Table Of ContentTHEORY OF HpS PACES
Peter L. Duren
Department of Mathematics
University of Michigan
Ann Arbor, Michigan
Academic Press
New York and London
1970
COPYRIGH0T 1970, BY ACADEMIPCR ESSI,N C.
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TO MY FATHER
William L. Duren
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 HPc lasses 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-Rogosinski-
Shapiro-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 tech-
“ “
niques 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
of functional 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 appropri-
ate 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 bibliog-
raphy and proofread the entire book. Renate McLaughlin’s help with the
proofreading was also host valuable.
In addition, I am grateful to the Alfred P. Sloan Foundation for support
during the academic year 19641965, 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, UniversitC 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 Ad-
vanced 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
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 sub-
harmonic 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
2 1 HARMONIC AND SUBHARMONIC FUNCTIONS
provide one measure of growth and lead to a particularly rich theory with
broad applications. A functionf(z) analytic in the unit disk JzI < 1 is said
to be of class HP( 0 < p Ia )i f Mp(r,f)r emains bounded as r -+ 1. Thus H"
is the class ofc bo unded anacly tic functions in the disk, while H2i s the class of
power series a,z" with la,12 < 00.
It is convenient also to introduce the analogous classes of harmonic func-
tions. A real-valued function u(z) harmonic in IzI < 1 is said to be of class
hp (0 < p 5 a)i f Mp(r,u ) is bounded. Since
+ +
ap5 (a b)P5 2P(ap bp), a 2 0, b 2 0,
for 0 < p < 00, an analytic function belongs to HP if and only if its real and
imaginary parts are both in hP. The same inequality shows that HP and hP
are linear spaces. Finally, it is evident that HP3 H4 if 0 < p < q I 00, and
likewise for the hP spaces.
Any real-valued function u(z) harmonic in IzI < 1 and continuous in 121 I 1
can be recovered from its boundary function by the Poisson integral
u(z) = u(reie)= -1 2nP (r, 0 - t)u(e") dt, (1)
2n 0
where
I-r2
P(r, 0) = +
1 - 2r cos 8 r2
is the Poisson kernel. Now replace u(e") in the integral (1) by an arbitrary
continuous function q(t) with q(0) = ~(27~T)h.e resulting function u(z) is
still harmonic in IzI <1, continuous in IzI 1, and has boundary values
u(e") = q(t).G eneralizing this idea, one is led to the notion of a Poisson-
Stieltjes integral. This is a function of the form
1
u(z> = u(reie>= -1 2=P (r, e - t)d p(t), (2)
2n
0
where p(t) is of bounded variation on [0,2x]. Again, each such function is
harmonic in IzI < 1.
THEOREM 1 .I.T hefollowingthreeclassesoffunctionsinI zI < 1a reidentical:
(i) Poisson-Stieltjes integrals;
(ii) differences of two positive harmonic functions;
(iii) h'.
The proof is based on the HeIly selection theorem, which we now state for
the convenience of the reader. (For a proof, see Natanson [l] or Widder [I].
Also, see Notes.)
1.1 HARMONIC FUNCTIONS 3
LEMMA (Helly selection theorem). Let (p,(t)} be a uniformly bounded
sequence of functions of uniformly bounded variation over a finite interval
[a, b]. Then some subsequence (pJt)} converges everywhere in [a, b] to a
function p(t) of bounded variation, and for every continuous function cp(t),
PROOF OF THEOREM 1.1. (i)* (ii). Expressing p(t) as the difference of
two bounded nondecreasing functions, we see that every Poisson-Stieltjes
integral is the difference of two positive harmonic functions.
(ii) * (iii). Suppose u(z) = ul(z) - u2(z), where u1 and u2 are positive
harmonic functions. Then
j2 n + 2n
lu(reie)l d8 4 ul(reiBd) 8 u2(reie)d B
0 0
+
= 2nL-u,(O) UZ(O)l,
so that u E h'.
(iii) 3 (i). Given u E h', define
Then pr(0) =0, and for 0 = to < t, < ... < t, = 2n,
Hence the functions pr(t) are of uniformly bounded variation. By the Helly
selection theorem, there is a sequence {r,} tending to 1 for which prn(t)- + p(t),
a function of bounded variation in 0 I t 5271. Thus
8 - t)u(r, ei') dt = lim u(r, z) = u(z).
n+ al
(Here, as always, z = reie.)
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 p(t). This is usually called the Herglotz representation.
The function p(t) of bounded variation corresponding to a given u E h' is
4 1 HARMONIC AND SUBHARMONIC FUNCTIONS
essentially unique. Indeed, if P(r, 8 - t) dp(t) = 0, analytic completion gives
where y is a real constant. Since
we conclude that
joz'ein' dp(t) = 0, IZ = 0, 5 1, 52, .
Since the characteristic function of any interval can be approximated in I,'
by a continuous periodic function, hence by a trigonometric polynomial,
this shows that the measure of each interval is zero. Thus dpis the zero measure.
1.2. BOUNDARY BEHAVIOR OF POISSON-STIELTJES INTEGRALS
If u(z) is the Poisson integral of an integrable function p(t), then for any
point t = 8, where p is continuous, u(z) + cp(f3,) as z +eioO.T his can be
generalized to Poisson-Stieltjes integrals : u(z) +p'(8,) wherever p is continu-
ously differentiable. Actually, it is enough that p be differentiable ; or, slightly
more generally, that the symmetric derivative
exist, as the following theorem shows.
THEOREM 1.2. Let u(z) be a Poisson-Stieltjes integral of the form (2),
where p is of bounded variation. If the symmetric derivative Dp(8,) exists at
a point B0, then the radial limit Iim,.+[ u(reieD)e xists and has the value
DP(80)-
PROOF. We may assume 8, = 0. Set A =Dp(O), and write
1 rlt
1
1 " 1
- - [p(t) - At] - P(r, t) dt.
2n
-z Kt
1.2 BOUNDARY BEHAVIOR OF POISSON-STIELTJES INTEGRALS 5
The integrated term tends to zero as r + 1. For 0 < 6 I It1 ln,
Hence for each fixed 6 > 0, u(r) - A - I, + 0, where
Given E >0, choose 6 >O so small that
Then
for r sufficiently near 1, as an integration by parts shows. Thus u(r) + A as
r + 1, 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' has a radial limit almost everywhere.
COROLLARY 2. If u is the Poisson integral of a function p EL', then
u(reie) q(8) almost everywhere.
4
By a refinement of the proof it is even possible to show that u(z) tends to
Dp(Bo) 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 < CI < n/2, construct the sector with vertex eie, of angle 2a, symmetric
with respect to the ray from the origin through eie. Draw the two segments
from the origin perpendicular to the boundaries of this sector, and let SJe)
denote the kite-shaped region so constructed (see Fig. 1).
" "
