy's inequality (Section 3.6): If
f E f) 1 and .f is the Fourier transform of its boundary function, then
r 0
l](t)l dt::;; t
t J<Xl -
lf(x)l dx.
CXl
(Hille and Tamarkin [3]. See Exercise 4 of Chapter 3.) 6. For fE f)P, 0 < p < oo, prove
f
lf(x
0
+ iy)IP dy ::;;f J<Xl IJCxW dx. -oo
(This analogue of the Fejer-Riesz theorem is due to M. Riesz [1].) 7. Show that if b(z) is a Blaschke product in the upper half-plane, then 11m y-->0
Joo -oo
loglb(x + iy)l d _ 0 X. 2 1 +X
Conversely, show that if f(z) is analytic in y > 0, lf(z)l < 1, and . I 1m y-->0
Joo -oo
loglf(x + iy)l d _ 0 X- , 2 1 +X
thenf(z) = ei(y+a•>b(z), where y is a real number, ex;;::: 0, and b(z) is a Blaschke product (Akutowicz [1]).
NOTES
Most of the results in Section 11.2 are due to Hille and Tamar kin [3], who considered only 1 ::;; p < oo. They proved the key result ~P c HP(D) using a lemma of Gabriel [2] on subharmonic functions. But in order to apply Gabriel's lemma, it must first be shown that a function/ E f)P tends to a limit as z---. oo within each half-plane y ;;::: ~ > 0. Hille and Tamarkin were able to show this only after a difficult argument proving the Poisson representation
NOTES
199
(Theorem 11.2) from first principles. Kawata [l] extended the Hille-Tamarkin results to 0 < p < l. The relatively simple approach via harmonic majorants, as presented in the text, is due to Krylov [1]. Krylov also obtained the canonical factorization theorems of Section 11.3. Theorem 11.9 is in the book of Paley and Wiener [11. For proofs of the Plancherel theorems and other information about Fourier transforms, see Goldberg [l ]. Theorem 11.10 is due to Hille and Tamarkin [l]; see their papers [2, 3] for further results. Kawata [l] proved theorems on the growth of 9Rp(y,f) analogous to those of Hardy and Littlewood for the disk.
This Page Intentionally Left Blank
THE CORONA THEOREM
CHAPTER 12
The purpose of this final chapter is to give a self-contained proof of the "corona theorem," which concerns the maximal ideal space of the Banach algebra H"'. After describing the result in its abstract form, we show how it reduces to a certain" concrete" theorem. Here the discussion must presuppose an elementary acquaintance with the theory of Banach algebras. However, the proof of the reduced theorem (which occupies most of the chapter) uses purely classical methods, and makes no further reference to Banach algebras. 12.1. MAXIMAL IDEALS
Let A be a commutative Banach algebra with unit, and let .A be its maximal ideal space, endowed with the Gelfand topology. In other words, the basic neighborhoods of a point M* E .A have the form 1111
= {ME ..H: lxk(M)- xk(M*)l < e,
k
= I, ... ' n},
where e > 0, the xk are arbitrary elements of A, and xk is the Gelfand transform of xk. That is, :X(M) = cPM(x), where cPM is the multiplicative linear functional with kernel M. Now let
Yk
= xk -
xk(M*)e,
202
12 THE CORONA THEOREM
where e is the unit element of A. Then Yk EM* (since Yk(M*) the equivalent form dlt
= {ME ..It: l.Yk(M)J < e,
= 0), and V/1 takes
k = 1, ... , n}.
It is well known that ..1{ is a compact Hausdorff space under the Gelfand topology. Associated with each fixed point (, J(J < I, the Banach algebra H"' has the maximal ideal M{
= {!E H"' :f(() =0}.
The problem arises to describe the closure of these ideals M{ in the maximal ideal space .It of H"', under the Gelfand topology. Are there points in .# which are outside this closure? To put the question in more picturesque language, does the unit disk have a "corona"? As it turns out, the answer is negative. CORONA THEOREM. The maximal ideals M,, J(J 0, we then have
f(z)
{1 = exp-
J -;-log + lf(e' )1 dtJ. 2
"
elt
1
z
2rc o e - z
.1
'
12.2 INTERPOLATION AND THE CORONA THEOREM
205
Now let wk =e2 "ikf• be the nth roots of unity (k = 1, ... , n), and let 1
{ f.(z) = exp-
n
L n
k=l
Wk + z } - - l o g lf(wk)l . wk- z
Elementary considerations show thatf.(z)---. f(z) uniformly in S. Let 1 ek = -- loglf(wk)l, n
so that 1 0 ::;:; ek ::;:; - - log p. n
= !5.,
say, where p. is the minimum of 1/(z)l on lzl = 1. Choosing n so large that !5. < t, let and define
Note that lakl = 1 if ek = 0, so that the corresponding factor in B.(z) is trivial. A calculation gives n
2logiB.(z)l = -2(1 -lzl 2 )
L1 ekl1- akzl- 2 + 0(!5/),
k=
uniformly in S. From this it follows that logiB.(z)l
= loglf.(z)l + 0(!5.),
uniformly in S. Hence logiB.(z)l --.loglf(z)l, which implies IB.(z)l --.lf(z)l, uniformly in S. Since B.(O) > 0, it also follows (by analytic completion of the Poisson formula) that B.(z) --.f(z) uniformly in each disk lzl::;:; r 0 < 1. Using Lemmas 2 and 3, we can now carry out the proof of the corona theorem. In fact, the argument will give Theorem 12.1 in the following sharper form. THEOREM 12.2. Let/1, ... ,j~ be H 00 functions with 11/kll ::;:; 1 (k = 1, ... , n)
and l/1(z)l
+ · · · + l.f..(z)l;;::: !5,
lzl < 1,
(2)
where 0 < t5 0}.
Suppose E does not divide the plane, and that the Dirichlet problem is solvable for D = H - E. Let aE denote the boundary of E, and let E*
=
{ilzl : z
E
E}
be the circular projection of E onto the positive imaginary axis. Let w(z) be the harmonic measure of E with respect to D. In other words, w(z) is the bounded harmonic function in D for which w(iy) = 0 (- oo < y < oo) and w(z) = 1 for z E aE. Finally, let
*
w (z)
1
f
=n p
x2
X
dt
+ (y -
Z =X+ iy,
t)2,
be the harmonic measure of E* with respect to H. For (x +iy) ED,
LEMMA 4 (Hall's Lemma).
w(x
+ iy) ;;::: 1w*(x -
ilyJ).
(5)
PROOF. Suppose first that E consists of a finite number of radial segments
k=1,2, ... ,n;
JOkl 0;
(6)
Re{z} > 0.
(7)
and U(z)
p
X::;:; p.
Now let 1/J(p) denote the total length of the part of E which lies in the disk J(- zl < p. Since 1/J(p) ::;:; 2p and M(p) is a decreasing function, integration by parts gives 1 Joo U(z)::;:; 2n o M(p) dljJ(p)
=-
1 Joo 2n o 1/J(p) dM(p)
n 2 3 ::;:; - -1 Joo pdM(p)=-1 Joo M(p)dp=-+- 0,
y < 0.
210
12 THE CORONA THEOREM
Hence by symmetry,
+ iy);;::: j-w*(x- iy),
w(x
X> 0, y > 0.
This proves the lemma for the special case in which E is a union of radial segments with nonoverlapping projections. For a general compact set E, choose e > 0 and consider the set S,
= {z: w(z) >
1 - e}.
Clearly, 8E c S,. Choose a set E c S, which consists of a finite number of radial segments with nonoverlapping projections, for which E* = E*. (To see that this is possible, cover E by open disks in S, and apply the Heine-Bore! theorem.) Let w(z) be the harmonic measure of E. By what has just been proved, w(x + iy) ;;::: j-w*(x - i lyl), since E* = E*. But the function [w(z)- w(z)] vanishes on the imaginary axis, is ;;::: 0 on 8E, and is ;;::: - e wherever it is defined on E. Thus by the maximum principle, w(x
+ iy) + e;;::: w(x + iy)
;;::: j-w*(x- 1iyl)
for (x + iy) E D. Now let e--. 0, and the lemma is proved for compact sets E. Finally, suppose E is closed but unbounded. Let E, be the intersection of E with the disk lzl ::;;; r, let E,* be its circular projection, and let w,(z) and w,*(z) denote the respective harmonic measures. Then w(x
+ iy);;::: w,(x + iy);;::: j-w,*(x- i/yl)
for each point (x + iy) E D. But it is clear from the integral representations that w/(z)--. w*(z) pointwise as r--. oo. This completes the proof. Hall's lemma will now be applied to obtain a special result needed in the proof of Carleson's lemma. Let R be the annulus p < lzl < 1, and let E 1 be a closed subset of R which does not divide the plane. Let w 1 (z) be the harmonic measure of £ 1 with respect to (R- EJ, and let El* = {eio: reio EEd
be the radial projection of E 1 onto the outer boundary of R. For fixed {3 < ref Ilog pi, let F 1 * be the part of E 1 * such that 101 :..:; flllog pl. Then the total length IF 1 *I of F1 * can be estimated as follows. LEMMA 5. If p 113
¢£1 ,
IF1 *1:..:; Ilog pl[cosh(rc{3/2)] 2 w 1 (p 1 ' 3 ).
12_4 CONSTRUCTION OF THE CONTOUR
r
211
PROOF. The multiple-valued function
C= ~ + i1] = iz-hr/log P = e"9flog P exp{in(!- log')} log p
2
maps R onto the right half-plane Re{C} > 0. Let z valued) inverse, and let
= cp(O denote the (single-
E= {(:cp(()EEIJ. Then E is a closed (unbounded) subset of H, and Hall's lemma may be applied. The harmonic measure of E with respect to D = H - E is w(() = w1(cp(()). Since cp maps the circular projection E* of E onto £ 1 *,the harmonic measure of E* with respect to H is w*(O = w 1*(cp(()), where w 1*(z) is the harmonic measure of £ 1* with respect to R. Thus Hall's lemma gives w 1 (cp(~
+ i1J))
: 1-. let S be the residual set of highest generation which contains z 2 . Then z2 E snk for some leading rectangle Rnk of s. If z1 ¢ snk' then vnk c a separates z1 from z2. If z1 E snk' then z1 E sml(j) for some leading block
12.5 ARCLENGTH OF
r
215
Rm 1(j) of Snk. But then the grating Gm 1(j) belongs to A and separates z1 from z2 , since z 2 belongs to no residual set of generation higher than that of S. Thus A separates .910 from !Ji 0 . Now let A be the set formed from A by deleting all interior points of .910 and of !Ji 0 . It is clear that A still separates .910 from !Ji 0 . Let n be the union of all the components of the complement of Awhich meet .910 , and let r be the boundary of n. Then .9/(e) c .910 c n; and flio II n = 0' since A separates .910 from flio. Consequently, e::;; IB(z)l ::;; er
a 2 , ••• be complex numbers such
that
L lanl 2 < oo. Then the series
(4) converges almost everywhere. Equivalently, the theorem says that for any square-summable sequence {an}, the series ±an converges for almost every choice of signs.
L
PROOF OF THEOREM. Let sn(t) = L~= 1 ak (/Jk(t). By the Riesz-Fischer theorem, there is an LJ. function ci>(t) such that
f lci>(t)- sn(t)1 1
lim n-+oo
2
dt = 0.
0
In particular, ci>(t) is integrable and (by the Schwarz inequality)
f sn(t) dt f ci>(t) dt, fJ
lim n-+oo
fJ
=
Cl
a:
05,~ m. Thus for
n > m.
In view of (5), we conclude that 0 = lim 11-+ 00
JPm [sn(t)- sm(t)] dt = JPm [(t)- sm(t)] dt. 11m
11m
Hence, because sm(t) is constant on (txm, flm), 1 sm(to) = flm _
t:Xm
JPm «m (t) dt ____. (t 0 )
as
m--. oo.
Thus {sn(t)} converges almost everywhere. If {1/ln(x)} is an orthonormal system in L 2 [a, b], and if oo, then for almost every choice of signs {en}, the series
COROLLARY.
L lanl 2
0)
forces L En an z" to be continuous in fzf ~ 1 for almost every choice of signs. For the proofs we must refer the reader to the literature (see Notes). NOTES
More information on Rademacher functions and related questions can be found in the books of Alexits [1], Kaczmarz and Steinhaus [1], and Zygmund [4]. Paley and Zygmund [1, 2] used Rademacher functions to prove various theorems in function theory. Proofs of the assertions in the last paragraph (above) may be found in Paley and Zygmund [1]. Theorem A.5 is due to Littlewood [4].
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MAXIMAL THEOREMS
APPENDIX B
The Hardy-Littlewood "maximal theorem" becomes clearer if it is stated first in discrete form. Let a 1 , a 2 , ••• , an be given nonnegative numbers, and let b 1 , b 2 , ••• , bn be the same numbers rearranged in nonincreasing order. For each fixed k (k = 1, ... , n), let 1 k rxk = max . L a; 1 s.is.k k- J + 1 i=j be the optimal average of successive a; terminating with ak; and let 1 flk =k
k
L b; i=l
be the corresponding quantities for the b;. Let s(x) be any nondecreasing function defined for all x ~ 0. Then, as Hardy and Littlewood showed, n
n
k=l
k=l
L s(rxk) ::;; L s(flk). Hardy and Littlewood interpreted the a; as cricket scores and s as the batsman's "satisfaction function." The theorem then says that the batsman's
232
APPENDIX B. MAXIMAL THEOREMS
"total satisfaction" is maximized if he plays a given collection of innings in decreasing order. In order to state a cdntinuous version of this theorem, it is necessary to define the "rearrangement" of a function. Let f(x) be nonnegative and integrable over a finite interval [0, a], and let 11-(y) be the measure of the set in whichf(x) > y. Note that 11-(y) is nonincreasing. Two functionsf1 (x) and f 2 (x) are said to be equimeasurable if they give rise to the same function 11-(y). It is then clear from the definition of Lebesgue integral that
If 11-(y) is associated as above with f(x), its inverse function f*(x) = 11-- 1 (x), normalized so thatj*(x) = f*(x+ ), is called the decreasing rearrangement of f(x). It is easy to see thatj*(x) andf(x) are equimeasurable. Finally, let A(x, ~)
1
X
= A(x, ~; f) = -->= J f(t) ~
X- 0. Since G(x) =
~
(g(:t) dt,
the continuous form of Minkowski's inequality gives
{f0aiG(x)IP dx
}1/P 5 a1 foa {foa Ig (Xt)IP a dx }1/p dt
= _P_{Jaig(u)IP du}1/p. p- 1 0
Now let a-+ oo to obtain Hardy's inequality. The next theorem is essentially a restatement of Theorem B.2 in a form convenient for certain applications (see Section 1.6). THEOREM 8.3. Letf(x) be periodic with period 2n, and supposefe U = U(O, 2n), 1 < p < oo. Then
F(x) =
sup O