Table Of ContentBorn in Erevan in 1953. Graduated from the Erevan State University in
1975. Received his Ph. D. (кандидат физ.-мат. наук) in 1981 from the
Leningrad University. Works at the Institute of Mathematics of the
Armenian Academy of Sciences in. Erevan.
Fields of research: complex .and harmonic analysis•
Dr. Faizo A. Shamoian
Born in Erevan in 1945. Graduated from the Erevan State University in
1966. Received his Ph. D.(кандидат физ.-мат. наук) in 1970 and his
D. Sc. (доктор физ.-мат. наук) in 1983, both in Leningrad. Works at
the Institute of Mathematics of the Armenian Academy of Sciences in
Erevan.
Fields of research: complex and harmonic analysis.
Djrbashian, Ashot E.:
Topics in the Theory of A
Faizo K. Shamoian. - 1. A
199 S.
(Teubner-Texte zur Mathem
NE: Shamoian, Faizo A..: ;
ISBN 3-322-00495-3
ISSN 0138-502X
©BSB B. G. Teubner Verla
1. Auflage
VLN 294-375/70/88 LSV 1
Lektor: Dr. rer. nat. Ren
Printed in the German Dem
Gesamtherstellung: Typodr
Bestell-Nr. 666 475 0
021 0C
TEUBNER-TEXTE zur Mathematik Band 105
Herausgeber / Editors: Beratende Herausgeber /Advisory Editors:
Herbert Kurke, Berlin Rüben Ambartzumian, Jerevan
Joseph Mecke, Jena David E. Edmunds, Brighton
Rüdiger Thiele, Leipzig Alois Kufner, Prag
Hans Triebei, Jena Burkhard Monien, Paderborn
Gerd Wechsung, Jena Rolf J. Nessel, Aachen
Claudio Procesi, Rom
Kenji Ueno, Kyoto
Ashot E. Djrbashian Faizo A. Shamoian
Topics in the Theory of A«Spaces
For the first time in mathematical literature the fundamentals of the
theory of area integrable analytic functions are presented in the book.
This theory arose in the mid-forties in the works of M. M. Djrbashian
and is developing rapidly during the last two decades. By now it be
came an important branch of complex analysis and has many applications
in various problems of analysis. Along with well-known results the book
contains also many new achievements in this field received in recent
years.
1
In diesem Buch werden die Grundlagen der Theorie flächenintegrierbarer
analytischer Funktionen erstmals in geschlossener Form dargestellt.
Die Theorie ging Mitte der 40er Jahre aus Arbeiten von M. M. Djrbashian
hervor und entwickelte sich während der letzten 20 Jahre stürmisch.
Mittlerweile ist sie ein bedeutender Zweig der komplexen Analysis mit
vielen Anwendungsmöglichkeiten auf Probleme der Analysis geworden.
Neben bekannten Resultaten enthält das Buch auch eine Reihe neuerer
Ergebnisse der letzten Jahre.
Pour la première fois dans littérature mathématique, les bases de la
théorie des fonctions ^analytique intégrables sur un domaine sont pré
sentées dans ce livre. Cette théorie est née vers les années 45 dans
les travaux de M. M. Djrbashian et s’est dêvellopêe rapidement durant
les deux dernières décades. En l’état actuel des choses, elle est de
venue une branche importante de l’analyse complexe et a beaucoup d’ap
plications dans divers problèmes d’analyse.
En parallèle avec des résultats bien connus, le livre présente de nou
veaux résultats de ce domaine reçus dans les dernières années.
Е книге впервые в математической литературе представляются основы
теории аналитических функций, интегрируемых по площади. Эта теория
возникла в середине сороковых годов в работах М. М. Джрбашяна и быстро
развивается в последние два десятилетия. И настоящему времени она пре
вратилась в важную ветвь комплексного анализа со многими применениями
в анализе. Вместе с хорошо известными результатами в книге представ
лены и новейшие достижения в этой области, полученные в последние
годы.
2
CONTENTS
page
Introduction 5
Chapter 1• Spaces of area integrable functions
in the unit disk
1*1. spaces of analytic functions 13
1.2. Integral representation for the class 15
1.3« The class A¿ 19
1.4. Further results 29
Chapter 2. Bounded projections, conjugate harmonic
functions, and bounded linear functioneds in A^
2.1. Bounded linear projections to A^ 33
2.2« Conjugate heLrmonic functions 36
2.3. The duals of aJ spaces 43
2.4. Further results 49
Chapter 3- Zeros of a£ functions
3.1. First results on zeros of a£ functions 54
3.2. Unions of a£ zero sets. Zero sets of
different a£ spaces 60
3*3. Division by Blaschke-type products. Subsets
Of a£ zero sets 64
3.4* Further results 70
Chapter 4. On zeros of analytic functions with
restricted growth
4.1. On M.MoDjrbashian's infinite products 75
4.2. Characterization of zeros of functions of
the class 85
3
page
4.3« On the zeros of analytic functions with
majorants of infinite order 93
4*4. Further results 99
Chapter 5. Division in and description of
closed ideals
5.1. Division in classes . The case ¿y > 1 101
5.2. Division in classes X^°. The case olvf < 1 110
5*3* Closed ideals of the algebra X^5 117
5.4« Further results 125
Chapter 6. aJ spaces of holomorphic functions
in higher dimensions
6.1. Integral representations for A^ spaces in
the unit ball and the polydisk of cn 128
6.2. Traces of functions from the class Hp(Dn),
0<p^.oo , on the diagonal of the polydisk 136
6.3« Diagonal mappings in spaces A^ 144
6.4« Representation of continuous linear functionals
in A^(Dn) spaces 147
6.5. Estimates for derivatives of HP(Dn) functions 154
6.6. Further results 159
Chapter 7. A^ spaces of harmonic functions
in the unit ball of Rn
7.1. Definitions and first results 162
7.2. Estimates for the kernel 167
7.3. Bounded projections and conjugate spaces 174
7.4. The rate of growth of fractional derivatives
and integrals of A^ functions 182
7.5. Further results 188
References 191
4
INTRODUCTION
The theory of Hardy spaces of analytic functions in the unit
disk is by now classical. However we begin by recalling some aspects
of that theory.
Suppose the function f(z) is analytic in the unit disk D =
= ^z: \z\<L and denote
Mp(f,r) = ( ^ y i l f(reie>lp de )1/p
for 0< P<CÖ and M^f^) = sup lf(z)( •
izi^r
The function Mp(f,r) is called the p'th integral mean of the
function f. Further, f belongs to the Hardy class Hp if and only
if sup M(f ,r) < oo , 0< p < oo .
0.sr<1 p
Closely related to the Hardy spaces is the Nevanlinna class N
of analytic functions satisfying the condition
sup — \ log+ If (reie)( d6c oo •
0<r<1 2ft Jjyi '
The fundamental property of all Hardy classes and the class N
(which contains Hp, 0«£p<oo) is the existence of boundary values,
i.e., any function from the class N (hence, also from Hp) has non
tangent ial boundary values.
The next property which we must mention here is the Poisson in
tegral representation formula: If the function f belongs to Hp,
1^ p < oo , then it has the representation
f(z) = [ Pr(9-vf) fie1*) dv^, z = re10 ,
-Tv
where Pr is the Poisson kernel and we have denoted the boundary
values of f by the same symbol.
The functions of the Nevanlinna class possess the following
factorization:
f(z) = B(z) exp(g(z)),
where B(z) is the Blaschke product formed by the zeros of the
5
function f and g(z) is a function constructed via the boundary
values of, f.
A natural step further from the theory of spaces is the
following. Suppose the integral mean Mp(f,r) is unbounded and
hence we leave the boundaries of Hardy space. What other natural
conditions must be imposed on that integral mean to receive a
reasonably rich class of analytic functions? One possibility is to
suppose that is integrable. Moreover, we may assume that
belongs to some with respect to some finite measure. Hence, we
will say that an analytic function f belongs to the class ,
0 < p, q-s. oo , if
^0 (Mp(f,r^ q d^(r)coo,
and dj* is a finite measure on [0,1).
However we will not use this general definition in what follows
and will restrict ourselves to the simpler "model" case when p * q
and d = (1-r)4 .rdr, o and will denote that class by
A^ , 0 < p<¿ oo , -1 celcoo . The reason is that the case p jí q adds
nothing new to the model case and although consideration of the case
of general measures for some cases gives principally new situations,
we prefer simplicity and richness of spaces A¿ •
Historical background. We may define the classes A^ also as
follows
\ |i(z)|P(1-lZ()^ dm2(z)< oo, (1>
JD
where dm^ is the plane Lebesgue measure. Hence, A$ is the class
of all analytic functions in the unit disk D which belong to the
class I? with respect to the measure (1- |z\ )otdm2(z)• In other
words, a£ is an area class of analytic functions. Of course, it is
almost impossible to find out who was the first who considered area
classes of analytic functions in D. The earliest result known to us
is in L.Bieberbach*s 19M paper [ll, where the space A^ = Aq was
present and the formula
t(z) = i \ — tSPl. dnl („) (2)
A (1-zi)2 2
6
for functions of that class was actually proved (see also J.L.Walsh
[1}, pp.322-328) .
Later on, in the 30's and 40* s the space A2 appeared many times
in the works of SoBergman and N.Aronszajn (not only in the unit disk
but also in other one- and multidimensional domains) in connection
with their theory of reproducing kernels (see S.Bergman (1,2j and
N.Aronszajnll ,2l) .Using that theory they in particular proved again
2
the formula (2) for A functions. However, their techniques were
based on the fact that this class is a Hilbert space and hence they
did not receive any results even in the case p ^ 2. Moreover, there
were no results in the weighted case.
The one thing we can claim definitely and with full responsibi
lity is that the first work where the classes a£ in the form (1)
appeared was M.M.DJrbashian*s 1945 short announcement [1^ and the
subsequent full publication [2^ in 1948. Here are the main results
found by him in these papers (for full and correct formulations see
Chapter 1)•
If an analytic function belongs to the class A^ in the unit
disk D, 1<p<oo,4?l>-1, then it has the integral representation
f(z) = ^ l (1-|w\2)* dm (w). (3)
X JD (1-zw)2^ 2
Following R.Nevanlinna, M.M.Djrbashian considered also the
generalization of the class N which we will denote by A . : a
*
function f belongs to the class A^ if
[ log+|f(z)l (1- \z[f dnu(z)<£ oo .
JD *
Obviously, C. A^ for any p and , 0<p^/oo, -1 < <1 < oo .
In the above mentioned papers [ll and [2} the following fundamental
factorization result was also established:
If f tA^ then it has the factorization
f(z) = Jl^z) exp(g(z)) , (4)
where JV* is a specific infinite product formed by the zeros of
the function f, and g contains the values of f in the disk in
its definition.
We know very well how important are the rcisscn representation
and factorization formulas in the theory of Hardy and Nevanlinna
spaces. Similarly, the formulas (3) and (4) play the same role in
the theory of and spaces. Moreover, as we will see, they
are the main working tools in almost of all our considerations and
constructions •
Unfortunately M.M.Djrbashian never returned to his results until
the most recent years and these important achievements remained un
known to the wide mathematical community. It was much later when a
new wave of interest in these classes arose and only beginning from
the early 70*s an increasing number of papers began to appear. And
this was perhaps the reason why in many recent publications the
spaces A^ are called "Bergman" or "weighted Bergman" spaces. We
consider this situation as fully unacceptable and in some of our
papers we have touched upon this problem of priority. However we
find it appropriate to return again to this problem here hoping that
the facts which we gave above will solve it once and for all.
In view of what has been said it is natural to call the spaces
A^ M.M.Djrbashian spaces and A^ the Nevanlinna-Djrbashian space.
Of course, we have no objection against the term "Bergman ker
nel" as a substitute for the term "representing or reproducing ker
nel". However, we want to stress again and for the last time, that
S .Bergman never considered the classes a£ in his works and hence
there is no scientific reason to call them "Bergman spaces"•
Description of the results. Our book is formed as follows. It
consists of seven chapters and conditionally may be devided into
two parts. The first three chapters may be called the "textbook"
part since there we have collected in some sense classical results,
and the remaining four chapters form the"advanced" part, where we
have presented more recent results.
The first chapter begins with some simple considerations and
then we formulate and prove the above mentioned two fundamental re
sults by M.M.Djrbashian: the integral representation (3) and the
factorization formula (4) for functions of the Nevanlinna-Djrba-
shian space.
The second chapter is about bounded projections, conjugate har
monic functions, and the duals of A*J spaces. Here we see many
contrasts with the case of Hardy spaces. The classical M.Riesz theo
rem states that the operator of "harmonic conjugacy" is bounded in
lP if 1<p<oo. This means also that the natural projection from
(on the boundary of the unit disk) to H^ is bounded under the
8
