Functions of a-Bounded Type in the Half-Plane (Advances in Complex Analysis and Its Applications)

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Author: Armen M. Jerbashian

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Functions of a-6ounded Type in the Half-Plane

Advances in Complex Analysis and Its Applications Volume 4

Series Editor: C.C. Yang The Hong Kong University ofScience& Technology, Hong Kong

Advisory Board: Walter Bergweiler Kiel University, Germany George Csordas University of Hawaii, U.S.A. Paul Gauthier University of Montreal, Canada Phillip Griffiths Princeton, U.S.A. Irwin Kra State University of New York, U.S.A. Armen G. Sergeev Steklov Institute of Mathematics, Russia Wolfgang Tutschke University ofGraz, Austria

Functions of a-Bounded Type in the Half-Plane

by ARMEN M. JERBASfflAN Institute of Mathematics National Academy of Sciences, Armenia

Springer

Library of Congress Cataloging-in-Publication Data A CLP. Catalogue record for this book is available from the Library of Congress. ISBN 0-387-23625-2

e-ISBN 0-387-23626-0

Printed on acid-free paper.

© 2005 Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline. com

SPIN 11161851

CONTENTS

PREFACE

vii

INTRODUCTION

ix

CHAPTER 1 THE LIOUVILLE OPERATOR AND HERGLOTZ-RIESZ TYPE THEOREMS 1. The Liouville integro-differentiation 2. Herglotz - Riesz type theorems 3. Proofs of Theorems 2.3 and 2.4

1 14 17

CHAPTER 2 BLASCHKE TYPE PRODUCTS 1. 2. 3. 4.

Elementary properties of factors Integro-differential properties of factors Blaschke type products A representation of the product

21 25 33 39

CHAPTER 3 EQUILIBRIUM RELATIONS AND FACTORIZATIONS 1. F. and R. Nevanlinna, Carleman and Levin type formulas 2. Some other formulas of Carleman and Levin type 3. Factorizations

45 55 65

CHAPTER 4 MEROMORPHIC FUNCTIONS WITH SUMMABLE TSUJI CHARACTERISTICS 1. Levin type formula and Tsuji type characteristics 2. Canonical factorizations 3. Descriptive representations

77 82 90

CHAPTER 5 BOUNDARY VALUES 1. Main results 2. Proofs of Theorems 1.1 and 1.2 3. Proof of Theorem 1.3

101 103 112

CHAPTER 6 UNIFORM APPROXIMATIONS 1. Main results 2. Proofs of Theorems 1.1, 1.2, 1.3

121 123

CHAPTER 7 SUBHARMONIC FUNCTIONS WITH NONNEGATIVE HARMONIC MAJORANTS 1. Main results 2. Representation in the semi-disc 3. Proofs of Theorems 1.1 - 1.4

129 134 141

CHAPTER 8 WEIGHTED CLASSES OF SUBHARMONIC FUNCTIONS 1. Green type potentials 2. Subharmonic functions with nonpositive Liouville primitives 3. The growth of functions from Sa

147 152 155

CHAPTER 9 FUNCTIONS OF o^-BOUNDED TYPE IN SPECTRAL THEORY OF NON-WEAK CONTRACTIONS 1. 2. 3. 4.

Factorization of regularized determinants Boundary behavior of regularized determinants 6p-perturbations of self-adjoint operators Completeness of operators in Cp

159 171 177 183

REFERENCES

189

INDEX

195

VI

PREFACE

This book is related to the theory of functions of a-bounded type in the halfplane of the complex plane. I constructed this theory by application of the Liouville integro-differentiation. To some extent, it is similar to M.M.Djrbashian's factorization theory of the classes Na of functions of a-bounded type in the disc, as much as the well known results on different classes and spaces of regular functions in the half-plane are similar to those in the disc. Besides, the book contains improvements of several results such as the Phragmen-Lindelof Principle and Nevanlinna Factorization in the Half-Plane and offers a new, equivalent definition of the classical Hardy spaces in the half-plane. The last chapter of the book presents author's united work with G.M. Gubreev (Odessa). It gives an application of both a-theories in the disc and in the half-plane in the spectral theory of linear operators. This is a solution of a problem repeatedly stated by M.G.Krein and being of special interest for a long time. The book is proposed for a wide range of readers. Some of its parts are comprehensible for graduate students, while the book in the whole is intended for young researchers and qualified specialists in the field. I am grateful to my father Prof. Mkhitar M. Djrbashian who in 1980's led me into the field of representations of regular functions. I am very glad to express my gratitude to Prof. Cabiria Andreian Cazacu and Prof. Chung-Chun Yang for encouraging me to write this book. Also, I am thankful to my wife Gohar Jerbashian for her valuable assistance in refining the text of the book.

ARMEN M. JERBASHIAN

INTRODUCTION

1. The fundamental role of integral representations and factorizations in complex analysis is well known. In this field, the basic results of Poisson, Cauchy, Weierstrass, and Hadamard where continued by outstanding achievements of Hardy, Blaschke, Fatou, Herglotz, F. and M. Riesz, Sego, R.Nevanlinna, V.I. Privalov and other classics of the contemporary complex analysis. These achievements partially were summarized in the widely known monographs of R. Nevanlinna [84], I.I.Privalov [86, 87], L.De'Branges [14]. A series of further results is due to Paley-Wiener [85], Hille-Tamarkin [47], V.I.Krilov [76], B.Ja. Levin [77], M.Tsuji [106] and others. The investigated non-weighted classes of functions mainly were connected with the following three classical problems, which depend on the specific geometry and technical apparatus of the disc and half-plane of the complex plane. Problem I. Describe some classes of regular functions, the growth of which is restricted by a rotation-invariant condition that does J^Q^ distinguish the boundary points of the unit disc, The unit disc on the Riemann sphere ^^ considers the whole boundary as a single point. The growth condition P2TT

sup / 0
\f{re'^)fdi}<-\-oo

defining Hardy's spaces H^ {0 < p < +CXD) of functions holomorphic in |z| < 1 is an example of such restriction. THE GLOBAL PROB Another example is the condition defining Nevanlinna's LEM IN THE DISC class of meromorphic functions of bounded type in the disc or a similar class of subharmonic functions possessing nonnegative harmonic major ants in \z\ < 1: /»27r

/ iJo 0
u+ (re''^)di? < +00

(w+ = maxfu, 0}).

(1)

Problem II. Describe some classes of functions which are assumed to be in a sense regular even on the boundary of the half-plane, except the point CXD, while the growth of these functions is restricted only near oo. Nevanlinna conditions defining a subclass of subharmonic functions possessing nonnegative harmonic majorants in the upper half-plane (i.e. providing Nevanlinna factorization in

INTRODUCTION The upper half-plane on the Riemann sphere

the upper half-plane) present an example of such restrictions: -t-oo

0

:dx < +00,

/

-co

liminf ^

RJo

THE LOCAL PROBLEM IN THE HALF-PLANE

l + x-

u-^{Re'^)smM^<-{-oo

(2) (3)

{u{z) is assumed to be subharmonic in a domain containing the closed finite half-plane Im z > 0). Another example is the Phragmen-Lindelof principle, where both quantities (2) and (3) are assumed to be zero.

P r o b l e m I I I . The growth of functions is restricted by a shift-invariant condition distinguishing only the point oo, while the finite points of the boundary are considered equivalently, i.e. the boundary is conThe upper half-plane on the Riemann sphere sidered as two points, oo and all finite boundary points. This is the similarity of the global Problem I in the 0 disc. The growth condition defining Hille-Tamarkin's [47] Hardy spaces in Im z > 0 is such an example: + CXD

sup y>o

/

\f{x + iy)\^dx < +00. -OO

THE GLOBAL PROBLEM

Another condition of such type defines V.LKrilov[76]E.D.Solomentsev's [103] classes 91 and 9 1 ^ of subharmonic functions with bounded Tsuji characteristics in Im 2: > 0: n-\-oo -I-00

/

u'^[x -h iy)dx < -hoc, -00

(4)

-foo

at"

sup ?/>0

/

\u{x -f iy)\ dx < +CXD. -00

One can note the differences between the analytic apparatus peculiar to the disc and to the half-plane. Particularly, representation as a Laplace transform in the half-plane corresponds to the Taylor expansion in the disc. 2. Further development of representation apparatus was necessary for applications in different fields of science. Particularly, the apparatus was extended to different weighted classes of functions regular in different senses. The canonical representations of weighted classes of holomorphic and meromorphic functions, established by M.M.Djrbashian [16] in 1945 were among first results in the field. His results actually revealed a new way of application of fractional integ-

INTRODUCTION

xi

ro-differentiation - in the representation problem of weighted classes of regular functions. One can find a detailed survey of these and other methods of fractional integro-differentiation in the monograph of S.G.Samko, A.A.Kilbas and O.I.Marichev [92]. The initial work [16] (see also [17] for detailed proofs and additional results) mainly aimed at improving Nevanlinna's result [84, Sec. 216] related to description of the density of zeros and poles of the weighted class of functions meromorphic in the disc \z\ < 1, the Nevanlinna characteristics of which satisfy the rotation-invariant condition

I

1

(l-r)^T(r,/)(ir <+oo

(5)

for a given — 1 < a < +oo. Namely, Nevanlinna has shown that the zeros and poles of functions from his weighted class (5) satisfy the modular condition ^(l-Kir+2<+oo,

J2i^-\b,\)-+^<+oo.

(6)

In [16] the following canonical factorization was found for meromorphic functions from Nevanlinnna's class (5):

F(z) = X exp^ ^ - ^

II

,2^a l0g|i^(C)l

(1 - IClT-^-^l^^daiO

y \z\< 1,

(7)

where Cx is the first non-zero coefficient of the Laurent expansion of F{z) at z — 0, ka is a. definite constant, {a^} and {bj^} are zeros and poles of F{z), and TTa are some Blaschke type products with factors of the form exp

r .1 ^i_tr^ 1 iicp(i-fi)"

=-1

1-Cz

Q

Note that later the same products were considered also by Tsuji (see [106, Ch. IV]) for integers a = 0,1,2, One can easily observe the Riemann-Liouville fractional integro-differentiation in the definition and in the structure of the canonical factorization (7) of Nevanlinna's class (5). Using that factorization apparatus, F.A.Shamoian [94] found some descriptive factorizations of the class (5) and particularly could prove that Nevanlinna's modular condition (6) completely describes the density of zeros and poles (and hence of any a-points) of functions from the mentioned class. One can find a series of further results related to these weighted classes and, particularly, a complete modular description of the zero-sets of more general classes in [95, 97, 99].

xii

INTRODUCTION

The papers [16, 17] also contain an investigation of the weighted Hardy type classes H^{a) (later known as A^) of functions holomorphic in \z\ < 1, the growth of which is restricted by the condition (1 — |CI)^l/(C)r^^(C) < +^^ {da is the Lebesgue area measure) / / ICKi JJ\c for given a > —1 and p > 1. Particularly, the widely known representations of functions f{z) G A^ were first found. The origins of investigations related to AQ can be found in a work of L.Biberbach [9] and in other references of [107], where the works [13, 73, 109, 43, 8] relating to ^ Q ^^^ even more general spaces are to be especially mentioned. For the later developments in the topics, see the monographs [15], [45] with their reference lists and [95, 97, 98, 100, 101, 90, 91, 68]. 3. Later the Riemann-Liouville fractional integro-differentiation was used [18], [19, Ch. IX] for construction of the factorization theory of the wider than (5) classes N^ {—l
D--u{re'')^^ r(Q^) Jo

D''u{re'^)=uire'^), D'^uire'^) = ^

{L)-(^-^)ix(re'^)} ,

(where 0 < a < +00 and p is the natural number deduced from p—l
xiii

INTRODUCTION

In assumption that f{z) is meromorphic and n{r, f) is the quantity of its poles in 1^1 < r, consider the following ce-characteristic functions and a-kernels: Nc.{rJ)^-f^ [\r-tr[n{tJ)-n{0,f)]^ l{l + a) Jo 1

m^{r,f)^—j

t

+

:^^[logr-k.], l[l + a)

4-

/•^'^

[D-^\og\f{re''>)\YM,

T^ir,f) =

mc,ir,f)+N„{r,f),

where ka = a X ] ^ i [ ^ ( ^ + ^)]~^' The class N^ (—1 < a < H-CXD) of functions of a-bounded type in |z| < 1 is defined by the condition sup Ta{rJ)

< 4-00,

0
and the following main theorem is true. Theorem 1. The class Na (—1 < a < -f-oo) coincides with the set of functions representable in the form /(z) = e ^ ^ + ^ ' = « z ^ | ^ | | ^ e x p | i - £ ' ' 5 „ ( e - ' ' z ) # w } , |^| < 1,

(9)

where j is a real number, A is an integer, ilj{'d) is a function of hounded variation in [0, 27r]; and Ba{z]a^) =YlAa{z]a^),

Boc{z]hy) -= J|Aa(2:;a^),

A,(z, 0=^^p|-/' W i-A+^- (f)

(10)

X

are convergent Blaschke type products vanishing in zeros {a^} C {\z\ < 1} and poles {bj^} C {\z\ < 1} of f{z), which satisfy the density conditions ^ ( 1 - |a^|)i+° < +00,

^ ( 1 - |6,|)i+" < +00.

Becoming the classical Nevanlinna factorization for a = 0, the above representation describes a wide scale of classes of meromorphic functions, such that A^a C A^ for - 1 < a < 0, and iV C A^^, for 0 < o; < +oo. The classes NQC (—1 < a < 0) which lie in AT have good boundary properties. This was shown in [25 - 30], where particularly was established that the exceptional set of functions from NQC C AT, on which no nontangential boundary

xiv

INTRODUCTION

values F{e^'^) — lim^_^et^ ^{.^) ^^^ly exist, is more delicate than the exceptional set of the class N (i.e. sets of zero Lebesgue measure). Namely, this set is of zero (1 + a)-capacity in Frostman's [33] sense. Theorem 2. 1°. / / F(z) G AT^, ( - 1 < a < 0), then the limit F{e'^) = lim F{z) exists and is finite for all d G [0,27r] with possible exception of a set £^ C [0, 27r] of zero (1 + a)-capacity. 2°. Let F{z) e Na ( - 1 < a < 0) 25 holomorphic and F{e''^) = 0 on a set E C [0,27r] of zero Lebesgue measure. If E is of positive {l-\-a)-capacity, then F{z) = 0 in \z\ < 1. 4- The above results in the disc relate to solution of Problem I for several classes of functions. Whereas, the construction of a factorization theory in the half-plane may relate to solution of Problems II and III for weighted classes of functions. This was done in author's works [49 - 56, 58 - 60, 69, 70]. The papers [65, 66, 67, 68] relate to Problem I while [61] and [71] deal with Problems I and III. This book presents several chapters of the author's Dr. of Sci., Math. Thesis to which only the Sections 1, 2 and 4 of Chapter 9 from author's joint work with G.M.Gubreev [40, 41] are added. The author is thankful to Prof. S.G.Samko for his remarks to Thesis, which are taken into account in this book. The book consists of 9 chapters. Chapter 1 ([55, 58]) starts by some main properties of the Liouville fractional integro-differentiation, which are used all over the book. These properties are used for establishing some similarities of the Herglotz [46] - Riesz [88] theorem on the descriptive representation of the class R of holomorphic functions having nonnegative real parts. The new theorems describe several classes of functions holomorphic in a half-plane, the real parts of which keep their sign after application of Liouville's integro-differential operator in the imaginary variable. For a = 0 the representations of these classes coincide with some well-known results of R.Nevanlinna [81] (see also [102, Ch. II], [2, Ch. Ill]) and I.S.Kats [72] (see also Addendum I in [4]) on subclasses of R, of functions representable by Cauchy integral. Chapter 2 ([49, 54, 59]) contains the construction and investigation of a family of Blaschke type products in the lower half-plane G~ = {w :1m w < 0}. These products are used as the main tool in the further factorizations. Chapter 3 ([55]) is devoted to solutions of Il-nd and Ill-rd Problems for some sets of functions holomorphic in a half-plane, which are larger than the classical ones. Using Liouville's integro-differentiation, a factorization theory of weighted classes of functions holomorphic in a half-plane is constructed. Depending on continuous a G (—1, +oo), for o; == 0 the obtained factorizations in essence coincide with the well known factorizations of R.Nevanlinna [83] and V.I.Krilov [76]. Chapter 4 ([56]) establishes canonical factorizations of some weighted classes of functions meromorphic in the half-plane, the Tsuji characteristics of which

INTRODUCTION

XV

are weighted-summable, similar to Nevanlinna's condition (1). Like [94], descriptive representations of the considered classes are established and the density of zeros and poles of functions of these classes is completely described. C h a p t e r 5 ([61]) is devoted to investigation of boundary properties of meromorphic functions which are of a-bounded type in the disc or in the half-plane. The main results in the half-plane in a sense are similar to Frost man's [33] theorem on the boundary properties of Blaschke products with "rare" zeros. Besides, a theorem on the boundary properties of Blaschke-M.M.Djrbashian product (8) in the disc improves the results of M.M.Djrbashian-V.S.Zakarian [28, 29]. C h a p t e r 6 ([60]) mainly is devoted to uniform approximation with arbitrary sharpness in the classes of meromorphic functions of a-bounded type in the half-plane. These approximations are done by finite Blaschke type products of Ch. 2 and, in a sense, are similar to the well-known theorem of S.Schur [93], stating that any function bounded and holomorphic in the disc (or in the half-plane) is uniformly approximated by finite Blaschke products. Besides, a necessary and suflficient growth condition is found, under which a function of ce-bounded type in the half-plane is a Blaschke type product. C h a p t e r 7. It is well known that the classical conditions (2)-(3) and (4) of Problems II and III describe only the growth of some subclasses of subharmonic functions possessing nonnegative harmonic majorants in Im z > 0. This was the reason for finding some general conditions [57, 62, 63] improving Nevanlinna factorization in the half-plane by description of all functions of bounded type in the half-plane. This led to solution of a new, IV-th problem, which is in essence a "symbiosis" of Problems II and III. Problem IV. Describe some classes of functions regular in Im z > 0, the growth of which is assumed to be restricted by a "global" (shift-invariant) and a "local" conditions simultaneously. It appears that a function u{z) subharmonic in the upper half-plane Im z > 0 has a nonnegative harmonic major ant in Im z > 0 if and only if simultaneously ~ I upper half-plane /'+^ . ^ dx the Riemann sphere hmmf / u'^[x + iy)z—,—5^ /

-R

u~^(x + iy) ^^1

dx :- < +00 +• X^-^

and

CX3

GLOBAL-LOCAL

liminf^ / R^+00 R JQ

u-^(Re''^)smM'd<+oo.

In particular, this result contains the classical Nevanlinna factorization [83] in the half-plane (see also [10]). Besides, Nevanlinna's uniqueness theorem and the Phragmen-Lindelof type

xvi

INTRODUCTION

theorem holding from a result due to L.Ahlfors - M.Heins [1] are improved by means of replacement of the traditional condition limsup

u{z)<0,

—oo < t < + 0 0 ,

z-¥t, Im z>0

by a less restrictive integral condition. In addition a new, equivalent definition of the classical Hardy spaces over the half-plane is found and a theorem on some weighted H^ classes is proved. C h a p t e r 8 ([64]): the methods of the previous chapter are used to investigate some weighted classes of subharmonic functions possessing nonnegative harmonic majorants in a half-plane after application of Liouville's fractional integration. Riesz type representations and complete growth characterization are established. In some natural sense, these growth characterization and representations completely describe all subharmonic functions, the Liouville primitives of which have nonnegative harmonic majorants in the half-plane. The considered classes of subharmonic functions are generalizations of the classes of functions of a-bounded type. C h a p t e r 9 gives the author's joint work with G.M.Gubreev [40, 41], which reveal a new way of application of the factorization theory in the disc and some of the results of Chapters 7 and 8 in the spectral theory of linear operators. The application of such results is of long standing interest for operator theory specialists. For example, soon after the monograph [19] was published, M.G.Krein mentioned at Moscow Mathematical Congress the necessity to find spectral interpretations of its results. Later, M.S.Livsic [79] and his pupils (L.Kh.Mehrabyan, Do Kong Khan and others) realized meromorphic functions of classes Na as transfer functions of some special linear systems in a series of works. However, the applications found in [40, 41] may turn to be more reasonable.

CHAPTER 1 THE LIOUVILLE OPERATOR AND HERGLOTZ-RIESZ TYPE THEOREMS

1. THE LIOUVILLE INTEGRO-DIFFERENTIATION 1.1. Let a generally complex-valued function U{v) be measurable on the ray —00 < V < d {d < +oo). Assuming that the below integral is absolutely convergent for almost all w G (—oo, d), for a > 0 we set W-"U{v)=-^

r r^

iv-tr-'U{t)dt /

Tic .a) Jo

a''-^U(v-ia)da.

(1.1)

U{v) is said to be of the class LQ,(—oo,(i) (0 < a < +00), if for any e > 0 (1 + |t|)"-^t/(t) e L i ( - o o , d - e).

(1.2)

Note that for any ai and a2 (0 < a i < a2 < +00) La2{~oo,d) C Lai(—oo,d).

(1.3)

Lemma 1.1. 1°. IfU{v) E La(—00, d) for some a G (0, +00), then the integral W~^U{v) is absolutely convergent for almost allv G (—oo,(i) and it represents a function summable on any finite interval (a^b) C (—oc,(i). 2°. If ai, a2 e (0,+oo) and U{v) e La,^a2{-oo,d), La2{-oo,d).

then

W-'^'lUiv)] G

Proof. 1°. Let - 0 0 < a < b < d. Then nb

I

oh

\W-'^U{v)\dv

Ja

 0 0 < (6 - t ) " - (a - t)^ < C(l + |t|)"-^ and

( 6 - t ) " < C ( l + |t|)^-i

if if

- oo < t < a, a
Therefore rb

[

\W-^U{v)\

dv < -^—-

/

(1 4- \t\r-'\U{v)\dt

< +00.

2°. We shall use the well-known formula f\y 7*2

_ fi)"'~'(
(1.4)

It is obvious that for any e > 0 f

{l + \t\)°'^-'^W-"'-'\U{t)\dt

But (1 4- |t|)^2-i < (7^^^ ^ d)«2-i for enough great Ci = Ci(a,c?,£) > 0 and any t G (—c? -f £, +00). Therefore, by (1.4)

J-d+e J-d

r(ai-ha2)

Besides, {d - :r)^i+^2-i < (^2(1 -[- |a;|)^i+^2-i (_oo < a: < c^ - £:) for some C2 = C2 {oLi -h a2, (i, e) > 0. Hence d—£

{l+\t\Y-'-^W-°"\U{t)\dt /

-00

= C i C 2 ^ / ^ ' ' ' ^ , / \ l + |:r|)^^+"^-^|C/(x)|dx < +00. r ( a i +0^2) y_oo L e m m a 1.2. Let f/('i^) G Jjoci+a2-\ Vocn,{~^^^d) /^^ some positive numbers aua2r'-otn. T/ien W-("i+"2+-+^-)t/(t;) an^ W^-^^W^-^^ ••-1^-^-^/(7;) are absolutely convergent for almost all v G {—oo,d), and a.e. in (—oo,d) | ^ - ( a i + a 2 + - + a . ) ^ ^ ^ ) ^ l ^ - « i I ^ - « 2 . . . W-'^-Uiv)

(1.5)

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

3

Proof. For n = 1 our assertion is trivial. Using (1.4), we come to the following equalities, which prove our lemma for n = 2:

A similar argument with an induction leads to the proof for any n > 1. R e m a r k 1.1. If a G [1,-hoo) and U{v) G La(—CXD,C?), then by Lemmas 1.1 and 1.2 the integral W~^U{v) is absolutely convergent for all v G (—00,0?) and it represents a function continuous on (—oo,d). By the same lemmas, if U{v) G Lp(—oc,c?) for any natural number p > 1, then W-PU{v) = f J—00

dti f'

dt2-" ['' ' U{tp)dtp

J—00

J—00

everywhere in (—oo,(i). Thus, W~'^U{v) is the p-th primitive

ofU{v).

1.2. By (1.3) and Lemma 1.1, for any a G (0,Q;O] the integral W~'^U{v) is absolutely convergent for almost all v G (—oo,c/), provided U{v) G LQ;O(—00,0?) for an ao G (0, +00). Besides, the following statement is true. L e m m a 1.3. Let U{v) G Lao(-oc,G!) for some ao G (0,-foo). Then for any a G (0, ao] the integral W~^U{v) is absolutely convergent in any Lebesgue point V G (—oc,c?) of the function \U{v)\. Proof. For a > 1 the assertion is obvious by the above Remark 1.1. Let 0 < a < 1 (a G (0, ao]). Assuming v G (—oo,(i) a Lebesgue point of |f7(f)|, we write

For evaluation of / i , choose a constant C = C{v, ao) G (0, +00) such that (^_t)^o-l
foj.

t < V - l .

(1.6)

Then il'^ < ^.7-r / a-^-'\Uiv - a)\da < ^ ^ / (1 + \t\r-'\U(t)\dt < +00 1 W JI I (a) y_oo since U{v) G L^o (—00, d). Thus, it remains to prove that also la is convergent. To this end, it suffices to show that lim^^-^+o/^ (^^~^\U{v — cr)|dcr is a finite number. For this, introduce the function

^,{G)=

r \u{v-t)\dt= r \u{T)\dT, Jo

Jv—a

4

CHAPTER i

which is continuous in 0 < cr < +00, and observe that for any a > 0 \Uiv)\

<- Tmv-

t)\-\Uiv)\\dt.

<7 J o

Besides, lim a''-^^Ja)

= \U(v)\

cr-)-+0

since v € {—oo,d) is a Lebesgue point of |^(i^)|. On the other hand, by the definition of ^y (a)

Js

Jd

= $,(1) - J"-^^^(5) + (1 - a) /

a''-Hy{a)da.

Therefore, the mentioned hmit is finite and the proof is complete. Lemma 1.4. IfU{v) G Lao(-oo,d) for some ao G (0,+CXD), then hm W~'^U{v) = U{v)

(1.7)

in every Lebesgue point v G (—00, d) of U{v). Proof. Let Of G (0, ao], 0 < a < 1, and let t' G (—00, d) be a Lebesgue point of U{v). Similar to the proof of the previous lemma,

In view of (1-6), \J^^^\<^^£^\l

+ \t\r'>-'\U{t)\dt^O

as a->+0.

Now introduce the function pa

F„(cr) =

pv

U{v- a)da = / Jo

U{T)d7

Jv—cr

which is continuous for a > 0. Since !» G (—oo, d) is a Lebesgue point of 17, \Fv{cj)

U{v) <-

I (^ Jo

\U{v~t)-U{v)\dt^o{l ) as a ^ -hO.

(1.9)

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

5

Hence liiRa^-^o (J"^Fy(a) = U{v). Therefore Fy{G)=a[U{v)-\-ujy{(j)], where the function a;^(cr) = a~^Fy{a) — U{v) is continuous on 0 < cr < +00 and such that LOV{-\-0) = 0. Further, for an e > 0 choose S = S{€) G (0,1) enough small to provide |cc;v(cr)| < e/b iov 0 < a < S. Then

T{a) Jo

r{a)

r ( l + a)

r(a) Js

r(Q^) Jo But one can verify that

T{a) io

I " r ( a ) 5 70

1 - ^ l\^[cj)cj^-Ha\<5--'^—^ r(Q^) J5 I

5r(l + a)' f\uy{a)\dcj

r(a)

JQ 1 — a

I

/ M

< 5V{a) ^_, , o
1 - Q

- r(a) + r(i + a)

- 1 \U{v)

e 1— a __-. 1 — a . ... -T^TT-—r+() T^. . max a;^;(cr) . 5 r ( l + a) r ( a ) o 0 was arbitrary. Hence our assertion follows by (1.8) and (1.9). R e m a r k 1.2. By Lemmas 1.1 and 1.2, in the requirements of Lemma 1.4 lim

W-'^U{v)^W-'^'U{v)

(1.7')

CK-^ai+O

for any ai G (0, ao) and almost all v G (—00,0?). This relation shows that the operator W~^ (0 < a < +CXD) is "continuous from the right" by a. In view of (1.7)-(1.7'), the natural extension of the definition (1.1) of W~^ (of > 0) to a = 0 is given by the identity W^U{v) = U{v).

(1.10)

6

CHAPTERi

For defining fractional derivatives, we assume that a > 0 and p > 1 is the natural number deduced from p — 1 < a < p. Then we set

W"U{v) = W-^P-''^i^£^U{v)Y

(1.11)

preassuming the existence of the latter function for almost all v G {—oo,d). This we call the a-th derivative of U{v). Observe that by (1.10) (IP

W'^Uiv) = —U{v), if p is a natural number, i.e. WPU{V) oiU{v).

(1.11')

coincides with the ordinary p-th derivative

1.3. Henceforth we apply Liouville's operator by the variable v to some functions U{w) = U{u-^iv) given in several domains which along with inner points w — u + iv contain the rays r(oo,w) = {s = w-i(j

: 0 < cr < +00}

(1.12)

directed from oo to w. We call such domains starshaped in respect to oc, or briefly oo-starshaped. Unless otherwise stipulated, we shall assume the function U{w) harmonic in a 00-starshaped domain D.^ with a possible exception of a countable set of points {wm}' Besides, we shall assume that any r(oo,if) C D contains not more than a finite set of points from {ifm}, and also that on T[oo^w) the function U{w){w — Wm) satisfies Holder's condition in neighborhoods of the points {wm} C r{oo,w). Preassuming absolute convergence in any interval disjoint from the points {wm} C T{oo,w) and existence of Cauchy's principal value in the rest of the integration contour, we call W'Uiw)

'miy

=:p^-3 /

{v- t)°'-'^U{u + it)dt

J —00

1 f^^ = —— / cr""^C/(^ - ia)da, 1 W Jo

w = u-{-iv eD,

(1.13)

the a-th (0 < a < +00) primitive of U{w). For o^ = 0 we set W^U{w) = U{w),

weD.

(1.14)

For definition of a-th {a > 0) derivative of U{w)j we assume that the derivative gp ——U{w),

w = u-\-iveD

{p — I < a < p)

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

7

satisfies the above requirements. Then we set WU{w)

= T^-(P-^) I ^ U { w ) I ,

w^^u + iveD.

(1.15)

Definition 1.1. A function U{w) given in a oo-starshaped domain D is said to be of the class Mp{D) (0 < /? < +00)^ if there exists an angular domain A((5o, Ro) = {w : |7r/2 + 8iVgw\ < Jo, \w\ > RQ} (0 < JQ < 7r/2; 0 < RQ < +00) such that -I-00

a^-^\U{w - ia)\da < +00 (1.16) sup / weK for any compact K C D D A((5o,i?o)- The function U{w) is said to be of the class K^{So,D) (0 < 7 < +00), if \U{w)\ < C\w\-^,

wGDn

A(Jo, i?o),

(1.17)

for some 60 {0 < 60 < 7r/2) and RQ {0 < RQ < +00). It is evident that for any /?i and P2 (0 < /3i < /32 < +00) M0,{D) C Mp^iD). Besides, it is obvious that for any 71, 72 (0 < 71 < 72 < +00) and 60 {0 < SQ < 7r/2) Thus, the classes Mp and K^ narrow down as P and 7 increase. At the same time, the below lemma is true. L e m m a 1.5. Let /? G [0,+CXD) and 7 G (/?,+00) be any numbers. any 60 {0 < 60 < 7r/2) K^{6o,D)cMp{D).

Then for (1.18)

Proof. One can verify that (a + b)~^ < max{l, 2^~'^}(a'^ + 6'*)"^ for any positive numbers a, b and K. Consequently, \w-ia\-^

< {\w-ia\'^)-i

= [\w\'^ + a'^{l-^2a-'^\lm

< {\w\'^ + a^)-i

< max{l, 2^-^/^}{\w\^

w\)]~^ < + a^)-^

if Im w <0 and a, 7 > 0. Thus, for any w (Im it^ < 0) and cr, 7 > 0 '-ia\~^

It; •

where

C^ - max{l, 2^-^/2}.

(1.19)

Now assume U{w) G K^{6o,D) (0 < JQ < 7r/2), where 7 G (/?,+00) (/3 > 0) and K is any compact in i^ fi A((5o, Ro). Then by (1.17) and (1.19) Ji

Ji

<

\w-ta\-y^ Ji +°° x^-'^dx CCJwf-'' 1^ ^^'/.-. l+:r7

|w;|T+0-T

8

CHAPTERi

for any w e K {if w e K, then Im w < 0). Hence (1.16) follows since \w\^~^ is uniformly bounded for it; G K. Thus, (1.18) is valid. 1.4' Lemma 1.6. Let U{w) be a continuous function in a oo-starshaped domain D, and let U{w) G Mp{D) for some (3 > 0. Then r-\-oo

sup /

a'^-^lUiw - ia)\da < +oo

(1.20)

we'KJo

for any a G (0,^] and any compact K C D. Proof. Choose the number HQ > 0 enough great to provide that whole compact K — iHo = {( = w — IHQ : w G K } is contained in the angular domain A(Jo, i?o) from the definition of Mp{D). Further, assuming w eK 3,n arbitrary point, we write /

a'^-^Uiw

- ia)\da =

/ =

+ / h{w)-\-l2{w).

It suffices to show that Ii{w) and hiw) end, observe that Uw) = / Jo

a " " ^ ( ^ " ^cr)|(ia-

are uniformly bounded in K. To this

a^'-^lUiw - ia)\da < ^ °

oi

'

max \U(C)\ < +00 «!!?-*°:, + 1

0<
since U{w) is continuous in D. For evaluating /2('W^), we use the obvious inequality {a -f- Ho)''-^ < CG^-^ {I < (J < +00), where C = C{a,p,Ho) is a constant. As K - IHQ C D D A{SO, RQ), -I-00

{a + HoT-'^\U{w - i{Ho + a))\da

/

+ 00

a'^-'^\U{w-i{Ho

/ < C

sup

/

+ (T))\da

a^-^\U{C - i(7)\da < -f-oc,

wG

Lemma 1.7. Let U{w) he a continuous function in a oo-star shaped domain D, and let U(w) G Mf3{D) for some p > 0. Then: 1°. For any u G (inf^^j;){Re w}, sup^^2){Re w}) and d < sup^^^{Im w} the function Uu{v) = U{u-\-iv) belongs to the class L/3(—oo,(i). 2°. For any a G (0,/3]; the functions W~°'U{w) and W~^\U{w)\ everywhere, finite-valued and measurable in D.

are defined

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

9

Proof. 1°. Let £ > 0 be arbitrary. One can see that if —oo
/

nd—e

{l + \t\f-^\U{u + it)\dt
{d-tf-'^\U{u + it)\dt

J — oo + 00

/

a^~'^\U{u-\-id-ia)\da

< +oc.

2°. It is obvious that along with U{w) for any N > 1 both functions fN{w) = T ^ / af'-'Uiw - ia)da, FN{W) = ^ [ a^-^\U{w - ia)\da r(a) Jo r(Q;) JQ are continuous in D. Besides, in view of (1.20) the following limits exist and are finite: lim / T V H = W-'^Uiw), lim FN{W) W'^lUiw)]. A/"—>oo

N^-oo

L e m m a 1.8. Let U{w) be harmonic in a oo-starshaped domain D, and let U{w) G Mp{D) for some /? > 0. Then also W~^U{w) is harmonic in D for any a G [0,/3]. Proof. Our assertion is trivial for a = 0 since Ty^ is identical. Consider the case ^ > 0, a G (0,/3]. By Lemma 1.7, W~^U{w) is finite-valued and measurable in D. Therefore (see, for instance, [11], Addendum, §1, Subsection. 1), W~^U{w) necessarily will be harmonic in D, if we prove the equality W-^'Uiwo) = ^ f 27r Jo

W-''U{wo+re'^)d^

for any wo ^ D and r > 0 such that Or{wo) = {C '• |C ~ '^o| ^ ^o} C D. It follows from Lemma 1.7 and (1.20) that W~^\U{w)\ is finite-valued and measurable in D. Also, it follows that 27r

X

W-"" \U{wo +re'^)\d^

< +00.

Therefore /.27r

^- /

27r ^"TT Jo

W-'^U{wo-i~re''^)d^

1

P+OO

= : F ^ / r(Q^) Jo

cr'^-^Uiwo - i(j)d(T - W-"/7(^o),

10

CHAPTER i

where all integrals are absolutely convergent. Thus, the proof is complete. Remark 1.3. The requirements of Lemma 1.8 can be weakened. Namely, its assertions remain valid if instead the uniform boundedness of (1.16) in any compact K lying in Dr]A{5o,Ro) one require only its uniform boundedness on a family of continuous curves {Ik} C D such that for any K, C D there exists /u > 1 for which Re K C Re Z^;- We omit the proof of this statement. 1.5. The following lemmas also will be useful. Lemma 1.9. Let U{w) be continuous in an angular domain A{6o^Ro) (0 < So < 7r/2; 0 < Ro < +oo); and let \U{w)\ < Ci\w\-^''-^^\

w e A(^o,i?o),

for some positive a and K. Then for a constant Ci > 0 W-'^\U{w)\

< Ci\w\-^,

w G A((5o,i?o).

Proof. If ^ G A{So,Ro), then by (1.19) + 00

<—;^7-^^—^k

T{a)

/

' '

:;

-— = Ci\w\ "".

l + x^+'^

Jo

" '

Lemma 1.10. Let U{w) be harmonic in a oo-starshaped domain D. 1°. / / U{w) G Mp{D) for some p = 0 , 1 , 2 , . . . , then for any a G [0,p] (a) W~^U{w) is harmonic in D; (b) —W-^P-°'^W-''U{w)} 2°. / / d/dvU{w)

= U{w),

w = u + iv€D.

(1.21)

e Mi{D), then for any a G (-1,0)

(a) W °'U{w) is harmonic in D; (b) W'='W-°'U(w) W^W-'^Uiw^^Uiw)= U(w) - lim Uiu-it),

w=u-\-iveD,

(1.22)

besides, for some constants ao, oi lim U{u —it) = aiu-\-aQ^

—OQ
(1.22^)

t—¥-\-oo

Proof. 1°. (a) is proved in Lemma 1.8. For proving (b), observe that by Lemmas 1.7(1°) and 1.2 ^ - ( P - « ) M ^ - ^ I 7 ( ^ + it) = W-W(u

+ it),

w =

u-hiveD,

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

ii

for any t such that u-\-it G D. Consequently, f}P

.

.

BP

^ ^ ^^=dvP W-^U{u + iv) dvP T ^ - ( p - ) {W-U{w)}

d r

= -— / U{u + it)dt = U{u + iv) ov J_^ 2°. (a) is obvious. For (b) observe that + iv) = WW-^^-^'^^^-Uiu

WW-'^Uiu

+ iv) - W-'^ — Ulu + i^;)

r9—U(u-\-it)dt^U(u-^iv)-

Hm1

//

L->-oo JAdt

Hm

t^+oo

Uiu-it)

^

^

where the hmit exists, is finite and depends solely on u. On the other hand, this limit is a function harmonic in D, from which (1.22') follows. 1.6. The assertions of the above lemmas can be inverted to similar statements related to an integro-differential operator with bounded integration contour. For an arbitrary point w E G~ (and z = w~^), consider the arc L{0,z) = [r{oo,w)]~^ ={C = uj-'^ :

ueT{oo,w)}

of the circle centered on the real axis, directed from the origin to 2:. A domain D containing L{0^z) together with any inner point z G D will be called 0starshaped. Note that L(0, z) is the inversion of the ray r(oo, w) = {(J — w — ia : 0 < a < +00} directed from 00 to tc, and both z = w~^ and the arc L{0,z) lie in the upper half-plane G"^, since w G G~ and r{oo,w) C G~. Unless otherwise stipulated, everywhere below we shall assume u{z) a function harmonic in a 0-star shaped domain T) C G~^ = {z : R e 2 : > 0 } , with possible exception of a countable set of points {zm}- Besides, we shall assume that for any z G D there is not more than a finite set of such points on L(0, z), also that u{z){z — Zm) satisfies Holder's condition in some neighborhoods of all Zm G L(0, z). Then we set I

/

w---u{z) ^ :pf^ / r(Q^)

W\{z)=u{z),

/

7,

1. y

a-l

f7 - - ) " CMOdC

JL(O,Z) V C /L(0,Z)

0 < a < +00,

(1.23)

ZJ

(1.24)

where p > 1 is thejnteger deduced from p — 1 < a < p. The convergence of the integral of W~^ {a > 0) will mean its absolute convergence on the

12

CHAPTER i

arcs 1/(0, C) C 1/(0, z), which are disjoint from the points Zm ^ Lifd.z)^ and convergence as Cauchy's principal value on the rest of L(0, z). Assuming that the requirements on u{z) are satisfied by the derivative

a(lm \Y

"^'^^9(£lfK{'^'^^'^"^} d{lm i) )' '^^^

where p > j_is the natural number from p — 1 < o; < p, we define the fractional derivative W^ (a > 0) as follows: Wul.z) =W'(^-''A—^^^u{z)\ liU{w)

,

0
(1.25)

is defined in a oo-starshaped domain D, then the function u{z) = U{w),

z-^ G D,

(1.26)

is defined in the 0-starshaped domain Y> = D~^. The connection between W^ and Liouville's operator is given by the following lemma. L e m m a 1.11. Let U{w) and u{z) he defined in a oo-starshaped domain D C G~ and in the Q-starshaped domain D = D~^ C G^ respectively. If these functions are connected by (1.26), then for any a G (—OO,+CXD) WU(w)

= W'^u{z),

z = t/;-i G D C G+,

(1.27)

where both sides exist simultaneously. Proof. Our assertion holds for a = 0, since both W^ and W^ are identical. Further, for any p > 1

d[lmz-^f-d{lmwy'

^

^ ^•^•

Therefore, it suffices to prove our lemma for a G (—oo,0). To this end, we assume it; G D an arbitrary fixed point and transform (1.1) by the change of variable Bue w -\- it = s: WU{w)

- - ^ ^

I

Further, the inversion s = (~^ gives

[i{s - ii;)]'"'-' U{s)ds,

weD.

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

13

and this change of variable is permitted by the properties of U{w) (see, for instance, [34], Ch. 1, § 3, Subsection 3.5). In the last integral

arg

t

^ \

._2 CMOdC

0.

Also, one can verify by a direct calculation that arg [i{C — z)] = i^o +19 — TT, dC= , ^ \ ^e^'^m\ I cos 'i/01

and arg [e'^l^l-i^V"" [i(C - z)]'"'"'C'"'"'^Cl = 0 L -'

for z = poe'^^ G G+ and C = pe^^ G L(0,z). Thus, (1.27) holds, and its sides exist simultaneously because of the invertability of our operations. The following classes in 0-starshaped domains are similar to the previously defined classes Mp{D) and Kj{6o,D). Definition 1.2. A function u{z) given in a 0-starshaped domain D is said to be of the class MpCD) {0 < P < H-oo)^ if there exists an angular domain A(5o,^o) = {z : |7r/2 - arg2:| < JQ, \Z\ < ro} (0 < ^o < 7r/2; 0 < ro < +oo) such that

sup/

iz-ci^-'icr'-^mdciK+oo

for any compact K c D fi X{So,ro)- The function u{z) belongs to Kj{Soy'D) (0 < 7 < +oc), if \u{z)\ 0 and some 6o (0 < Jo < 7r/2) and ro > 0. One can verify that if U{w) and u{z) are connected by (1.26), then the inclusions U{w) G Mp{D) and U{w) G K^{5Q^D) are equivalent to u{z) G M/3(D) and u{z) G K^{5Q^Y>) ( D = D~^) respectively, whatever be /3 G [0,+oo), 7 G (0,+cx)) and Jo G (0,7r/2], i.e. M^(D) = {u{z) = U{z-')

: U{w) G Mp{B-')}

K^{6o, D) = {u{z) = U{z-^) : U{w) G K^{6Q,

,

D"^)}

.

(1.28)

Therefore, all lemmas proved for Mp{D) and K^{So,D) can be equivalently stated for Mp(D) and i^^(Jo5D) (E) = Z)~-^). Omitting these statements, we only note that by (1.27) and Lemma 1.5 Mp{-D) C K^{5o,T>) for any P G [0, +oo), 7 G {p, +00) and Jo G (0,7r/2].

(1.29)

14

CHAPTER i

2. HERGLOTZ-RIESZ TYPE THEOREMS Below we give some well-known results on descriptive representations of several subclasses of the Herglotz-Riesz class by Cauchy integral. After that, we state the main results of this section. 2.1. Everywhere below R means the Herglotz-Riesz class in the lower halfplane G~ = {w : luiw < 0}, i.e. the set of functions holomorphic in G~, having there nonnegative real parts. Definition 2.1. R° is the set of functions holomorphic in G~, representable in the form n ^ ^ ^ l T M l „,G-, m J_^ w - t where fi{t) (—oo < t < +CXD) is a nondecreasing, hounded function. 7 < 2) is the set of those F{w) G R for which

I

(2.1) R7 (0 <

+°° Re F{-it) ^^ ^ -dt < +00.

1

ty

The class R° was first investigated by R.Nevanlinna [81]. One can find a detailed account of its properties in [4] (Addendum I) containing the below Theorem 2.1. The following statements are equivalent: 1) F{w)

GR°;

2) F{w) e R , supt\F{-it)\ < H-oo; t>o 3)F{w)eIl, sup t Re F {-it) <-\-oo, t>0

lim F{-it)

= 0.

t-^+oo

The classes R'^ (0 < 7 < 2) were considered by LS.Kats [72] (see also [4], Addendum I). The next theorem follows from his results. Theorem 2.2. R'^ (0 < 7 < 1) coincides with the set of functions representable in the form F{w) - iC + - T " ^ ^ , weG-, 7nJ_^ w-t where C is a real number and fi{t) is a nondecreasing function ^+^ d/2{t) < +00. l + lth

I

(2.2) satisfying (2.3)

J —(

R e m a r k 2 . 1 . It is well known (see [4], Addendum I), that R° C R'^ C R for 0 < 7 < 2. One can see that by Theorem 2.1 formula (2.1), where /^{t) is a nondecreasing, bounded function, is a descriptive representation for the class of functions defined by any of statements 2), 3) of the same theorem.

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

15

2.2. For stating our theorems, we need the following definitions. Definition 2.2. R^ (—1 < OL < +CXD) is the set of those holomorphic functions in G~, which are representable in the form

F(nj) = £ ( i ± ^ e - ^ f (^+-) r ^ r ^ % - ,

w G G-,

(2.4)

where fi{t) (—oo 0 is the integer from p — 1 < a
(b) l ^ - ^ R e F H > 0 , T^"^

(c)

/

weG-; df

W^-^Re f{-it)

— < +00,

if

a=p.

(2.5)

2°. / / - l < a < 0 , then (a) Im F'{w) e M i ; (b) T^-^Re F ( ^ ) > 0,

WGQ-.

Theorem 2.3. Let F{w) he holomorphic in G~. 1°. / / 0 < a < +00 and p > 0 is the integer from p — 1 < a < p, then the following statements are equivalent: (i)

F{w)GR^;

(ii) Re F{w) G Mp, T^-"Re F{w) > 0 for w e G', supt^^^\F{-it)\ t>o

< +00,

/ Ji

W - " R e F{-it)—

< +oc t

{if a y^ p, then the convergence of the last integral is not required); (iii) Re F{w) € Mp, W'^'Re F{w) >0 s u p t W ^ R e F{-it)

< +cx),

t>0

forweG', lim F{-it)

= 0.

t-).+oo

2°. If — 1 < a < 0; then the following statements are equivalent: (i) F{w) e R^; (ii) Im F'{w) G Ml,

W-'^Re F{w) > 0 for w e G',

supt^+^|F(-i^)| < +00,

supt^+^llm F'{-it)\

t>0

t>0

< +oo;

16

CHAPTER i

(iii) Im F\w)

e Ml, W - « R e F{w) > 0 for w e

sup^l^-^Re F{-it)

< +00,

t>0

G',

lim F{-it)

= 0.

t->+oo

T h e o r e m 2.4. H^ {—l0

^„.,C +

from p — 1 < a
ffi±i^e-«-./;;^;^.

..0-.

where C is a real number and ^i{t) is a nondecreasing function f+°°

2°. For - K a FH

(.6,

satisfying

d^l{t)

<0

= a^u. + ao+iC+

I i i ± ^ e - f (^-) / " ^ ^

^

^

(2.8)

{w G G~), where ai, a^ and C are real numbers and fi{t) is a nondecreasing function such that

Remark 2.2. 1°. If a function F{w) holomorphic in G~ admits any of representations (2.4), (2.6)-(2.7) or (2.8)-(2.9), then +^

W-'^Re F{w) - M /

dfi{t) {u - t)2 + i;2 '

w = u-hiveG~.

(2.10)

2°. If a function F{w) is representable in the form (2.8)-(2.9), then aiu-\-ao=

Hm ReF(u

— it),

—oo
(2.11)

t—>+oo

These assertions will be established in Sec. 3, in Proof of Theorem 2.4. Remark 2.3. It is obvious that strictly RS C Ra,

-l
+00.

(2.12)

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

17

3. PROOFS OF THEOREMS 2.3 AND 2.4 3.1. First we prove some lemmas which will be used in this and some other sections. Lemma 3.1. Let ii{t) he a function of locally hounded variation on the real axis, and let , , h 1 f^"^ dfi{t) iw m J_^ w — t where h is a real numher. Then: 1°. / /

W{t)\

^+-

i + it|i+"-p

< +00

(3.1)

for a given a G [0, +00) and the integer p >Q (p—1 < a < p), then for any w = u + iv G G~

'dvP

^

'

^

^\(tw)i+«

T:J_^

[i{w-t)Y+<'\

2°. If

for some a G (—1,0); then W^^^iw) = r ( l + a)!. - 4 ^ + - r°°

,., "^^^H,^^ ],

weGT

(3.2')

Proof. We shall use the well-known formula /

— —

=r(7)r(l-7)(i^)^-\

7 6(0,1),

ZGG-,

(3.3)

and the following estimate which is true for any compact K C G~: sup

(f±^!^|=C^(K)<+oo,

- o o < t < + oo weK

\ \W — l\

AG(-l,+oo),

(3.4)

^ ^ J

where C^(K) > 0 is a constant depending only on A and K. 1°. If a = p > 0 is an integer, then (3.2) is trivial since VK~(P~^) = W^ is identical and (3.1) is true. If a > 0 is not an integer (i.e. 0 < p — a < 1), then it suffices to show that W-^P-^^^iw) ^

= r ( l + a - p) ( j — ^ '\{iwy+°'-P

+ - /^°°—-^4wz n J_^[i{w-t)]^+°'~P}

1

(3-5) ^ '

18

CHAPTER i

To this end, first we prove that the integral W~^^~^'^^{w) is absolutely convergent. Indeed, one can verify that by (3.4), (1.19) and (3.1)

J_^

\w-t\-

r{p-a)

Jo

1+x

y_^ l + |t|i+"-p'

where the last integral is convergent. Hence, by (3.3) a change of integration order gives (3.5). The assertion 2° is proved similarly. Lemma 3.2. Let P > 0 be not a natural number, and let p > 0 be the integer deduced from p—l
(3.6)

Proof. If /3 = 0, then p - ^ = 0 and W'^^ is identical, and !^\U{-it)\f < p—l<(3< +00 since U{w) G Mo{G~}. Now let /3 > 0 be not an integer (i.e. p). Then the change of integration order gives

< Lemma 3.3. Let v[t) be a function of locally bounded variation on the real axis, and let

1°. If a e [0, +oo)^ p>0

is the integer from p — 1 < a

< +00,

/

-oo

l"

then Re (fai'w) G MQ for a = 0, and (fai'i^) G Mp for a > 0. Besides, gp _ V F - b - « ) ( ^ o ( ^ ^ ) = (p^{w),

W-'^^aiw)

= ^o(^^),

2°. If ae (-1,0) and ^i-oo +/

-oc

\du{t)\ 1

+ \t\1+a

< +00,

w=u+

iveG-.

LiouviLLE O P E R A T O R AND H E R G L O T Z - R I E S Z T H E O R E M S

19

then '^'a.{^) ^ -^1- Besides,

P r o o f is obvious. 3.2. P r o o f of T h e o r e m 2.4. 1°. Let F{w) G R^ for some a > 0. Then the function W~^Re F{w) is harmonic in G~ by Lemma 1.10(a) with D — G~. Besides, this function is nonnegative, and by Lemma 3.2 +°° /

rif W-^'Re F(-it)-rr

< +00.

Thus, the function ^{w) = W~^Re F{w) + iV{w) (where V{w) is the conjugate harmonic of ^ " ' ^ R e F{w)) belongs to R^+*^~^. By Theorem 2.2, ^{w) is representable in the form (2.2)-(2.3) for 7 = 1 -}- a — p. Therefore, the representation (2.6)-(2.7) follows by the assertions 1° of Lemmas 3.1 and 3.3. Conversely, let F{w) admit the representation (2.6)-(2.7) for some a > 0. Then, by Lemma 3.3(1°) Re F{w) G Mp. By the same lemma, also (2.10) is valid, and hence W-'^Fiw) > 0 for w e G'. On the other hand, by (2.10) the function ^{w) is representable in the form (2.2)-(2.3), where 7 == 1 + a — p. Hence (2.5) holds for integers a = p > 0 by Theorem 2.2. 2°. For — 1 < Q; < 0 our assertion and formulas (2.10) and (2.11) are proved similarly, using Theorem 2.2 and Lemmas 1.10 and 3.1, 3.2, 3.3. P r o o f of T h e o r e m 2.3. In both cases 1° (0 < a < +00) and 2° {-1< a < 0) we prove the implications (i) => (ii) ^ (iii) => (i). 1°. Let a > 0 and (i) is true, i.e. F{w) admits (2.4). Then, by Lemma 3.3 Re F{w) G Mp and (2.10) is valid. Hence W^-"Re F{w) > 0 for ^ G G". Besides, (2.5) follows from (2.12) and Theorem 2.4. Now observe that by (2.4)

t^+^\F{-it)\ < £ i l ± ^ / " dfi{t) < +00, ^

t> 0.

J-00

Hence (i) ^ (ii) is true. Let (ii) be valid. Then \F{-it)\ < Cot"^"" (0 < t < +CXD) for some Co > 0. Hence F{—it) = o(l) as t -^ +CXD and for any t > 0 tW-'^Re

F{-it)

^

< - ^

/

^ - r(a) Jo {1+x)

l+a

< +00.

This proves (iii). Now let (iii) be true. Then, by Lemma 1.10(a) the function Ty-^Re F{w) > 0 is harmonic in G' and W ^ R e F{-it) < Cit''^ {t > 0) for some Ci > 0. Consequently, for any 7 > 0 the above function ^{w) is of R^. Using Lemmas 3.1 and 3.3 we conclude that F{w) is representable in the form (2.6). As it follows from (2.10), +00

IL

dfi{a) = sup tW'^Re

F{-it)

< +00.

20

CHAPTER i

Thus, the function fi{t) is bounded. On the other hand, it is obvious that \C\ < \Fi-zt)\

+ ^

^

J_^

^^ _ 2 | / + . = 0(1)

as

t->+oo.

Hence C = 0 and the representation (2.4) is true, i.e. (i) is valid. 2°. The proofs of the case — 1 < a < 0 (including formulas (2.10) and (2.11)) are quite similar. NOTES. Some of the auxiliary statements on Liouville's fractional integro-differentiation, which are given in Section 1, are well known. One can find them, for instance, in [92].

CHAPTER 2 BLASCHKE TYPE PRODUCTS

We shall consider a family of Blaschke type products in the lower half-plane G~ = {w \ Iva w < {)} and investigate some of the properties of these products. The factors of our products depend on a parameter —\
b^{w,Q=exAj{[r

+ i{'w-0]

' " + [t(u; - C) - r] ' " V ^ d r l (1)

For a = 0 this expression becomes ^^^, and the inversion z = w ^ {zk = Wj^ , k = 1,2,...) transfers the properties of (1) into ones for several products in the upper half-plane G+ — {z \ Im. z > 0}, which for a == 0 coincide with the widely known Blaschke product

^(^'^^^»=nH7l' ^^'^>^^'' ^

< +00.

(2)

Zk

1. ELEMENTARY PROPERTIES OF FACTORS 1.1. We start by another representation of hoc{w, (). Assuming C, — (^-[-ir] ^ G~ a fixed point, for any a G (—1,-foo) we introduce the function ^.KC)=/ Jo

—— -ZZ7TT^+ [r + i{w - C)j Jo

7=^ TiT^[i{w - C) - ^]

(1-1)

The changes of variables t = —{r + rj) and t = r-i-rj in the above integrals give »aKC)=r'

J-\n\

../'"^'"'"^"l..

[t{w

-l
(1.10

-O-t]

By (1) and (1.1) ba{w, C) = exp{-nc{w,

C)},

- 1 < a < +00.

(1.2)

22

CHAPTER 2

Sometimes we shall write ilai^^X) Qaiw,

i^ the form

0 = C / a ( ^ , C) + Va{w, ( ) ,

" K

a < +00,

(1.3)

where

Theorem 1.1. T/ie function ba{wX) (~1 < Q^ < +00) is holomorphic in the region C \ { ^ -\- ih : 0 < h < +00} anc? z;f vanishes only at the point ( G G~, which is a first order zero. Proof. By (1.2), (1.3), (1.4)

w—(

^

^

The functions Ua{w, () and Uo{w^ C) obviously are holomorphic in C\{^ + ih : 0 < h < -\-oo}. Thus, it suffices to show that

F4w,C)=Vo{w,0-Va{w,0

-r{i

i\v\-tr [i{w - ^) - t]^-^""

i{w-i)-t

•dt

is holomorphic in the same domain. To this end, we write

^--HL-Q[,^^^w^-^-^and separately prove the holomorphity of la

and la . First observe that

is holomorphic in C\{^ -\-ih : 0 < h < +CXD}, and for —00 < t < \r]\ the same function is holomorphic in the smaller domain C\{^H-i/i : —\rj\ < h < +CXD}. Next, for any compact K C C (fai'wX^t) = 0{t~'^)

as

t ^

~oo

uniformly in respect to IL' G K. Therefore, by uniform convergence, the integrals la {w^ C) and la {w, () represent holomorphic functions in the domains

BLASCHKE T Y P E P R O D U C T S

23

C\{^ -\-ih : 0 < h < +00} and C\{^ + ih : —\r]\ < h < +00} respectively. But la {w, C) is a constant. Indeed, considering this function on the ray w = ( — ih {0 < h < +00) and taking t — —ah + \r]\ we get

1.2. The below recurrent formulas for Ua{wX) and Va{w^() lead to some representations of Qai'i^X)- We shall assume that a > 0 and p > 1 is the natural number deduced from p — 1 < a < p. First observe that integration by parts in (1.3') gives Ua{w, 0

= -

r

(1^1 + trd[i{w

- C) - r "

= -

( ^ )

^ - Ua-l{w,

C).

Hence

u^{w, c) = E ^ 3 ^ ( ^ j

+ {-iru^-Aw, c).

Similar to this,

^"KO = - E ^ r ^ ( ^ )

+^^-.(-.0.

(1.40

By these recurrent formulas and (1.2), (1.3), (1.4)

^

<J

a-p+j

\w-^

1.3. Lemma 1.1. If a e (—l,+oo) and ( — ^-\-ir] £ G~ are arbitrary, and w = u-\-w is an arbitrary point from the cut complex plane C\{C -\-ih \ 0 < h < +00}^ such that \w — £^\> \r]\, then:

l"-(^'C)l < ^fia(w,C) <

J . i.i+.y'"'".

(1-6)

(l^-^l-W) 2+.^\vr"-

(1-6')

Proof. Evidently \i{w - 0 - ^1^'*'" > i\w - CI - IvW'^" for any t € [-|7?|, |??|]. Hence by (1.1') we come to (1.6). For proving (1.6'), one have to repeat the same argument after a differentiation of (1.1') by f = Im w.

(1.4)

24

CHAPTER 2

Lemma 1.2. If a e ( - l , + o o ) , then for any compact K C G~ 2 g - i f (l+a) |^|l+a

Oc('a;,C + i7/)

(it;-0^+'' 1 + a

+ 0(|77|2+«)

as

rj^-0

(1.7)

uniformly in respect to it; G K and <^ G (—00, +00). Proof. Integration by parts in (1.3') gives

t / . ( ^ , C 1) -+ TQ;^\i(; ( -—^ ^

^.(^,0-

z?7

C/l^a('?^,C), l+a

+ ^1+^(^,0.

1 4- a yu' — ^

Hence by (1.3)

where Raiw, () = — C/i+a(i(;, C) + Vi^ai"^^ C)- We shall prove that there exists a constant C depending on a and on the compact K, for which \Ra{wX)\
as

rj-^-0

uniformly in respect to if; G K and ^ G (—00,+00). To this end, we denote p = min^^j^ |Im u^l and observe that if \r]\ < p/2, then |if; — ^| — \r]\ > p/2 for any if G K and ^ G (—00, +00). The desired estimate for i?^ follows from (1.3'). Remark 1.1. If \w\ > Ro > 4|C|, then \w - ^\ - \ri\ > \w\ - 2|C| > \w\/2 since 2|CI > 1^1 + 1^1- Hence by (1.2), (1.6) and (1.6') 22+a

|log|6„(«;,C)|| < |i)a(ii^,C)l < f - — | r ? | ' + " k r ( ' + ° ^ l + a d_ < — log|6a(^,C)| dv Qa{w,C)

(1.8)

for any w G C\{(-\-ih : 0 < h < +00} {\w\ > RQ). More simple representations than (1.5) are true for ba{w,() for a = p, where p is an even or odd number (see [54], formulas (1.6)-(1.80). 1.4- Assuming that 5 = 5 + iA is an arbitrary fixed point from G^ and ( = ^ -\-ir] e G~ — s~^ (G G~), consider the ray r*[^, 00] = {w = ^ + ih: 0
+00}

(1.9)

BLASCHKE TYPE PRODUCTS

25

and the arc L*[r\0]

= {T*[^, 00)}-^ ={z = W-' : w G r [ C , o o ) } .

(1.10)

One can see that L*[^~-^,0] is contained in the closed half-plane G~ = {w : Im. w <0} and for ^ 7^ 0 it becomes a semi-circle connecting ^~^ = k P / ^ with the origin, and the center is on the real axis. Now introduce the functions ,

ba{z,s) =ba{w,C)\

- 1 < a <+00,

(1-11)

and observe that their properties are simple restatement of those of 6a(^^',C)• Particularly, (1.12) Further, in view of (1.2), (1.2') and (1.5) 6„(z,s)=exp<;- /

N^"'l

{\lm

... ' ,

s-'\-\t\)"dt

^ '

ly

.,!+, >

|Im s-i| [«(-2~-' - Re S-1) - t]

(1-13)

for — 1 < a < 4-00, and the following similarity of Theorem 1.1 is true. T h e o r e m 1.1*. The function ba{z^s) (—1 < a < +oo) is holomorphic in C\L* [|5p/J, O] and vanishes only at the point s G G^, which is a zero of first order. 2. INTEGRO-DIFFERENTIAL PROPERTIES OF FACTORS Assuming ( — ^ -{- irj a, fixed point in G~, we shall apply Liouville's integrodifferential operator W~^ (formulas (1.13)-(1.15) in Ch. 1) to logba{wX) = —Qa(^, z) and consider the properties of the function W-"^ log \ba{w, 01 = - W - ^ R e 0 « ( ^ , 0 ,

- 1 < a < +oo.

2.1. Introduce the following line intercepts:

-\v\
(2.1)

Lemma 2.1. The function W "log6a(w,C) (~1 < a < +00) is holomorphic in the domain C\[C, C]; where W~" \ogbo.{w, C) = -W-'^^o.iw,C)

I"! (|7?| 1 f™ (M -- \t\rdt = :p7T—^ / T ^ T{l + a) 7_|^| t - i

(2.2)

26

CHAPTER 2

Proof. As W^ is identical, for o; = 0 our assertion is trivial by (1.1') and (1.2). The cases a > 0 and a <0 have to be considered separately. (a) 0 < a < -foe. By ( l . l ^ , (1.2) and (1.13) of Ch. 1 W^-"log6,(^,C) = -W^-"Oa(^,C) r»+oo

p\rj\

(")yo

J-Mli{w-()-t

+ t,]'+'

for w e C\{C -{- ih; 0 < h < +00}. It is obvious that the right-hand side integral in this equality is absolutely convergent. Therefore, using the wellknown formula I +CXD

I

+^

/o

a^'-^da (z + a ) i + -

1 az

z+a

a=0

1 az^

a > 0,

(2.3)

we come to (2.2) for w G C\{C + ih : 0
f"-^ . . r'^' {\v\-\t\rdt ^ r .'4. r r ( l + a) io ^ ^^ J-\r,\ [i{w - 0 - ^ + ^l""^" 1+^

for w G C\{C + ih : 0 < h < +oc}. Hence we again come to (2.2). Remark 2.1. From (2.2) it follows that 2 \w-^\~\ri\r{2

\W-''log\b^{w,C)\\<

|77|i+^ + a)'

-1 < a < +CXD,

(2.4)

for any w eC such that |w — ^| > \r]\. The proof of this estimate is similar to that of (1.6). Lemma 2.2. 1°. For any a € (—l,+oo) and any w € C\[C,C] M^-"log|6„(w,C)| = - R e

W-"naiw,(:)

_ 2Imw /"I"! \w-^\'^-t^ ~T{l + a)Jo |(w_^)2+i2|2<

- t)°'dt.

(2.5)

2°. For any a G (0, +oo) and any w £ C W-"log\bc.{w,0\

= -Re 1

r(

W-'^n^iwX) rM ^ — it — w {\v\-trdt ^ -\-it — w w—s

5i/I-

r(a)

W

(2.6)

27

BLASCHKE TYPE PRODUCTS

where X(^{s) is the characteristic function of the interval ( C O ^^^ da{s) is the area element. Proof. 1°. Denoting z = i{w — ^), from (2.2) we obtain

Hence (2.5) follows. 2°. Observe that Re {dt/[t - i{w - ^)]} = dt log \t - i{w - C)\ ^ov w e C and t G (—oo,+oo). Therefore, integration by parts in (2.2) gives hi

W-^\og\bc.{w,C)\

1

d{\v\-ty

r ( i + cx)j_ rlnl

^ — it — w t)'^dt ^-\-it — w w—s \^-s\'^-^Xc{s)da{s), w—s

and the proof is complete. From (2.5) it follows that W~^ log \ba{WyQ\ is harmonic outside the intercept [C, C], and also that this function vanishes on the real axis, i.e. W

'^\og\ba{uX)\=0,

-oo
u^i

a >-1.

(2.7)

Further, for [it; — (^| > \r]\ the integrand in (2.5) is nonnegative. Hence, for \w-i\ >\r]\ W - ^ log 16a (if;,01

< 0, if weG~, if we G+,

{> 0,

1 < a < +CXD.

(2.8)

2.2. For further analysis of W ^\og\ba{w,()\ we use some well-known properties of Cauchy type integrals. From (2.2) it follows that for z 0 C\[—[ryl, \r]W

w-^'iogKi-iz + tOl

r(i + a)

Im ^cx{z),

-l
+00,

(2.9)

where ^.

1

r'"' (1^1 -1^1) -dt. I'?!

(2.90

t-z

In view of these formulas W-''log\b^{-iz

+ ^,0\=-W-"log\ba{iz

+ ^,0\,

ZGC.

(2.10)

28

CHAPTER 2

The function ^a{z) is holomorphic in C\[—[r/l, I?;!]. Besides, ^a{t) = {\v\ ~ \A)^ satisfies the Lipschitz condition (for a = 0 and a > 1 of order 1 on [—17/|, |ry|], for 0 < Q; < 1 of order a on [—\r]\, \r]\] and for —1 < a < 0 of order 1 on (—|ry|,|77|)). Therefore, the below statements are true by the well-known theory of Cauchy type integrals (see, for instance, [34], Ch. 1). 1°. The Cauchy type integral

^o.{x) 0 =-^ - 1^ I M^^^dt, I

- 1 < a < +00,

t — X

is a continuous function of Lip A in (—Ir/I, Ir/I) for some A G (0,1). 2°. For any a G (—1, +00) and any x G (—|ry|, \r]\) the limits \im ^oc{z)

= ^t{^),

lim

z^-x, zEG'^

^a{z)=-^-{x)

z-^x, zEG~

exist, are finite and connected by the following formulas: *+(a;) - ^-{x)

= {\v\ - \t\r,

^t{x)

+ ^-{x)

= 2$„(x).

3°. For a > —1 these limits are continuous functions of Lip A on (—1^|,|^|) with some A G (0,1). 4°. If a > 0, then ^a{z) is continuous at the points —1?7| and \r]\, as a function of complex variable. 5°. If — 1 < Q; < 0, then the following representations are true in enough small neighborhoods of the points —\rj\ and \rj\:

where '^^(2:) and il^l,{z) are holomorphic in the same neighborhoods. In view of (2.9) and (2.9') we come to the following L e m m a 2.3. The function W~^ log \ba{wX)\ (—1 < a < +00), which is harmonic in C\[C,C]; ^^ continuous on [C,C] /^^ a > 0, and is continuous on iCX) for -1 < a <0. Besides, for -1< a <0 W^-"log|6«(^,C)i f r ( l — a) I 1'^ — Cr^osfaarg i{w — Q] -{-u^{w),

rn-a)

-

-

\w — CTcos [aarg i{( — w)] -\-ul^(w),

^^-^^^

BLASCHKE T Y P E P R O D U C T S

29

in enough small neighborhoods of ( e G~ and ( G G'^, where ul^{w) and ul^{w) are harmonic. In contrast to the properties of log |6o(tf, C)!? it appears that the entire intercept [C, C] is the support of the mass of W~^ log \ba{w, Q\. Theorem 2.1. 1°. For a G (0,+CXD) the function W~^ log \ba{wX)\ ^^ (continuous in the closed w-plane, harmonic out of [C^C]; subharmonic in G~ and superharmonic in G^. Besides, < 0 for i y - ^ l o g | 6 , ( ^ , C ) l | T n Zi > 0 for

w eG~ IZ^J we G+,

0
(2.12)

2°. For a G (—1,0) the function W~^ log \ba{wX)\ ^^ continuous in the closed w-plane, except the points ( G G~ and ( G G'^, is harmonic out of [C,C]; superharmonic in G~\C and subharmonic in G'^\(. Besides, for — 1 < a < 0. ,

.

.J

<^

for

weG~,

\w-i\>\r]l

t>0

for

weG^,

\w-^\>\ri\.

3°. For any a G ( - 1 , +oo) W"" log \bcc{u, 0 1 = 0 ,

-oQ
+00.

(2.14)

Proof. Whatever be a G (—1,-foo), by (2.2) W~^\og\ba{wX)\ is continuous in the closure of i(;-plane (by (2.2) W~^\ogboc{ooX) — 0), with possible exception of points C, and C- Therefore, 3° follows from (2.7). 1^ Let a > 0. Then in view of (2.9) and (2.9') VK"^ log 16^(^,01 is harmonic in i(;-plane, except [QX]- Because of continuity of this function, it suffices to prove that for any 5 G [C, C] ^^^ enough small /? > 0

1- j ^ W-^\ogK{s-\-pe'\0\dd

Hence our assertion will follow by the maximum and the minimum principles of subharmonic and superharmonic functions. Before proving (2.15), observe that by (2.10) and (2.14) 1 C"^^ ~ j W-'^\og\b^{^ + pe'\C)\d^

= W-'^\ogK{U)\=Q

(2.16)

for any a e (—1, +oo) and p > 0. Now assume that ^ < h < \r]\ and 0 p. Hence

^

rW-^\ogK{i-ih^pe'\0\dd 1 (['-' • /'^'^ r(i + a) l^y_|,, 7;,+,;

{\v\-\t\r (i^. t-/i

Consequently, by (2.2) we obtain that 1

/>27r /"^^

- y

27r

W^-" log |6„(^ - ih+pe''', 0\d^ - W-" log |64^ - i/i, C)| ir^M^^ r(i + a)Jh-p t-h

(2.17) ^

^

for a G (—l,+oo), 0 0.

(2.170

for any a > 0 and p G (0, Ir/I). Now observe that

''-"'{\v\-\t\r.._ Jh-p h-p

t-h i — II

r Jo Jo

i\v\-x-hr-{\v\+x-hr dx. X

Consequently, ^27r

^ y ' " w - ^ log |6,(^ -ih + pe'\ C)\d^ - W - ^ log |6«(^ - 2/1,01 27r./o M « - (\n\ - ^ _ M « 11 ZPP *^ n-nl 4- ^ - {\rj\-hx-h)^-{\rj\-x-h)'' -da: r ( l + a) Jo X > 0, n. 0n <<- arv <

1 <0,

-l
(2.18)

for any a G (—l,+oo), 0 < h < \r]\ and 0 < p < max{/i, Ir/I — /i}. Hence the estimates (2.15) follow by (2.17') and (2.10). 2°. As we have shown, the function W~^ log \ba{wX)\ (—1 < a < 0) is continuous in the closure of w-p\3,ne, except the points ( and (. Besides, this function is harmonic out of the intercept [C, C] ^^^ the estimates (2.13)

BLASCHKE T Y P E P R O D U C T S

31

(see (2.8)) are true. Therefore, it suffices to see that by the second inequahty in (2.18) and (2.10) W~^ log \ba{w, ()\ is superharmonic in G~\C and subharmonic in G"^\C2.3. L e m m a 2.4. For any a G

(-1,4-CXD)

+00

IW"^ log \bJu + iv, C)\\du = 0.

/

(2.19)

Proof. By (2.2)

for any u, v G (—oo,+oo). Hence, changing the order of integration we get + 00

W-''\og\ba{u /

+

iv,0\du

-OO

r\ri\

—

^

/

{\rj\-\t\rsign

{t +

v)dt^U\rjlv).

For calculation of the last integral, observe that sign {t+v) — sign v for \v\ > \r]\ and —\rj\ \r]\ TT

lailvlv)

= (sign v)

r\v\ Z"'^'

/

27r

(|r/| - \t\rdt = (sign

\l+ot

v)———\r]\'

On the other hand, U\r,lv)

= (sign ^ ) p ^ ^ [|r?|^+" - {\v\ - k l ) ' + 1

for \v\ < \r]\. Consequently, for any v G (—oo,+oo) + 00

W-''\og\ba{u /

+ iv,C)\du

-OO

2n

(sign ^)p.^_^ .1^1^"^^

if 1^1 > 1^1,

(signi;)|,^^^[|r7r-^--(M-|H)'-^1

if

(2.20) H < M-

As the function W'"^ log |6a('?i;, C)l (<^ > 0) keeps its sign in G' and G+, (2.20) implies (2.19). For proving (2.19) with a G (—1,0), note that in view of (2.10) it suffices to prove (2.19) as t* —> —0. To this end, suppose v < 0 and

32

CHAPTER 2

denote by S^ the part of the Hne {w = u -}- iv : —oo 0, and by S~ the remaining part of the same hne, where W^-"log|6a(^,C)l < 0 . Then /

IW-'^loglbaiu

+ iv.Olldu^

(J

= f 2 [ - f

^ - J

^W-^'loglbaiu

^W-'^loglbaiu

+ W

+ iv^Oldu.

(2.21)

But the last integral in the right-hand side of this formula vanishes as i; -> — 0. Therefore, it remains to show that lim /

W-'^log\ba{u-\-iv,C)\du

= 0.

To this end, observe that in virtue of first inequality in (2.13) S^ is contained in the semidisc {w : \w — ^\ < \rj\^ w e G~} for \v\ < \rj\ {v < 0). Hence 0< /

W-'^loglbaiu-^-iv.OlduK

/

IW-^'loglbaiu-^-iv^QWdu.

(2.22)

Further, by Theorem 2.1 W~^ log \ba{w, Q\ is continuous in the closed quadrat {w = u -\- iv : \u — ^\ < \r]\, r]/2 < f < 0}, and therefore is bounded there. Thus, by Fatou's lemma and (2.14) lim sup /

\W~'^\og\ba{u-\-ivX)\\d'^ < /

lim \W-'^log\ba{u

+ iv,C)\\du = 0.

2.4. Note that by (2.20) and (2.21) r^

+ OO

/ |W^-" log \b4u + iv,0\\du< ^^^ ^ ^^ | 7 ? r " (2.23) for any — 1 < a < -f-00 and —00 < v < +00. Further, note that the results of this section can be inverted to similar results on properties of W~^ log \ba{z, s)|, where W~^ is the integro-differential operator mentioned in Ch. 1 (formulas (1.21)-(1.23)) since by our notation (1.11) W-^\og\b^{z,s)\\

^

^ =W-^\og\b^{wX)l

zeC.

(2.24)

BLASCHKE TYPE PRODUCTS

33

3. BLASCHKE TYPE PRODUCTS Now we proceed to the convergence and main properties of the products Bc,{w, {wk}) = Yi ^ociw, Wk),

- 1 < a < -f oo,

(3.1)

k

with zeros {wk} in the lower half-plane G~. These will result in similar properties of the product Boc{z, {zk}) = WK{Z,

Zk),

-l
+00,

(3.2)

k

the zeros Zk — W^^ (/c = 1, 2 , . . . ) of which lie in the upper half-plane G+. S.l. For p < 0 we set G~ — {w \ \v^w < p) and G^ — {w \ Im tt; < p}. Theorem 3.1. 1°. Let {wk^T satisfying

^ ^~

^^ ^^2/ sequence of complex numbers

oo

^|Im^^|'+"<+oo

(3.3)

k=l

for a given a G (—1, +oo). Then for any p < 0 the infinite product oo

•

oo

N

k=l

^

k=l

J

25 absolutely and uniformly convergent in the closed half-plane G'^, and the function Ba{w, {wk}i^) is holomorphic in G~, where {wk}^ are its zeros. 2°. If {wk}^ C G~ is a bounded sequence, and for a given a G (—1, +oo) the product (3.4) is absolutely and uniformly convergent inside G~, then {wk}i^ satisfies (3.3). Poof. 1°. Let p < 0 be a fixed number. Since by (3.3) Im 1^;^; -^ — 0 as fc -> oo, one can choose a natural number Np > 1 such that Wk ^ G'^ ioi k > Np -\-l. Besides, Wk G G^, then \w — Re Wk\ > |Im w\ > \p\ > |Im Wk\ for k > Np-\-l. Therefore, by (1.6)

for any Wk G G^ and k > Np-{-l. Together with (3.3), this implies the absolute and uniform convergence of the series oo

y^ logba{w,Wk) k=Np+l

oo

= -

^ fla{w,Wk) k=Na+l

(3.5)

34

CHAPTER 2

in G~^. Hence, also the product (3.4) is absolutely and uniformly convergent in G~^. Consequently, Ba{w^ {'^k}T) ^^ holomorphic in G~ and {wkj'i^ are its zeros. 2°. The function Ba{w,{wk}f) ^ 0 is holomorphic in G~ since (3.4) is assumed uniform convergent inside G~. Besides, Im Wk -> 0 3.8 k -> oo since the sequence {wk}i^ is bounded. Observe that the absolute convergence of the product (3.4) in any fixed point a G G~ is equivalent to the absolute convergence of the series (3.5) (where p = lin a and Np is chosen by p as above) in the same point. Besides, by our requirements sup^ |Re Wk\ < sup^ \wk\ = M < -hoc. Hence 2g-if(l+a)|j^^^|l+a

>(|a + M| + |a-M|)-^-"'l^"^"'=l'''"

(a-Reif;fc)^+^(l + a)

Since Im IC'A; —> 0 as fc -> oo, (1.7) and the last inequality lead to \fta{ci,Wk)\ > C\lm Wkl^'^'^,

k > fco,

for enough great ko > 1, where C = C{a, a, M) > 0 is a constant independent of k. Therefore, the absolute convergence of (3.5) in w = a implies (3.3), and the proof is complete. Remark 3.1. Considering the convergence oiBa{w, {tffc}i°) on compacts K C C, one can show that under (3.3) Ba{w, {^k}T) {^ > ~1) admits holomorphic continuation into the strip {w : a < Re w < b, 0 < Im w < +00} C G"^, through any interval (a, b) disjoint from the points Uk = Re Wk (AJ = 1, 2 , . . . ) . Let {wk} C G~ be a bounded sequence (finite or infinite) satisfying (3.3) for a given a G (—1, +00). Denote sup;. \wk\ = M (< 4-oo) and assume w G G~ a fixed point such that \w\ > 4M. Then, in view of absolute convergence of (3.5) and (1.8) \\0g \Ba{w,{Wk})\\

<

^\\0g\ba{w,Wk)\\ k

< 1 1 ^ Yl ii"i "''^i'^" I kr^'+"^-

(3-6)

3.2, The below lemmas are to be used in investigation of the properties of W~^\og\Ba{w,{wk})\' Let the sequence {wk}i^ C G~ satisfy (3.3) for a given a G (—l,+oo). Then lui Wk -^ 0 3,s k -^ 00 and, as it was mentioned above, for any closed half-plane G^ = {w : lin w N -\-l. Introduce the function N ija{w)

=\og\Bcy{w,{Wk}T)\

-^\0g\bc,{w,Wk)\ k=l

00

=

Yl k=N+l

iog\ba{w,Wk)\.

(3.7)

BLASCHKE T Y P E P R O D U C T S

35

For k > N -\-l the summands in the last series are harmonic functions in G^. On the other hand, from the proof of Theorem 3.1 it follows that this series is absolutely and uniformly convergent in G^. Therefore '0a ('^) is harmonic in G~. Using the inequalities (1.6') one can show that in G^ there is also another absolutely and uniformly convergent representation: — -^^{w)= 9(Im w)

V — log \ba{w,Wk)\. ^ J ^ ^ a(Im w)

(3.70

Lemma 3.2. If (3.3) is true for a given a G (—1, +00)^ then the series 00

J2

W-'^loglbaiw^Wk)]

k=N-\-l

is absolutely and uniformly convergent in the closed half-plane G^. Proof. We have Wk ^ G^ for A: > A^ + 1 by the choice oi N = Np. Besides, for /c > AT + 1 \W--\og\bo.{w,Wk)\\<

2 llm wi.\^'^^ ^^^_^^^^ ^^^^'^ ,

wueG^,

(3.8)

where 6 — max/c>Ar+i |Im W]^\ < \p\. Hence the desired assertion follows. Lemma 3.3. / / (3.3) is true for a given a G (—1, +oc); then oo

l^->,(^)=

J2

Wlog|6a(^,^fe)|,

wGGp,

(3.9)

k=N+l

where the series is absolutely and uniformly convergent in G^. Proof. Setting 6 = maxfc>iv+i |Ini Wk\, preliminarily we give two estimates which are consequences of (1.6) and (1.6 ') and are true for u G (—OO,+CXD), t G (-00, p) and fc > AT + 1: \logKiu+

it,w,)\\

< (|,|_'^)i^J^";^'i]^",

— log\ba{u + it,Wk)\ ^ ( | , | _ j ) 2 + J I " ^ ^ ^ l ' ^ " (a) Let a > 0, then from (3.10) it follows that the integral

i{w)^-— i [a)

{v-tr-' 7_oo

Yl i^ogKiu+it,wk)\\dt i,_Ar_Li k=N+l

(3.10) (3-10')

36

CHAPTER 2

is convergent for w = u-hiv G C^. Indeed, since —oo
"»)^Ti^fibClj--ii/-'

(|«|-<5)i+«

dt < +00

by (3.10) and the convergence of the series (3.3). Hence, iov w = u + w G G~^ -J

pv

W-''Mw)=——

00

{v-t^-^

l^y ^ - 0 0 00

=

Y. ^og\ba{u + it,Wk)\dt k=N+l

^

pv

YJ W^ {v-t)''-Hog\baiu k=N+l ^"' '^-°°

+

it,Wk)\dt

00

= Y,

W-^\0g\ba{w,Wk)\

k=N+l

in view of the definition (3.7) of '0a(tt') and the convergence of I{w). (b) Let — 1 < a < 0. Similar to previous case, (3.10') imphes the convergence of \og\boc{u + it,Wk) dt k=N-\-l

iov w = u + iv G Gp . Hence, at any w =

= J2 fvTTTT^ /

u-\-iv£Gp

iy-tT^^ogK{u

+ it,Wk)\dt

CO

=

^

W-"log|6,(^,^,)|.

k=N+l

3.3. The following theorem is a consequence of formula (3.7), Lemmas 3.2 and 3.3 and the properties oiW~^ log \hoc{w^Wk)\ stated in Theorem 2.1. Theorem 3.2. 1°. If a sequence {wk} C G~ satisfies (3.3) for a given a G (0,+oo)^ then the function W~^\og\Ba^{w^{wk})\ is continuous and subharmonic in G~ and harmonic in the domain G~\ Uk [wk^Re Wk), and

W-^log|5,(i/;,{^^})|<0,

weG-

BLASCHKE T Y P E P R O D U C T S

37

2°. / / {wk} C G~ satisfies (3.3) for an a e (-1,0); then the function W~^\og\Boc{w,{wk})\ is continuous and superharmonic in G~\{wk} and is harmonic in the domain G~\Uk [wk^Ke Wk)- Besides, W-''\og\Ba{w,{wk})\<0 for any w G G~ such that \w — Re Wk\ > |Im Wk\ (fc = 1,2,...). R e m a r k 3.2. By a similar argument, one can also show that under (3.3) the series H^-^ log \Ba{w, {wk})\ = 5 ^ 1 ^ - " log \ba{w, Wk)l a G ( - 1 , +oc),

(3.11)

k

is absolutely and uniformly convergent in any compact K C C disjoint from condensation points of the sequence {Re Wk}- Therefore, by Theorem 2.1 W~^ log \Ba{w^ {'^/c})| permits continuous extension from G~ through any interval (a,6) containing not more than finite number of points Uk = RewkBesides, ^ - ^ log \Ba,{u, {wk})\ = 0 , a
l i y - ^ log \Boc{u + iv, {wk})\\ du = 0.

/

Proof. By (3.7) for w G

G-\{wk}

\W-''\og\B^{w,{wk})\\
where the series is convergent. Therefore, by (2.23) + 00

/

-oo

(3.12)

-oo

I W " log \Ba{u + iv, {wk})\\du

for any v <0. Hence, by (2.19) and Fatou's lemma + 00

{W-"^ log \Boc{u + iv, {wk})\\ du /

-oo + 00

/

{W'"^ log \ba{u + iv, {wk})\\ du = 0. -oo

(3.13)

38

CHAPTER 2

Remark 3.3. Since W~^ log \Ba{w, {wk})\ {a > 0) is subharmonic in G~ and satisfies (3.12), one can use the well-known Littlewood's theorem on boundary values of subharmonic functions (see, for instance, [105], Ch. IV, Sec. 10, Theorem IV.34) to prove that for almost all ix G (—oo, +oo) T^-" log \Ba{w, {wk})\ - 0,

lim

(3.15)

where l{uo) is the image of a radius for any conformal mapping of |2:| < 1 onto G-. Lemma 3.4. If {wk} C G~ is a bounded sequence {i.e. sup^ \wk\ = TQ < +CXD) satisfying (3.3) for a given a G ( - 1 , 4-oo)^ then for w e G~ {\w\ > 4ro)

\W-'^log\B^{w,{wk})\\ < |

^ ^ ^ E | I ^ ^ ^ I ' ^ 4 H " ' -

16) (3-

Proof. Assume w G G and \w\ > 4ro. Since \w — Re Wk\ — |Im Wk\ > \w\ - (|Re Wk\ + |Im Wk\) > \w\ - 2\wk\ > \w\/2 + 2(ro - \wk\) > \wk\/2, (2.4) implies 4

l+a.

1-1

Therefore, (3.16) follows from (3.11). 3.4- Now we pass to the properties of the products Ba{z, {zk}) = Y[ba{z, Zk),

-l< a < +00,

(3.17)

k

with zeros in the upper half-plane G^. Some of^these properties are described by means of the integro-differential operator W~^ (formulas (1.23)-(1.25) in Ch. 1). One can verify that the inversion w — z~^ {wk = z^^^ fc = 1,2,...) transforms Theorems 3.1, 3.2 and 3.3 into the following equivalent assertions. Theorem 3.1*. 1°. Let {zk^'i^ he a sequence of complex numbers in the upper half-plane G^ and let oo j

E

^ I l+a

Im —

< +00

(3.18)

Zk

for a given a G (—1, +oo). Then the infinite product Ba,{z,{zk}T)^\{K{z,Zk)

(3.19)

k=l

is absolutely and uniformly convergent inside G~^, and Bct{z^{zk}) morphic function in G^, with zeros {zk}-

is a holo-

BLASCHKE TYPE PRODUCTS

39

2°. / / an infinite sequence {zkyf^ C G"^ lies out of some neighborhood of the origin, and for an a G (—1,+CXD) the product (3.4) is absolutely and uniformly convergent inside G^, then {zk}^ satisfies (3.18). In the next theorem I/[5,Re s~^) {s G G"^) means an arc with the endpoints Zk and Re Zk~^, which is the part of a tangential to the imaginary axis circle centered on the real axis. Theorem 3.2*. 1°. / / a sequence {zk} C G^ satisfies (3.18) for a given is continuous and subharmonic in a € (0,-hcxD)^ then W~^\og\Boc{z^{zk})\ G^ and this function is harmonic in G^\ U^ L[zA;,Re Zk). Besides W-^\og\Bo^{z,{zk})\<^,

zeG^.

2°. / / a sequence {zk} C G'^ satisfies (3.18) for some a e (—1,0); then is continuous and superharmonic in G~^\{zk} and is W~^\og\Ba{z,{zk})\ harmonic in G~^\ Uk L[zk,Re Zk)- Besides, W-^log\B^{z,{zk})\<0 for any z G G'^ such that \l/z — Re l/zk\ > |Im l/zk\ (/c = 1 , 2 , . . . ) . Theorem 3.3*. If {zk} C G+ satisfies (3.18) for an a E ( - l , + o o ) , then sup

/ \w-''log\Ba{Rsm^e'^,{zk})\\

d^ l+a < 4-00,

-r(2 + a)4lim

r

\w-^\og\Bo,{Rsm^e'\{zk})\\ -

(3.20)

Zk

^

= 0.

(3.21)

R^-\-oo

4. A REPRESENTATION OF THE PRODUCT 4.1. Below we shall prove a representation which will be used later, in Ch. 5, for investigation of boundary properties of our Blaschke type products. Theorem 4.1. Let a G (—1,1) be any number, and let {wk} C G~ be a sequence of complex numbers such that simultaneously ^^\lm

Wk\^^'^ <-^00

k

and ^

|Im tt;^! < + 0 0 .

(4.1)

k

Then for any w G G~ Ba{w, {wk}) = Bo{w, {wk}) +00

- - { ^ ^ - - - / r i ^ ^ } ' <"'

40

CHAPTER 2

where fi{t) is a nonincreasing, bounded function in (—00, +00) for —1 < a < 0^ and for 0 < a < 1 this function is nondecreasing in (—00, +00) but such that

J-00

1 + 1*1"+^

<+oo

(4.3)

for any e > 0. Besides, whatever be a G (—1,1) there exists a sequence dn iO by which fi{t) = lim / W-"" log

Bcc{u-i6n,{wk}) BQ{u-i8n,{wk})

du,

-cxD
(4.4)

4.2. Before establishing some necessary lemmas, note that for a = 0 l0gba{wX)

=-^a{w,C),

W,CeG~,

Q; > - 1 ,

becomes the main branch of the logarithm. Lemma 4.1. Let o; G (—1,1) and ( = ^-^ir] e G~ be any fixed numbers. Then the function ^„KC)=W^-'*log^^

(4.5)

is holomorphic in C\{^ + i/i: 0 < h < +00} where r ( l + a)Mw,

0= J\r,\

+ /

\[a + i{w^

or'

-[
(^"da -"

\^[a-i{w-C)]''-[a

+ iiw-C)]~''^a"da.

(4.6)

Proof. Preliminarily we shall show that VF-"log6o(w,C) = -fiTTa)i

{W + i{^-Or'-[
+ i{u>-Or'}<^''da

(4.7)

ioi w ^ {(-\-ih : 0 < h < +00}. Indeed, for o; = 0 (4.7) is trivial since W^ is identical operator. For 0 < a < 1 integration by parts gives

r ( l -i-a)

JQ

O-

+ i{w - C)

which implies (4.7). If - 1 < o; < 0, then

w-'^iogboiw.o = w-(i+") {[i(^ - 0]"' - [i{w - o r ' } .

BLASCHKE TYPE PRODUCTS

41

Hence (4.7) follows. The representation (4.6) providing the holomorphity of (Pa{wX) in the required domain follows from (4.7) and (2.2). Lemma 4.2. Let a G (—1,1) and ( = ^-{-ir] e G~ be fixed. Then the function Re (pa{wX) ^5 harmonic in G~ and continuous in G~. Proof. By Lemma 4.1, it suffices to prove the continuity of Re (fai"^, C) ^^ the point If; = ^ or, which is the same, the continuity of
(4.9)

+ a)^^{w + tC)

at the origin. To this end, we subtract from (4.6) the similar formula for ^o{wX) (= 0) multiplied by \rj\^. This gives ^aiw,C) =

\[aJ\rj\

\v\-hiw]-'

~ [a + \rj\+iw]-H{a^

^

+ /

-

\rjnda

^

{W- \v\ - H ~ ' -W- \v\ + H " ' } K - IvHdcT.

But \a^ — \rj\^\ < CaW — \rj\\ (0 < a < +oo), where Ca depends only on a and \r]\. Therefore, estimating the real parts of integrands and using the Lebesgue theorem on dominated convergence, we conclude that Re cp'^{w,Q is continuous in G"~ and /'+00

lim R e ^ ; K 0 - 2 M /

a _ I la

^ r ^ d a .

-^0

Im w<0

4^3. Lemma 4.3. For any ( ^ G Re(/:^*(ii;,C)

I /I

and w G G~ < 0,

if

- 1 < a < 0,

> 0,

if

0
(4.10)

Proof. We shall initially prove these estimates iov w = u G (4.9) and (4.6)

J^ +J

ja^dlog

a — \r]\ -f m = ( j + IT^I +iu

-oo,+oo). By

Il{u)-\-l2{u),

Besides, if 0 < cr < |?7|, then _9_ log (J — \r]\ -{- in cr-f- \rj\ + m da

\{a + m)^ — 7/21

But a"^ < Ir/I'" for a > 0, and a"^ > Irj]"^ for a < 0. Consequently,

-i^riog

2\rj\+iu

> Ii{u)

if

- 1 < a < 0,

if

0
OO < U < + 0 0 .

42

CHAPTER 2

On the other hand, integration by parts gives 2|r/|-hm

l2{u)^\ririog

/'4-00

-a

a — \rj\ i-iu cr+ \r]\ -{-iu

log J\ri\ 7l

^a-l

da.

The above estimates of Ii{u) imply (4.10) for Re (^*(i(;,C) (Im if = 0). For extending the inequalities (4.10) to all w G G~, it suffices to use the PhragmenLindelof principle and the estimate

IV:KC)I<3

1 1^1l+a

\v\ \W 1-a

l + a \w\

,

If; G G " , \w\ > 2\r]\,

which follows from the representation (4.7), (4.9) of ^ ^ ( ' ^ ' 0 by evaluation of modules of integrands. Remark 4.1. The last estimate of (/?* (ti;,C) implies that \ipa{u+iv,C)\ <

3r(H-a)

\rj\ , 1 |77|i+, v<-2\r]\. ii;|i-" l+a \v\

(4.11)

On the other hand, using (4.6) and (4.10) one can verify that if i; < 0 is enough small, then for any rj € (—1,0) CalvM'^^" |r/|i+"|t;ri

\Re ipaiiv,ir])\ >

if if

0
(4.12)

where Ca and C^ are positive constants depending solely on a. 4-4- Assuming {wk} C G~ an arbitrary sequence satisfying the conditions (4.1) for an Q; G (—1,1), consider the following function which is holomorphic in G~: Ba{w,{wk}) Bo{w,{wk})

^a{w,{Wk})=W-''log

= E^-°>»«^-E-(»-). ("3) k

^

^

k

Note that the last two series are absolutely and uniformly convergent inside G ~ (this follows by (4.11) and by the convergence of the series of VK"^'! log ba\)- On the other hand, the first two equalities in (4.13) are true by Fubini's theorem (see (1.8) and (2.4)). Lemma 4.4. Let a G (—1,1) be arbitrary, and let a sequence {wk} C G~ satisfy (4-V' Then ^oc{w,{Wk})

1 /"^^ d^i{t) = — 7^^ J-oo W — t

w G G~,

(4.14)

BLASCHKE T Y P E P R O D U C T S

43

where ^{t) is a nonincreasing and hounded function (—00, +00) for — 1 < a < 0; and for 0 < a < 1 this function is nondecreasing in (—00, +CXD) but satisfying (4-3) for any 5 > 0. Besides, for any a G (—1,1) there exists a sequence 6n i^ by which (4-4) holds. Proof. By (4T3), (4T0) and the Herglotz-Riesz' theorem, in both cases — 1 < a <0 and 0 < a <1 1 f^"^ ( I t ] ^a{w, {wk}) = ipw-\- — < 7 -h :——^ > dfi{t) +iC, TTZ J_^ [w-t 1+t^ }

w eG

,

where p and C are some real numbers and the function //(t) is nonincreasing for —1 < a < 0, nondecreasing for 0 < a < 1 and such that

/.

' \i^ < +00, 1 H- r^

- 1 < a < +00.

Besides (see, for instance, [4], Ch. I, Sec. 4), there exists a sequence J^ i 0 by which (4.4) holds. Now observe that by (4.11) and (4.13) \^a{u + iv,{Wk})\ <

3 r ( l + a)

ui^Eii--^i + a+.)r 7r:^

-'""

for V < —2in3,Xk\lT0[iWk\' Consequently, supy^Q\v^a{iv,{wk})\ < +00 for — 1 < a < 0. Hence (4.14) and the boundedness of fj,{t) follows (see, for instance, [4], Addendum I, Sec. 4). As to the case 0 < a < 1, it is obvious that

lim ^a{iv, {wk}) = 0

and

f^"^ dt / \^a{~it, {wk})\ —— < +00

for any e > 0 (we cannot get rid of e in view of the below estimate for Re $«? which follows from (4.12)). Hence we come to the representation (4.14) and to the relation (4.3) with obligatory e > 0 (see [4], Addendum I, Sec. 3). 4.5. Lemma 4.5. Under the conditions of Lemma 4.4; W'^^aiw.iwk})

= log J^)^' e l l ^

Imw < -max|Imt<;fe|.

Proof. We start by the case when the sequence {wk} consists of a single term C = ^ + ir/ G G~ and use the formula r(7 + n ) r ( l - 7 ) 1 ,+°° <^-''da [ ( . + a ) » + i = nl ^ ^ ' /

^ / nl -^(—'0]'

,A ,<,^ (4.15)

44

CHAPTER 2

which particularly is true for any integer n > 0 and any 7 G (—n, 1). Let - 1 < a < 0 and Imw
= ~r(i-f-a)r(i-a) y_|,|^l^l" I^I^^^Vo

[i(^-0-t + a]2

==log6a(if;,C)Besides, W^«lpy-«log6o(u;,C) = -(l-a)

dt

—

-7—

T7^-—=logbo{wX)

since by formula (1.13) of Ch. 1 and (4.15) /"'"^l df W-^ log6o(., C) = - r ( l - a) j_^^^ [ , ( ^ _ ^ ) _ , , , _ . • For 0 < a < 1 and lmw
one can obtain the same equalities in a similar

W>a(^,C)=log^4^'

Initi;
aG(-l,l).

Hence the desired formula follows by (4.5) and the absolute and uniform convergence of the series Ba{w,{Wk})

bo{w,{wk})

^ Y^i

^

ba{w,Wk)

bo{w,Wk)

for Im If; < — max/c |Im Wk\ (see. (3.5)). Proof of Theorem 4.1. In view of Lemmas 4.4 and 4.5, it is sufficient to prove the equality ,

w eG

,

under the assumption that ii{t) is as required. One can directly verify this equality using (4.15).

CHAPTER 3 EQUILIBRIUM RELATIONS AND FACTORIZATIONS

1. F. AND R. NEVANLINNA, CARLEMAN AND LEVIN FORMULAS In this section F. and R. Nevanlinna type formulas are established in two starshaped domains: in the semidisc G~^{R) = {z : Im z > 0, \z\ < R} {0 < R < +CXD) and the half-plane G~ = {w : Im w < p} (—CXD < p < 0). Then some limit passages lead to a Carleman type formula for the semidisc and a B.Ja.Levin type formula for a disc that is tangential to the real axis. 1.1. For w e C we set V'lw, oo) = {C = w -h i(T : 0
: w e D, 0
We shall use the following estimates for the products of Ch. 2: |log|5.HI| < ^ d

I 9(Im w)

\og\Bc,{w)\

|Im w\ -l-a

(T\lmw,\'^A

(1.1)

-2-oc

(1.10

\W\ - l - a

(1.2)

<2^+" [^|Im^A:|^+^] \lrm

which are true for Im. w < — 2maxA; |Im ifj^l and

\\og\B^{w)\\<'^(y^\\mwk\'^A d \og\Boc{w)\ 9(Im w)

< 2^+^ Yl ii^ Wk

l+a

\W -2-a

(1.20

which are true for sup;. \wk\ = M < +cx) and \w\ > 4M {w G G~). The proofs of these estimates are similar to that of inequality (3.6) in Ch. 2.

46

CHAPTER 3

1.2 Lemma 1.1. Let F{w) be meromorphic in a oo-starshaped domain D, and let F{w) have finite sets of zeros ak and poles bn in the oo-starshaped continuation of any domain Di C D for which Di\{oo} C D. Then: V. If log\F{w)\ G Ma{D) for some a > 0, then W'"^ log \F{w)\ is continuous in D and harmonic in D\{[UkT'^[cik, oo)] U [UA;r*[6n, oo)]} . 2°. / / d/d{lm w)\og\F{w)\ G Mi^a{D) for some a G (-1,0); then the function W~^log\F{w)\ is continuous in D\[{ak}U {bn}] and harmonic in D\{[UkT*[ak,oo)] U [UA;r*[6n,oo)]}. Besides, in a neighborhood of each point A G {ak} U {bn} W " ^ log \F(w)\ = ^ A ^ ^ ^ ' ^ V - ^ r cos{aarg[i(^ - A)]} + V^a(^, A), a where da is the order of zero {then da > 1) or pole {then da < —1) in A, and ' ^ ^ ( K ; , ^ ) is a harmonic function. Proof. Assuming e > 0 and p > 0 arbitrary numbers, introduce the domains D^ = {w e D : Oe{w) = {C :\C-w\ <£} C D} and De,p ^ D^D G". Since F{w) has a finite set of zeros and poles in Ds,p^ the function Y\

ba{w

-ip,bn-ip)

Yl

ba{w

-ip,ak-ip)

F{w)

Uoc{w) = l o g

- 1 < a < +00,

ak^De,p

is harmonic in D^^p, and W-'^ log\F{w)\ = W-'^Uo^iw) +

J2

^ ~ " ^°g l^"(^ - 'P''"' - ^P^\

akeDe,p

-

Y^

W-'^\og\ba{w-ip,bn-ip)\,

-l
(1.3)

bn^Ds^p

For proving 1°, observe that by (1.2) |log \boc{w -ipX-

22+a ^p)\\ < C{1 + a)——\rj - - n|l+«|^|-(l+«) p 1 -\- a

{\w\ > RQ, Im W < 0) for enough great RQ > 0 and any C = ^ -^ '^V ^ G^ . Hence log \ba{w -ipX-

ip)\ e Ma,{De,p)

and

Ua{w) G Ma{Ds,p).

Consequently, by Lemma 1.8 (Ch. 1) W~^Ua{w) is harmonic in D^^p, and by and Theorem 3.2 of Ch. 2 W~^ log |-F(tf^)| is continuous in D^^p and harmonic

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

47

in Ds^p\{[UkT*[ak^oo)] U [Unr*[6n5 oo)]}. Hence our assertion follows since e and p are arbitrary. The proof of 2° is similar. It is based on the inclusions • log \ba{w -ipXaim w

ip)\ ^ ^2+a ( ;7^£,p ) C Mi+a(D^,p) \2

which follow from (1.2'). R e m a r k 1.1. If the zeros and the poles of F{w) from Lemma 1.1 are such that the sets {yjk^*[ak^ CXD)} and {Unr*[6n, oo)} are disjoint, then (1.3) and the properties of l^~^log|6a| (see Theorem 3.2 in Ch. 2) provide the validity of the following additional assertions: (a) if a G (0,4-00), then W~^\og\F{w)\ is subharmonic in the points w G {UA;r*[aA;,cxD)} and superharmonic in the points w G {Unr*[6n, CXD)}, i.e. for enough small r > 0

SI ^ ^

^1

\>W-^\og\F{w)\,

«;G{U„r*[6„,oo)};

(b) if Q; G (—1,0), then in the same sense the function W~^\og\F{w)\ is superharmonic in the points w G {UA;r*[aA;, oo)\ {ak}} and subharmonic in the points W G {Unr*[&n,00)\{6n}}.

1.3. The following two lemmas will be necessary. Lemma 1.2. Let a function u{z) he harmonic in the semidisc G^{R) = {z : Im z > 0, |z| < R} {R > 0) and be continuous in its closure, with possible exception of a finite set of points {cm}i C dG^{R) {cm 7^ 0, 1 < m < g) m neighborhoods of which U{z) = dm log \Z - Cm\ + ^m{z)

u{z) =Em\z-

Or

Cm\~^\cmzr COS {7arg [i{z~^ - c~^] } -h ^^miz)

(0 < 7 < 1); where dm and Em CLTC real constants, and ^m{z), harmonic functions. Then for z G G'^{R)

^TT IdG+iR) JoG+(R)

2Tr Ja dG+{R)

«(C)

^" 1

[C-z

1

C-z

/

r

\R^-Cz

T'

R'^-Cz

(1-4)

^m{z)

are

M

dC,

(1.5)

where G{(,z) is the Green function ofG^{R), d/dn is the differentiation along inner normal, and ds is the element of the curve length.

48

CHAPTER 3

Proof. u{z) can be written as a Poisson integral in dG'^{R) since it can have only integrable singularities (1.4) on dG^{R). For a similar situation, one can find a detailed proof in [38] (Theorem 1.1, Sec. 1, Ch. 1). Lemma 1.3. Let U{w) he harmonic in the half-plane G~ {p < 0) and continuous in its closure C^ = {w : Im. w < p}, with possible exception of a finite set of points {Am}i C dG~ {Am ^ oo^ 1 <m < q) in neighborhoods of which U{w)

= dm log \W-Am\+

U{W) =Em\w-

Am\-^

(fm{w) COS {^[i{w

Or - Am)]}

+ ^m{w)

(0 < 7 < 1)

(1-40

where dm CLTid Em are real constants, and cpmi'w), i^mi'^) (^f^ harmonic functions. If + 00

/

\U{u + iv)\du < 4-00, -oo

then + CXD

/

\U{u + ip)\du< 4-00

(1.6)

-oo

and 7-T/ X

U{w)^^-

k-pl

/»+oo

f

U(u-\-ip)du

^ / ) ^^ TT j _ ^ {u - ty + {v-

.

T2 < + ^ ' w = u^iveGp.

^_

PY

P

,^ ^.

1.7

Proof. The function V{w) = U{w-\-ip),

weG-,

(1.8)

is harmonic in G~. Besides, by the properties of U{w) V[u^)JAr^

^^f

^,

w = u^-iveG-,

(1.9)

where p{t) is of bounded variation in (—00, +00) and +00

/

/'+00

f{u)V{u + iv)du^ -00

/

f{u)dii{t) J-00

for any f{u) continuous in (—CXD,+CXD) and such that \miu-^±oo f{u) — 0 (see, for instance, [36], Ch. 1, Theorems 5.3 and 3.1(c)). It suffices to prove that in our case the measure dp{t) is absolutely continuous and equals V{t)dt. To this end, for an arbitrary segment [a, 6] G (—00,+00) introduce a sequence {fn{u)}T of finite, continuous functions in the following way: fn{u) = 1 for u € [a, b], fn{u) = 0 for t^ G (—00, a — 1/n] and u £ \b -\- 1/n, +00), and fn{u)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

49

to be continued to the remaining intervals as a linear function. Then, by the properties of V{w) +00

/*+oo

/

fn{u)V{u)du= lim / for n > 1. On the other hand, pb

p-\-oo

fn{u)V{u^-iv)du= /'+00

I V{u)du — lim /

fn{u)d^{u)

ph

fn{u)V{u)du

= /

«/ —oo

*J a

I

dii{u).

*J d

Therefore, in (1.9) dii{t) = V{t)dt and V{t) G L^(—oo,+CXD). Hence our assertion follows by the definition of V{t). 1.4' Now we turn to F. and R. Nevanlinna type formulas. Theorem 1.1. Let f{z) he meromorphic in the closed half-plane G+\{0} = {z : Im z > 0, z 7^ 0}; and let for some RQ > 0 this function has no zeros and poles in the semidisc G+(i?o)\{0} = {z \ Im z > 0, \z\ < RQ, Z ^ 0}. Further, let a > —1 and K, > 0 be any numbers for which one of the following conditions is satisfied: r . log|/(z)| G ^ a + / . W 2 , G + } , if ae [0,+oo), 2^ 9/a(Im l/z)\og\f{z)\

G ^ i + a + 4 7 r / 2 , G + } , if a e ( - 1 , 0 ) .

Then for any —l 0 and any z G G~^{R) which does not belong to the set of zeros {ak} or to the set of poles {bn} of f{z), W - " log | 6 c ( C , a , ) | ^ ^ ^ d s - W^-" log |6„(Cafc

- J2 \^ I

W--l0g\UC,bn)\^^^ds-W-"l0g\MC,bn)\\

= 1/" w--log\f{0\^^^ds-W-'^\og\f{0\, 27r JdG+{R) on where G{C,z) is the Green function

(1.10)

ofG~^{R).

Proof. The function •^"(^) ^ W~^—T~r

T-^(^)

( - K a < +oo)

(1.11)

is meromorphic in G+(i?)\{0} and it can have only a finite set of zeros and poles is harmonic in {j4m}f c dG'^{R). Therefore, by Lemma 1.1 W~°'log\fa(z)\

50

CHAPTER 3

G+(i?)\{0}, except the points {Am}i, where one of representations (1.4) holds (the first one for a = 0, the second one with 7 = |a| for — 1 < o: < 0, and the second one with 7 = 0 for 0 < a < +00). On the other hand, by (1.2)-(1.2') for 5 G G+ log \ba{z, s)\ e Ki^a{7r/2, G^} d/d{lm l/z)\og\b4z,s)\

if

0 < a < +00

e ^2+aW2,G+}

if

- 1 < a < 0.

Hence, by Lemma 1.9 of Ch. 1 and our requirements in both cases 1° and 2°, W-"^ \og\fa{z)\ -^ 0 as z -^ 0 remaining in G+, i.e. W-"" log\fcc{z)\ is continuous at z = 0. Thus, this function satisfies the conditions of Lemma 1.2, it admits the representation (1.5) which is easily transformed to (1.10). Note that for Of = 0 (1.10) coincides with R. and F. Nevanlinna formula (see, for instance, [38], Ch. 1, Theorem 2.1) for the semidisc G~^{R). Besides, if a > 0, then (1.10) is true in all points z G G'^{R). Also the following useful remark is true. Remark 1.2. For a > 0 the function W~^ log \ba{z^ s)\ is subharmonic in G+. Hence, for a > 0 the terms of left-hand side sums in (1-10) are nonnegative. Theorem 1.2. Let F{w) be meromorphic in the lower half-plane G~, and let for some p < 0 this function have not more than a finite set of zeros and poles in the closed half-plane G^ = {w : lui w < p}. Further, let a G (—l,4-oo) be any number and r . l o g | F ( ^ ) | GMa{G-}

if ae

2°. d/d{lmw)log\F{w)\

[0,+oo),

GMi+a{G~} if a e (-1,0).

// sup /

IW'"^\og\F{u

+ iv)\\ du < +00,

v

then W ^ log \F{u-{-ip)\ G L-^(—oo,+oo), and for any w = u-\-iveGp, which does not belong the set of zeros {ak} or to the set of poles {bn} of F{w) - ^

W'^^loglbociw-ip,ak-ip)\-\-

^

Im ak
W'^loglbaiw

- ip.bn - ip)\

Im bn
Proof. The function rTlrr. h

^nbai'W

F . H = i^""''"<\°

-

Ip,

bjl

"

ip)

F(u;)

(-l
(1.13)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

51

is meromorphic in Gp , except the sets {Re ak -\-ip : Im ak < p} C dGp and {Re bn -\-ip ' Im bn < p} C dG~. And this function can have not more than a finite set of zeros and poles {qm}i C dG~ {qm y^ oo, 1 < m < N) which are some of zeros and poles of F{w). By Lemma 1.1 and Theorem 3.2 in Ch. 2, W~^log\Fa{w)\ is harmonic in G^, except a finite set of points on 9G~, where one of the representations (1.4') is valid. Besides, by (3.14) of Ch. 2 +00

/

IW-"^ log \Fa{u -\-iv)\\du<

+00.

-oo

Thus, W ^log\Fa{w)\ satisfies the conditions of Lemma 1.3. Consequently, W~^ log \F{u-\-ip)\ G L-^(—oo, +oo) and (1.12) holds. This completes the proof. Note that for a = 0 (1.12) becomes the formula of F. and R. Nevanlinna (see, for instance, [38], Ch. 1, Theorem 1.1) in G~. Besides, for a > 0 formula (1.12) is true in all points w € G~. Besides, formula (1.10) of Theorem 1.1 as well as formula (1.12) of Theorem 1.2 can be written in equivalent forms by means of inversion w = z~^ and similar changes of variables. 1.5. The following two lemmas will be used later for proving a Carleman type formula. For any a G (—1,4-CXD), r > 0 and w, ( E G~{r)

= {w : Im w <

0, \w\ > r} we set jr = {s = re'^ : -TT < i? < 0},

(7^ = [T^

1-1

(1.14)

(we assume that this arc is directed clockwise) and

s—w

w r^ — sw

w

w r^ — w

ds.

(1.15)

Lemma 1.4. For any a G (—1, +00), r > 0 and ( = ^ -\-ir] £ G (r) Ia{C.r) = l

lim \v\Uiv,C,r)

2 t^^-oo

= - [ ]v-^logba{re'\C)^inM^,

(1.16) (1.17)

L [2 -\- a)

Proof. The function W~^logba{s^C) is continuous by s in G~, with possible exception of the point ( G G~{r) (see Theorem 3.2 in Ch. 2). Therefore, using the equality lim \v\

1 s — IV

IV s -\-iv

IV

+ r^ +

isv

2i{

^

1

(1.18)

52

CHAPTER 3

we get laiCr)

= ^ J W-'^log\b4s,C)\

( ^ - l ) ds

= - / W-"\og\ba{re'^,0\smM'd. •^ Jo On the other hand, by formula (2.2) of Ch. 2 I

hm \v\W-^\og\bc,iiv,0\

I"! (1^1 - l
= l

\v\ t-\v\+i^

\V\1+a r(2 + a)' Lemma 1.5. For any a > —1, r > 0 and ( = ^-}-ir] ^ G~{r) J.2

r\r]\

a j

(1.19) |77|--^r2-{2

Ia{(i,r)

'n1 + a

T^d^

rRe /

r(2 + a)

(T-icy Jo l+a i^|i+a _ (i^i _ y;:T3:e^)

z/ 1^1 < r.

(1.190

Proof. Since W " log |6a(re^^,C)l == Re {T^-« log6a(re^^,C)}, (1-16) can be written in the form Ia{C,r) = ^Re J W-'^logbais^ol^-Ads.

(1.20)

Substitute the representation of W~^ \ogba{s, () as a Cauchy type integral (i.e. formula (2.2) of Ch. 2) and change the order of integration (for the validity of this operation see, for instance, [34], Ch. 1, § 7). This gives

Changing the variables: \T]\ —t = T in J^

/a(C,r-) = | ; ^ Y T ^ /

and Irjl+t = T in /_, ,, we get

Re{J(T,C)Kdr,

(1.21)

where ds s — {C -{-ir)r~'^'

^^^->-rJ,JH

(1.21')

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

53

Now set

j(

.. ^ J_ f

ds

^_ r

ds

= J i ( r , 0 + J2(T,C),

(1.22)

and calculate Ji and J2 in the assumption that r G (0, |7/|). Two cases are to be considered. (a) 1^1 = I Re CI ^ '^- The integrands in Ji and J2 have first order poles at •50 = (C + ^'^)^~'^- Besides, the integrand of J2 has a second order pole at s = 0. Consequently, 1

r2

since in case |5o| > |Re 5o| = |^^~''^| ^ 1- Hence (1.19) holds by (1.22) and (1.21)-(1.210. (b) 0 < 1^1 = |Re CI < r. In this case the pole SQ = {(-\-ir)r~^ of the integrands in Ji and J2 is out of the unit disc for 0 < r < jr/j — y^r^ — ^2^ ^nd n lies inside the unit disc for \r]\ — \/r^ — (^ < r < \(]\. Hence ^^(^'0=^(7^^.

0<|e|
0
(1.23)

if 1^1 - \/r2 - (^2 < r < IT^J. And ^i(^.C) = ( res + r e s ) ^ ^ hijX) if Iryl - ^/r^~^

= -res

—

. . , . . _ n = 0,

r-T—T = - 1

< r < [ryl. Thus, by (1.21)

J{r, C) = - 1 ,

0 < 1^1 < r,

Iryl - Vr^ - ^
\vl

(1.230

Formulas (1.22), (1.22'), (1.20) and (1.20') imply (1.19'). Remark 1.3. For a = 0 (1.19) and (1.19') coincide and become

Ioi(:,r) = -r'^4-

(1-24)

54

CHAPTER 3

Besides, if |^| > r, then ]{( - r\ > r for r G [0,\r]\]. And if |^| < r, then \i( — r | > r for T E [0, \r]\ — ^Jr^ — ^\. Hence, one can obtain from (1.19) and (1.19') that for any a G (—1, +CXD) and ^ G (—oo, +oo) |^,l+a

l^(C,r-)l< r(2 + a ) '

C = ^ + i»?€G-,

Kl>r.

(1.25)

i.^. Theorem 1.3. Let a function f{z) be meromorphic in the closed halfplane G+\{0} = {z : Im z > 0, z :^ 0}, and let for some RQ > 0 this function have no zeros and poles in the semidisc G+(i?o)\{0} = {z : Im z > 0, \z\ < Ro, z ^ 0}. Further^ let a G (—1, +oo) and K, > 1 be any numbers, and let r , \og\f{z)\

G ^ c . + . W 2 , G + } ifae

[0,+oo),

2 ^ 5/S(Im l / z ) l o g | / ( z ) | G ^ i + , + 4 7 r / 2 , G + } , ifae

(-1,0).

Then for any R> 0

E• |afc|
1 r ( 2 + a) l+a

+ /a 4:- / ^R Jo

W-''\og\f{Re'^)\smM^

+sr(^-s^)"'""'°^i«')«-'>i'«- (>• 26) where ak are the zeros and hn are the poles of f{z) and the terms left-hand side of sums are nonnegative. Proof. f{z) satisfies the conditions of Theorem 1.1. Hence (1.10) is true. Taking z — iy {0 < y < R) in (1,10), let y ^ +0 after multiplying both its sides by {2y)~~^. This gives (1.26) by Lemma 1.4 and (1.17). It remains to see that the terms of sums in (1.26) are nonnegative by (1.25). Remark 1.4. Since W^ is identical and (1.24) is true, for a = 0 formula (1.26) coincides with the well-known Carleman formula for the semidisc (see [10], Sec. 1.2.2 or [77], Ch. V, formula (5.02')). Theorem 1.4. LetF(w) be meromorphic in G~ =^ {w : Im ti; < 0}; and let the sets of zeros and poles of F{w) in the closed half-plane G^ = {w : Im w < p} be finite for some p < 0. Further, let a G (—1, H-oo) be any number, and let 1°. \og\F{w)\ € M „ { G - } ifae

[0,+oo),

EQUILIBRIUM RELATIONS AND FACTORIZATIONS 2°. d/dilm

w) log \F{w)\

GMI+„{G-}

55

z/aG(-l,0).

+00

I W ^ log \F{u -}-iv)\\du<

/

+00,

(1.27)

-CX)

then W-"" log \F{u + ip)\e L\-oo,

+00)

(1.270

and for any —1 < a < +CXD l+a Im ak
^

Im On
1 /""^^ 1 = — / W-''\og\F{u-\-ip)\du-lim T^-^log |F(2i;)|, (1.28) 27r J_^ 2 i;-^-oo where ak and bn are zeros and poles of F(w), and the limit exists and is finite. Proof. F{w) satisfies the conditions of Theorem 1.2. Hence (1.27') and (1.12) hold. Taking w = iv {v < p) in (1.12), multiply both sides by \v\/2 and let V ^ —OQ. In view of (1.17) this passage gives (1.28). Remark 1.5. For o; = 0 and w = \/iz (1.28) becomes a version of B.Ja.Levin's formula (see [77], Ch. IV, §2). 2. SOME OTHER FORMULAS OF CARLEMAN AND LEVIN TYPE Here we prove some other similarities of Carleman and Levin formulas which will be written in a semi-ring and in the domain {z : \z — iR/2\ < i?/2, \z\ > Ro} (0 < RQ < R < +oo). One has to mention that in some cases these domains turn to be more preferable (see, for instance, [38], Ch. 1, Theorems 3.1 and 3.4). Also some applications of obtained formulas are given. 2.1. We start by two preliminary lemmas. Lemma 2 . 1 . For any r, ro (0 < r < ro < +oo), any ( = ^ + irj e G~ (r < Id < TQ) and any —1 < a < +CXD

"^"•^ " " • ' W ^ ~ " ^°^ Kiroe'',

O l j sin^di? = :^^—^-

(2-1)

Proof. Set Q^'\r,ro,a,0=li'^+lP, li'^ = ^ y ^^^ = ~ h £

(2.2)

W"' log |6„(roe*^ C)l (^o + M sini^d^, ('•o -^')^W^~"log|^'a(r•oe^^C)|sin^d^.

(2.2')

56

CHAPTER 3

For writing la and la in more suitable forms, observe that \(\ < VQ, and by formula (2.2) of Ch. 2 W~^ \ogba{w, Q is holomorphic outside [C, C]- Integrating by parts and using Cauchy-Riemann equalities we get 0

/

^a^ = ^ J

/i'^ = ^ J l m

{W-^ logbairoe'^,C)}Re

^2

(roe '^

Re{W-"log6a(roe^^C)}Im

~^^ ^ dd. roe"

(^voe'^ - ^

)

d^.

(2.3)

From (2.2) of Ch. 2 it follows that the function Re {W-'^\ogba{roe''^,()} is odd and Im {VF""log6a(roe*^,C)} is even for i? G [—7r,7r]. Other multipliers in the integrands of (2.3) have similar properties. Using these propoerties we come to some new representations for / i and la , which differ from (2.3) by the multiplier l/47r instead of l/27r and by the integration contour that becomes the whole [—7r,7r]. This results in the representation

'i" + '^' = ^ - {2^ /„„. (' - ?) "'"" '»«"'-<»• « * ) • Inserting here the formula (2.2) of Ch. 2, changing the integration order and then calculating residues, we arrive at (2.1). Lemma 2.2. For any p <0, ro > \p\ any ( = ^-j-irj e G~ (|C| < ^o) o,nd any — \
1

= ^

\p\

/"— arcsin ^

/

ip)\)

r

[

\ W"logl^c(roe'''

X A ^ - « l o g | 6 , ( r o e ^ " ' -ip,C-

-ip,<:ip)\\rod^

ip)\sm^

- (rosin7? - p )

= ^^f^^fj^'^-

(2-4)

Proof. An argument similar to the proof of the previous lemma gives Q(^){p^ro,W-^\og\ba{w

-ip,C

-ip)\)

= -^Re

j W-^\ogba{s,C

- ip)ds,

where F is the union of the arcs 7'^(p,ro) (i.e. the part of \s — ip\ = ro in G+) and 7~(p,ro) (i.e. the part of \s + ip\ = ro in G~). Further, one has to insert into this formula the representation (2.2) (Ch. 2) of W~^ log6a(«5, C), to change the integration order and to count residues. 2.2. Theorem 2.1. Let a function f{z) be meromorphic in the closed halfplane G+\{0} = {z : Im z > 0, 2: 7^ 0}; and let for some Ro > 0 this function

57

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

have no zeros and poles in the semidisc {z : Im z > 0, \z\ < Further, let a G {—1, 4-cx3) be any number and let r . log\f{z)\

G M^{G-^} if ae

RQ} U

{—RQ^RQ}.

[0,+oo),

2°. d/d{lm l / z ) l o g | / ( z ) | G Mi+a{G+} if a e (-1,0). Then for any R > RQ and any — 1 < a < -hoo

E

Ro<\ak\
r ( 2 + a)

Im

1

|l+a

+ /a

Ctfe

E flo<|tn|
I

1+a

r ( 2 + a)

Im —

r

/ 1

1

On

W-'^\og\f{Re'^)\smM-d

+ hC{¥~h)

^"" ^°^ i/(*)/(-*)i'^*+^^'^

where ak are the zeros and bn are the poles of f{z), nonnegative and as R ^ +cx) Q('^^Q^'\R,Ro,W-'^log\f\)

(2-9)

the terms of sums are

^ - g jf'^|l^-logi/(i?oe'^)|(^ + - l

+ ' i ^ ~ i ^ ) •^^W-'^^og\f{Roe'^)\ymM^

= 0{l).

(2.10)

Proof. It is evident that fa{z) defined by (1.11) is meromorphic in the closed semidisc G~^{R)\{0} = {z : Im. z > 0, \z\ < R, z j^ 0}, where it can have only a finite set of zeros and poles {Amjl C dG'^{R)\{0}. Further, Fa{w) = fa{w~^) satisfies the conditions of Lemma 1.1. Therefore, u{z) = W~^ log |/a('2;)| is harmonic in G+(i?)\{0}, except some points {Am}l, in neighborhoods of which one of representations (1.4) is true, as these are integrable singularities. Repeating the argument which proved Theorem 3.2 in Ch. I of [38], one can verify that u{z) admits the representation (3.2) of that theorem, i.e. ^ / 27r

(^~^)

[''(^) + ^(-^)]^^ + ^

PuiRe'^)

sin7?^^

--QW(i?,i?o,ix), where Q^^\R,RO,U)

58

CHAPTER 3

Taking here u{z) = W~^ log |/a(^)| and granting that Q « ( i ? , Ro, W-'^ log K{z, 01) = Q^'HR-\

RO\ T ^ - " log

K{z-\C')\),

we come to (2.9). It remains to observe that by Theorem 1.3 the terms of the sums in (2.9) are nonnegative and (2.10) is true. Remark 2.1. As W^ is identical, from (1.24) it follows that formula (2.9) for a = 0 coincides with the Carleman formula in the semi-ring (see, for instance, [38], Ch. I, formula (2.1)). Theorem 2.2. Let f{z) be meromorphic in the upper half-plane G^ and also in the neighborhoods of points —RQ and RQ for some RQ > 0. Further, let f{z) have no zeros and poles in the semidisc {z : Im z > 0, \z\ < RQ}, let a € (—1, H-oo) be any number and let r . log|/(z)| G Mc^iG-^} ifaG 2\ a/9(Im l/z)\og\f{z)\

[0,+cx)),

e Mi+a{G+} if a e (-1,0).

Then for any R > RQ

r(2 + a)

E

1.1

a f c - i f <^

r(2 + a)

6.-if|
/>7r—arcsii

1 2^

E

/

I^-"log|/(i?sm^?e^'')|

-^0

./arcsin -^

+

1+a

Im

bn

M Rsm^§

Q^^\R,R^,W-''\og\f\),

(2.11)

where {a^} and {hn} are the zeros and poles of f{z) and as R -^ +oo

Q(2)(i?,i?o,W^-^log|/|)^^ / ^ ^ Jar

d

sin?? RQ

RJ

URQ

'^'T^-'^log|/(i?oe-)|(-^ ,^ ^'O

W - ^ log |/(i?oe^^)| }d^ = 0(1).

(2.12)

Proof. The function F{w) = f{w ^) is meromorphic in the lower half-plane. Assuming p = —R~^, define Fa{w) as in (1.13). The latter function is meromorphic in G^ — {w : lui w < p}, where it can have only a finite set of zeros and poles {cmjl C {w = u-\-ip : \u\ < y/r^ - p^} (ro = ^o""^)- Further, Fa{w)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

59

satisfies the conditions of Lemma 1.1. Therefore, u{z) = W~^ log\Fa{z~^)\ is harmonic in {z : \z — iR/2\ < i?/2, 2: 7^ 0}, except the boundary points {Am}I = {(^m^}i (l^ml > ^ 0 , 1 < '^ < ^)- Bcsidcs, u{z) admits one of the representations of (1.4) in neighborhoods of points Am- Repeating the argument proving Theorem 3.5 in Ch. I of [38], for u{z) we obtain formula (3.4) of that theorem, i.e. -j

/"TT —arcsin - ^

^^ ^arcsin ^

ia

^

R Sm^ 'U

where Q^'^\R^RQ^U) is the expression of (2.12) with u — VK~^log|/|. Hence (2.11)-(2.12) holds by the definition of u{z) and (2.4). And it remains to observe that the quantity Q^'^\R^ i?o, W~^ log |/|) is bounded as i? -> +00. Remark 2.2. For a = 0 (2.11)-(2.12) coincides with B.Ja.Levin's formula (see, for instance, [38], Ch. I, Theorem 3.4). 2.3. The value distribution of functions which are meromorphic in the closed half-plane is one of the main fields where Carleman and B.Ja.Levin formulas are applied (see [38], Ch. I, § 5, Ch. Ill, §3). But the generalizations of these formulas given by Theorems 1.3, 2.1 and 1.4, 2.2 are valid under additional restrictions on the behavior of the considered meromorphic function f{z) near the origin. Namely, when f{z) has no zeros and poles in some semidisc {z : Im z > 0, \z\ < RQ} (0 < JRO < +00) and one of the inclusions 1°, 2° of these theorems is true for a given a G (—1, +00). These assumptions permit to apply W'^" to log |/(>^)|. Our next lemma shows that one can get rid of those additional restrictions if f{z) is meromorphic in a neighborhood of the origin. Lemma 2.3. Let f{z) be meromorphic in \z\ < RQ. Then, multiplying f{z) by definite rational functions R{z) one can attain any of inclusions r. 2°.

\og\R{z)f{z)\ G K^{n/2,G^} for a given (3 G [0,+oo), d/d{lm 1/z) log \R{z)f{z)\ G Kp{7r/2, G+} for a given (3 G (0,1).

Proof. It suffices to consider the case when f[z) (/(O) 7^ 0) is holomorphic in \z\ /3 be a natural number, and let f{z) = ^t^^^z^ (ao 7^ 0, 1^1 < RQ). Observe that, multiplying f{z) by the n-th partial sum Pn{z) of the Taylor expansion of l/f{z), one can find f{z)Pn{z) = 1-\- bnz'^ + 6n+i2:^+^ + • • • {\z\ < Ro). Then \og\Pn{z)f{z)\^ l^nJkr^ as ^ -^ 0, where ni > n. Therefore, 1° holds by the definition of Kp. 2\ If f{z) = Et^o^kz'

OlomzA d{lml/z)

_ ^^^

(ao 7^ 0, \z\ < i?o), then

L . a . + 2 a . . - f 3 a 3 . - + . . . | ^ ^^^^^^^^ ^^ ^ ^ ^^ \

ao-\-aiz-h

a2z'^-\-'

60

CHAPTER 3

where ni > 2. Thus, 2° automatically holds for any p G (0, 2). The following lemma contains a useful estimate. Lemma 2.4. Let F{z) he holomorphic in G~\{0} = {w : Im w < 0, w ^ 0}, and let this function have no zeros in {w : Im w <0, \w\ > r} for some r > 0. Further, let log \F{w) G Ma{G-\{0}} for an a e (0, 4-oo), and let \F{w) < e"l^l~^'^'''\

1^1
we G^\{0},

(2.14)

\w\
weG^\{0},

(2.15)

for some numbers R, a and A. Then W - ^ l o g | F H | <(Ti\w\-^,

where G\ is a positive number and Ri = min{i?/2,i?otan(5o,i?o + 1} (with RQ and So from the definition of Ma)Proof. By Lemma 1.1, W~^log\F{w)\ \w\ < i?i, set W-^log\F{w)\=—-

/ =

is continuous in G~\{0}. Assuming

+ /

+ /

]t--'\og\F{w-it)\

Il{w)-\-l2{w)-\-Is{w).

Since the compact K{Ro,Ri) = {( = w — iRo : w G G", \w\ < Ri} lies in A{So,Ro) = {C : l a r g e + 7r/2| < ^o, |CI > Ro}, \Ii{w)\<—-

/

{r +

Ror-'\\og\F{w-iRo-iT)\\dr

r(Q^) Ji

<

max{l,(i?o + l ) " - ' }

sup

^^"" / r ^ - 1 |log |F(C -iT)\\dT^Ni<

:iRo,Ri)

Jl

+c

by the definition of M^. Further, it is obvious that J2H <

^ '

;) ^

r(a)

^

/

log F ^ - z t d^.

y^^

The last integral is continuous as a function of w in G~\{0} 4- iRi = {w : Im w < Ri, w ^ iRi}. Hence |/2('^)| < -^2 < +00. Further, pRi

j.a—1

r-hoo

„ Q ! - 1 c?x

\w\-^

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

61

by (2.14), and (2.15) follows from the estimates of / i , /2, I3. 2.4' It is natural to consider the Carleman and Levin type formulas of Theorems 1.3, 2.1 and 1.4, 2.2 as equilibrium relations between the density of zeros and poles and the growth of the meromorphic function. These formulas can be written also as equilibrium relations for some Nevanlinna and Tsuji type characteristics. Below we give only the relations arising from the Levin type formula. Let f{z) be meromorphic in the upper half-plane G^ and also in some neighborhoods of the points —RQ and RQ for an RQ > 0. Further, let f{z) have no zeros and poles in the semidisc {z : Im z > 0, \z\ < Ro}, and let f{z) satisfy the remaining conditions of Theorem 2.2 for a given a G (—1, +00). Assuming n(t, / ) the number of poles (according to their multiplicities) of /(^r), which lie in {z: \z- it/2\ ^ t/2, \z\ > RQ}, set

'^«<'''fl = r(TT^£(i-s)°»(''«?' «»<«<+»• Observe that, being everywhere constant the function n(t, / ) grows stepwise at t = p^/ sini/jn, where the jumps are equal to the quantity of poles bn = Pn^^^"^ of f{z), which lie on the arcs {z : \z — it/2\ = t/2, \z\ > RQ}. Therefore, integration by parts gives Y^

1

f smijjn

1^

^^'^^\.M\<^ ^" ^ 1

<-R /I

1 \ l+«

r fi iv^°' 'Ro

Hence, formula (2.11)-(2.12) can be written in the form Nc{R,r^)--N^{RJ) I'-i^iR)

-I

{R)

^

^,9

W--log|/(i?sin^e^^)|-^^+0(l), Rsm 'd

(2.16)

27rL where K,{R) = a.Tcsm{RQ/R). For a+ = max{a, 0}, a = max{—a, 0} set 1

n7V-K,{R)

+

JQ

ma(i?,/) = - / {T^-"log|/(i?sin^e^'')|} - — ^ ^T^ JK{R) ^ ' Rsm^'d Ca{R,f) = mc.{R,f) + N4R,f). Then (2.16) becomes m a ( i 2 , / - I ) + N „ ( i ? , / - I ) - m „ ( E , / ) + N«(i?,/)+ 0(1) or, which is the same, the following equilibrium relation Cc,{R,r')

= Cc,{R,f) + Oil).

62

CHAPTER 3

2.5. One can use the boundedness of above mentioned Nevanlinna and Tsuji type characteristics for defining some general classes of meromorphic functions and finding their representations by passages in R. and F. Nevanlinna formulas of Sec. 1. Nevertheless, we prefer a different way for establishing our factorizations. As our first step, from Carleman and Levin type formulas we find out some sufficient conditions under which zeros {zk} G G^ of a holomorphic function satisfy the condition

E Im —

l+a

(2.17)

< +00

Zk

for a given a G (—1, +00). If zeros {zk} G C of an entire function f{z) satisfy (2.17) for a = 0, then it is said that f{z) is of the class A. Similar to this, we shall say that a function holomorphic in the upper half-plane G'^ is of the class Ac,{G^} ( - 1 < a < 4-00), if its zeros {zk} € G+ satisfy (2.17). Note that the Blaschke type products of Ch. 2 are of Acx{G^}. T h e o r e m 2.3. Let f{z) be holomorphic in the closed half-plane G+\{0} = {z : Im z > 0, z 7^ 0}; and let for some RQ > 0 this function have no zeros in the semidisc {z : Im z > 0, \z\ < RQ} U {—RQ^RO}- Further, let a G (—1,+oc) be any number and let 1°. log 1/(^)1 G M^{G+} if aj [0,+oo), 2°. a/5(Im 1/^) log 1/(^)1 G M^{G+} if a G ( - 1 , 0 ) .

If liminf - i - /

R^+OOTTR

Jo

lw-°'log\f(Re''^)\y L

J

sinM^ < +00

(2.18)

and (2.19) then f{z) G A^{G+}, 2°.

/'+00 r-\-oo\

-RQ p-Ro

(

di_

+ fRo

/

+/

^-00

JRo

(2.20)

< +00,

W-"log|/(i)| J '

lim R-^-\-oo -foo

/

W-'^\og\f{t)f{-t)\

dt

(2.21)

63

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

3°. The following relations hold

1+a

lim

(2.22)

VF~^log|/(i?e^^)|sin^rf^ = / i ^ o o , _4^ / R Jo

— lim

.

Im

,?.ir(2 + a)

\zk\
/ 1

1

^fe

r(2 + a;

E

1+a

Iml

< +00.

(2.23)

Proof. We start by 2°. Let {Rn}T ^^ ^ sequence by which the lower limit (2.18) is attained. Then by formula (2.9) (where the terms of left-hand side sums are nonnegative) r—Ro

(£>!"

)(^i5){"'"°'°*l«"l}"^'

-Ro

<

'

RQ

As 4/3(1/^2 - l/i?2) > 1/^2 for iJo < \t\ < i?„/2, we have

(/-;>/:)(^^)F-"'-'«'"}"^ Hence the convergence of (2.19) for {W~^ \og\ f(t)\}~ follows. Together with (2.19), this provides the convergence of (2.20). For establishing (2.21) (where the right-hand side integral is already proved to be absolutely convergent), set m = Evidently rR

f^"^ ~ W-^\og\f{x)f{-x)\ Jt

f^ [^-^)w-''^og\f{t)f{-t)\dt-1^

dx — , ^

Ro
w--\og\mf{-t)\ dt '^2

64

CHAPTER 3

Hence (2.21) follows since 1 /•« ^{R) - ( - ^ ) ^(^o) = o{l) and -^ I til){t)dt = o{l)

as

i? -> +oo.

1°. If (2.18) is attained on {Rn}T^ ^^en by (2.9)-(2.10) and (2.18), (2.21)

S lr(2 + a) \zk\
Im

| 1 + CK

1

J^ _1_

+ Ia

Zk

0(1) as n -> oo.

(2.24)

Rn

The terms of this sum are nonnegative and

\Zk

(2.24')

0.

lim Ioc[ —, — R^+oo

R

Fixed R> RQ, observe that Rn> R for enough great n>l.

"^.Siri^

Im

\zk\
1

""Vzfc' Rn) )

Zk l+oc

<

E

Im

r ( 2 4- a)

\zk\
Hence

|l+a

-\-Ia

Zk

ji_ _i_ Zk^

0(1).

Rn

The passage n -> oc gives that for any R > RQ 1 r ( 2 + a)

1

|l+a < M < +00.

Zk

Zk\
Hence (2.17) follows, and f{z) G A^{G+}. 3°. (2.23) will hold if i2\zk\ +oo. To prove the latter relation, for an arbitrary e > 0 choose Re > Ro enough great to provide jl+a 1 <£. Im r ( 2 + a) Zk

E

Zk\>Re

Then, by (1.25)

E ^« \zk\
1_ 1 Zk'

R

'"i^i

+ e.

for any i? > R^. Consequently limsup^^^.^ \Y.\zk\
^) <shy

Q('\R,Ro,W-''log\f\)

it—>+oo

d

:^'')|};

27rJo {W^-"log|/(iZoe**)|i?o" + ^ W ^ - " l o g | / ( i ? o e ^ ^ ) | } sin7?d,9,

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

65

where the right-hand side integral is absolutely convergent. Hence (2.22) follows by (2.9) and the finiteness of the limits (2.21), (2.23). The proof of the next theorem is quite similar. Theorem 2.4. Let F{w) be holomorphic in G~ = {w : Im w < 0}, and let for any p < 0 this function have not more than a finite set of zeros in G~ = {w : Im w < p}. Further, let a E (—1, -{-oo) and let r . log\F{w)\ e Ma{G-} if ae [0,+oo), 2°. a/a(Im w)log\F{w)\ e Mi+a{G~} if a G (-1,0). +00

/

IW'"^ log \F{u + ip) I du < -f oo,

- o o < p < 0,

(2.25)

-oo

then: 1°. Zeros {wk} C G~ of F{w) satisfy ^\lm

Wk\^'''' < + o o .

(2.26)

k

2°. The following relations hold:

Im Wk
k

+ 00

|M^-" log \F{u + ip)\\du = p^

/

Hy^oo.

-oo

3. FACTORIZATIONS Theorems 3.2, 3.5 and 3.6 of this section present some solutions of Il-nd and Ill-rd Problems (see Introduction) for several weighted classes of functions holomorphic in a half-plane. 3.1. We start by a representation of harmonic functions, which follows from Nevanlinna factorization in the half-plane (see [83] or [10], Sec. 6.5). Theorem 3.1. Let u{z) be a real-valued harmonic function in G'^, and let this function be continuous in G+ = {z : Im 2: > 0}^ with possible exception of a countable set of real points {cm}i {if ^ = 00^ then lim \cm\ = 00), in which m—>oo

u{z) admits one of representations (1.4)- If liminf — / then

u^iRe''^) smd < +00

f

'^^ rfr^dt

J —c

and

< +00

/

- — ^ d t < +00,

(3.1)

CHAPTER 3

and u{z) = hy-\-

1 r^

u{t)dt z = X

(3.2)

"hiyeG^,

71" J-oo

where h is a real number deduced from h=-

lim

4 /

(3.3)

u(Re'^)sin^.

Theorem 3.2. Let f(z) be holomorphic in G+\{0} = {z : Im z > 0, \z\ < RQ^ z ^ 0}; and let this function satisfy one of the following conditions: 1°. log|/(z)| G ^a+^{7r/2,G'+} for some a G [0,+oo) and

K>l-a+p,

2°. 5/a(Im l / z ) l o g | / ( z ) | G ^ i + a + « W 2 , G + } for some a G (-1,0) and K> 1 — a.

If liminfi?-^ /

\w-'^\og\f{Re'^)\y

R^+oo

^

Jo

sinMi} < -hoo

(3.4)

^

and

r+-{f^-log|/(^)|}+ /

-c?t < + 0 0 ,

(3.5)

(it < + 0 0 ,

(3.6)

J —C

then /•+00 | y--"al;o g | / ( i ) | | +°o |W

|ih(l + | t | 2 - )

i-oo

and ^/le zeros {zk} C G^ of f{z) satisfy the condition

E

Im

1 ll+o:

< +00.

(3.7)

Besides, for z G G'^ f{z) = Boc{z,{zk})eyi^l

iC+ +00

+ r ( l + a)e-^5(i+") / i ^ i + " +

IL

TT/-a M^-" log 1/(01 dt + ^ 0 ^ , (3.8)

i2(l/2-l/i)l+^

where C and h are real numbers, and h=-

lim

i /

TT iJ->+oo i t J o

W-^'loglflRe'^MsinM^.

(3.9)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

67

Proof. By Lemma 1.9 of Ch. 1 |M?-^log|/(z)|| < 0 ( | z r )

as

z-^0

(ZGG+).

(3.10)

In view of (1.27) in Ch. 1, f{z) satisfies the requirements of Theorems 2.1 and 2.3. By Theorem 2.3, f{z) G Aa{G"^}, i.e. (3.7) is true. Hence the product Bot{z^ {zk}) is convergent and the function Uz)^

/^f

(^Q,

zeG^)

(3.11)

is holomorphic in G+\{0}, except some points {(Re l/zk)~^}' On the other hand, l o g | / , ( z ) | G ^^+^{7r/2,G+} if a > 0, and a ^ I m l/z)\og\fo.{z)\ G i^i+c.+7W2,G+} (7 = min{/^,l}) if - 1 < a < 0 by (1.2), (1.2') and our requirements 1°, 2°. Therefore, by Lemma 1.1 and Theorem 3.2 of Ch. 2, the function u{z) - W-^ log \f^{z)\ = W-^ log \f{z)\ - W-^ log \B^{z, {zk})\

(3.12)

is harmonic in G+\{Q}, except the points {cm}i^ = {(Re l/^m)""^}!^ ( ^ < oo, {zm}i^ C G+\{0}, Re Zm 7^ 0) which are the zeros of f{z) in (—CXD,+OO). Besides, u{z) has the representations (1.4) in the neighborhoods of these points, and by Lemma 1.9 of Ch. 1 u{z) is continuous at the origin, besides u{0) = 0. By (1.16), (1.19)-(1.190 and Remark 3.2 in Ch. 2 lim

4 /

W-''\og\Bc,{Re'^,{zk})\smM^

= 0.

(3.13)

Hence by (2.11') of Ch. 2 u{z) satisfies the conditions of Theorem 3.1 and therefore is representable in the form (3.1)-(3.3). Besides, by (3.15) the number h in (3.1)-(3.3) can be deduced by (3.9). Further, by (3.10) the integral (3.6) is convergent since the last term in (3.12) vanishes on the real axis. Thus, W--\og\Uz)\

= hy+^

r^T'^l^^f^^?^^ TT J_^ [X - ty + y^

z = x + iyeG^.

(3.14)

For proving the remaining formula (3.8), introduce the function U{w)=\og\fa{w-^)\

(3.15)

which is harmonic in G~. As W~^U{w) ^ W~^ log\fa{w~^)\,

[ ZW

7TI J_^

W-t

J

(3.14) becomes

68

CHAPTER 3

On the other hand, (3.6) impUes r+oo 1^—iog|/(t-iMl

1 + 1*2-/« 1

/ J —(

dt < +0C,

where 2 — / ^ < l + a — p i f Q ; > 0 and 2 — / ^ < l + Q^if—1
=

hm_log|/(z)| - 0 ,

for w G G~ we come to the representation

Hence (3.8) follows by (3.15) and (3.11). Remark 3.1. Similar to the case a == 0, it is natural to call the number h in the factorization (3.18) a-mean-type of f{z). Along with (3.9), there is another relation for /i: _^ /i = limsup2/-^W"log|/(i2/)|. (3.9') y-¥-{-oo

Remark 3.2. Let f{z) be holomorphic in G+, where this function is of finite order p > 0, i.e. p = limsup^°g^7jf'^\ ie^+oo

M(i?,/)=max|/(i?e-)|.

log i t

0<^<7r

Then for any a > max{p — 1,0} by the coefficients of Taylor's expansion of f{z) in z = 0 one can construct a polynomial Pn{z) of order n == [a] + 3, such that f{z)Pn{z) admits a factorization of the form (3.6)-(3.9). Proof. Denote p' = max{p, 1}. If a > (p—1) + , then a = p'+£ —1, wheree > 0. By the assertion 1° of Lemma 2.3, multiplying f{z) by a definite polynomial Pn{z) of order n = [a] + 3 one can attain the validity of the requirement 1° of Theorem 3.1 for f{z)Pn{z). On the other hand, it is obvious that for any number 7 G ((e — 1)~^,^) |/(i?e^^)Pn(i?e^^)| < exp {i?^'+^} ,

i? > i?o, 0 < ^ < TT,

for enough great RQ > 0. Hence T^-" log \f{Re'^)Pn{Re'^)\

< aiR^-'^\

R>Ro,

0 < 7? < TT.

EQUILIBRIUM

69

RELATIONS AND FACTORIZATIONS

where a i is a positive constant and 7 — e + 1 G (0,1). By the properties of W~^ \og\ f{z)Pn{z)\, which follow from Lemma 1.9 of Ch. 1 and Lemma 1.1, the function f{z)Pn{z) satisfies all conditions of Theorem 3.2. Hence the desired factorization holds. 3.2. Consider two classes of functions harmonic in the lower half-plane. Definition 3.1. The classes ii and ii^ are the sets of functions U{w) which are harmonic in G~ and satisfy the following conditions: +00

/

t/+(w + iv)du < +00,

(3.16)

\U{u + iv)\du < +00.

(3.17)

-00 +00

/

-00

Obviously il"^ C i t . Considering the function f{z) = exp{U{—z) + V{z)} (where U{w) G i t or U(w) € it"^ and V{z) is the conjugate harmonic of (f{z) = U{—z) in G"^), which is holomorphic in G"^, one can state the below theorems as direct consequences of Theorems XVII and XX of [76]. Theorem 3.3. The class ii in the form

coincides with the set of functions representable

(/(»)= «W + t ! £ ° ° _ ^ f f l _ , „ = . + i„,G-,

(3.18)

where a is a nonpositive number and fi{t) can be written as the difference ii{t) = Ml (^) — M2 {t) of two nondecreasing functions for which /

"^//i(^)<+oo

and

/ " Y ^ < + O O .

(3.18')

Theorem 3.4. The class ii^ coincides with the set of functions representable in the form (3.18), where a = 0^ and /i{t) is of bounded variation in (—00, +00). We also use the following subclass of i t . Definition 3.2. it^ ( - 1 < K < 1) is the set of those U{w) G it"^ for which

I

+°°

dt U-i-it)^iTJ^<+'^-

(3.19)

Using some results from [72], for U{w) G it"^ one can show that \U{—it)\ G L{t-^^-^'^^dt, (1, +00)) for any K G (-1,1). Hence, it"^ C n_i<«
70

CHAPTER 3

L e m m a 3.1. For any K G (—1,1) the class il^ coincides with the set of functions representable in the form (3.18), where a = 0 and fi{t) can be written as the difference fx{t) = (Jii{t) — /i2(0 ^f ^'^^ nondecreasing functions such that j y

dn,{t)<+<x^

and

/_^°°i+7^{iL<+°°-

(3-18")

Proof. Let U{w) e il+ for some KG (—1,1). Then U{w) is representable in the form (3.18)-(3.18') since ilj^ C it"^. For proving that in this representation a — 0 and ^i2{t) satisfies the second requirement in (3.18''), consider the harmonic functions rr . N

1^1 f^""

Ul{w) = ^ ''

J —c

dfii{t)

.

y{u-t)^-\-v^' —-^^^^p-^,

C/2(^) = - a | i ; | + M

/ +"^

W=^U +

dfi2{t)

^ ZVeG

w = u-\-iv e G

{u - t)^ 4- v^ '

Obviously Uj{w) > 0 {j = 1,2) and U{w) - Ui{w) - U2{w). Besides, U{w) = U'^{w) — U~{w), where Ui{w) > U'^{w), U2{w) > U~{w). Therefore, setting ^{w) — U\{w) — U"^{w) we conclude U-{w) + ^{w) = U2(w),

w G Q-.

(3.20)

But 0 < ^{w) < Ui{w), and for Ui the integral of (3.19) is convergent in view of Theorem 1 in [72]. Therefore, the same integral is convergent also for ^{w). Further, by (3.20) the integral (3.19) is convergent also for f72. Hence, by same theorem a = 0 and fJ^2{t) satisfies the second requirement in (3.18''). The converse statement is obvious since f^"^

0
dt

f^"^

U-i-it)-^<j^

dt

C/2H*)^<+oo.

3.3. Definition 3.3. OT^ (—1 < a < +oo) is the set of those functions holomorphic in G~, which satisfy the following conditions:

F{w)

(i) The zeros of F{w) lie in a semidisc {w G G~ : \w\ < RQ] (RQ > 0). (ii) If 0 < a < +cxD^ then log\F{w)\ G Mp{G~}, wherep is the integer deduced from p-l
sup /

[T^-^log\F{u + iv)\ydu<

+oo.

(3.21)

(iv) If a = p> 1 is a natural number, then additionally

I

+°°

df

[W-''log\F{-it)\]

f

<+oo.

(3.22)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

71

Definition 3.4. The class OT^ (—1 < a < +CXD) is the set of those functions F{w) holomorphic in G~, which satisfy the following conditions: (i) F{w) has not more than finite number of zeros in any half-plane G~ = {Im w < p} {p< 0). (ii) If 0 < a < +00, then log\F{w)\ G Mp{G-} {p > 0, p - 1 < a < p). If -l
I W " log \F{u -\-iv)\\du<+00.

/

(3.23)

-oo

L e m m a 3.2. Let F{w) be holomorphic in G~, and let {wk} = {uk +ivk} C G~ be the zeros of this function. If F{w) G O^a or F{w) G 0^^ for some a G (—1, +oo), then

k

k

Proof. One can be convinced that for F{w) G ^ ^ (3.24) follows from Theorem 2.4. Let F{w) G O^a- Similar to Proof of Theorem 1.2, introduce the function

which is holomorphic in G~.

By Lemma 1.1, W~^\og\Fa{w)\

= U{w) is

harmonic in G^, with possible exception of a finite set of boundary points, where one of representations (1.4') is true. By (3.14) of Ch. 2 + 00

/

/•+00

U^ {u-{-iv)du < SM^) I -oo

\W~^ \og\Foc{u-{-iv)\\

du

'^

/

\W~^ \og\ha{u + iv^Wk — p)\\du < +00. -oo

Therefore V{w) = U{w + ip) G i t . Hence V{w) is representable in the form (3.18)-(3.180, where dfi{t) = W'"^ log \Fa{t + ip)\dt. Consequently,

Y^

"^^oglbaiw-ip^Wk-p)]

Vk
< V-P\ TT

f'''^[W-^l0g\F^{t /

—? /•-j-oo

.xo , /

+ ip)\]'To—dt-W

y^-e..\r,(.\ log i^a(^)

y_^

{u - ty -{-{v- PY

r /

[w-''\og\F^{t + ip)\Ydt-w-^\og\F^{w)\

72

CHAPTER 3

for any w G G~. Now let w = —i{Ro + 1) (see. Definition 3.3(i) of let p G (-1,0). Then, by (2.5) of Ch. 2 and (3.21)

^

^

OTQ,),

and

Vk
where C is a constant independent of p and

4..,,) = ^'""*' J^^LtI±4!±ii^,|., ,, _ ,.„ , 0. Obviously lim Al '(p) = /

—!^

^(\vk\

^

-t)°'dt

13(i2o + 1)4 1 + a Consequently, for any po < 0 13(i?o + 1)3 r ( l + a) ^

'"""I

< r ( l + a)

^

P^'^o^'^

^^^ ~ ^-

Hence our assertion follows by the arbitrariness oi po < 0. 3.4- The representations of the below theorems are solutions of Ill-rd Problem of Introduction for the weighted classes O^a and 0^^. T h e o r e m 3.5. ^a (—1 < a < +oo) coincides with the set of functions admitting one of the following factorizations in G~: for 0 < a < +oo

and for — 1 < o; < 0 F{w) = Ba{w) exp< aiw + ao

/n t/iese factorizations Ba{w) are convergent Blaschke type products with a hounded sets of zeros satisfying (3.24), ^, c^i; ^^^ G are real numbers and fi{t) = p,i{t) — iJ^2{t), where p>i{t) are /J^2{t) nondecreasing functions satisfying

j^

dpi{t)<+00, I

^ /ffi+i-p < +^ ^/ 0
EQUILIBRIUM RELATIONS AND FACTORIZATIONS

73

{p > 0^ p — 1 < a < p) or

J " dfiiit) < +00, J ^ iflJlT^ < +^ ^^ - K a < 0. (3.260 / / (3.26)-(3.26')

is valid, then

lim log \F{u —it)\ = aiu-\-ao^

—oo
(3.26'')

Proof. If F{w) G Ola ( - 1 < a < +oo), then the zeros {wk} C G~ of F(it;) satisfy (3.24) by Lemma 3.2. Hence, by the product Ba{w) is convergent and the function F(w)

is holomorphic in G~. First assume o; > 0. Then log \Fa{w)\ (which is harmonic in G~) belongs to Mp{G~} since log |i^(tf^)| and log \Ba{w)\ are of Mp{G~}. Consequently, by Lemma 1.10 of Ch. 1 also the function U{w) = W-"" log \Fa{w)\ = W"^ log | F ( ^ ) | - W-"" log \Ba{w)\

(3.27)

is harmonic in G~. By (3.21) and formula (3.14) of Ch. 2 +00

/

r+oo

[U{u + iv)]'^du 1 is a natural number, then by (3.22) and nonpositivity of W-^'loglBaiw)] in G- (see Theorem 3.2 in Ch. 2) + 00

n + OO

7.

1.

/

[u{~it)r j<j^

[w-'^iog\F^{-it)\y I < +00.

If Of > 0 is not a natural number, then by Lemma 3.2 of Ch. 1

J^^^ m-itr ^

< ^^°° w"^ iiog \F4-it)\\ ^

< +^.

Thus, U{w) e it+_„ since all requirements of Definition 3.2 are satisfied. Hence, U{w) is representable in the form (3.18), (3.18"), where a = 0, i.e.

W-'^log\F.{w)\=Re

{-

r ° ° ^ } , W ^ / '

weG'

74

CHAPTER 3

where /x(t) is as required in (3.25)-(3.25"). In view of Lemmas 1.10 and 3.1 of Ch. 1, applying (dP/dvP)W~^^~^^ to both sides of the last formula we come to (3.25). Now let - 1 < Of < 0. Then the proof of formulas (3.26)-(3.260 and (3.26") is quite similar to the previous case. The only difference is that instead of nonpositivity of W~^ log \Bot{w)\ one has to use its continuity in G~\{wk} and the fact that the points {wk} (i.e. zeros of F{w)) are integrable singularities. Also the inclusion W~^ log \Ba(w)\ G KI{7T/2, G~} has to be used, which is a consequence of (1.2') and Lemma 1.9 of Ch. 1. For proving the converse statements, first assume that a > 0 and (3.25)(3.250 is valid. Then log|F('?i;)| G Mp{G-} by Lemma 3.3 of Ch. 1. The function U{w) defined by (3.27) is harmonic in G~ and representable in the form (3.18)-(3.18'0, and U{w) G il^_p by Lemma 3.1. Therefore, by the estimate (3.14) of Ch. 2 sup /

[W-''\og\F{u

+

iv)\]^du

v<0 J-oo +00

/

p-\-oo

W^[u + iv)du + sup / -oo

IW~^ log \Bot[u -{-iv)\\du < +oo.

^'<0 J—oo

On the other hand, by (1.2) and Lemma 1.9 of Ch. 1

j^^[W--\og\F{-it)\]--^^ U-{-it):^r^,

+/

\w--\og\B^{-it)\\

-^^^^ < +00.

Thus, F{w) G O^a (0 < a < +Q;), as all requirements of Definition 3.3 are satisfied. For — 1 < o; < 0 the proof is similar. Remark 3.3. Let F{w) be holomorphic in G~ and also in a neighborhood of oo. If \F{w)\ < e x p j - ^ ^ l ,

^ G G-,

|Im w\ < ro,

(3.28)

for some p, ro, CQ > 0, then for any a > p there exists a definite rational function R{w) such that Fi{w) = R{w)F{w) G Ota? and hence Fi{w) admits the factorization (3.25)-(3.250. Proof. By Lemma 2.3, | l o g | F i ( ^ ) | | < ci|tt;p-W (|^| > RQ) for a rational function R{w) defined by the coefficients of the expansion of F{w) at oo. Therefore, by Lemma 1.9 of Ch. 1 |l^-"log|Fi(7i;)|| <-^,

weG^,

\w\>Ro.

(3.29)

EQUILIBRIUM RELATIONS AND FACTORIZATIONS

75

Besides, the estimate (3.28) remains true for Fi{w) (only CQ changes). Assuming V < 0 any number, set + 00

/

[W-''\og\Fi{u

+

iv)\]^du

-oo

fRo

([ + + // ° ) [W-^\og\Fi{u \J\u\>Ro J-RoJ /\u\

+ iv)\]^du

= h{v)+l2{v).

(3.30)

From (3.29) it follows that Ii{v) < cs = 2c2/Ro (—CXD < i; < 0) and also that l2{v) is a continuous function of v (—00 < v <0), such that /2(—00) == 0. For evaluating this function as t? —> —0, observe that for —ro/2 < t' < 0

, , , ^ 2J?o fcii?o-(^+M-") ,

CO p V " " ! ^

,,

where 1

/"^o

(^(t;) = — - r / J- (<^J J-Ro

/'-Ro

du

\\og\Fi{u-hi{v-(T))\\da Jro/2

is continuous by v on the whole interval (—00,ro/2). Therefore, /2(f) < C4 {—To < V < 0) and /i(t') + h{y) < C3 + C4 = C5 for —CXD < t; < 0. Hence by (3.30) the relation (3.21) holds for Fi{w), and Fi{w) e O^a since this function satisfies all the remaining requirements of Definition 3.3 of Ot^T h e o r e m 3.6. OT^ (—1 < a < +00) coincides with the set of functions admitting in G~ the factorization (3.25)-(3.26), where Ba{w) are convergent Blaschke type products with zeros satisfying (3.24), <^; ^ i ; ^^^ C are real numbers, and ii{t) is a function of hounded variation on the whole axis — 00 < t < 4-00. Besides, the relation (3.26") is true. P r o o f does not essentially diff'er from that of Theorem 3.5. Therefore, we shall just outline the main stages. As in the proof of Theorem 3.5, consider the functions Fa{w) and U{w) = W~^\og\Fa{w)\. One can be convinced that U{w) € il"^ and hence is represent able in the form (3.18), where a = 0 and the function /i(t) is of bounded variation on the whole axis — CXD < t < +CXD. In view of Lemmas 1.10 and 3.1 of Ch. 1, application of OP/dyPW-^P-"^^ (if a > 0) or W^ (if — 1 < Q; < 0) to both sides of this representation leads to the desired representations. For proving the converse statements, one has to verify that the function U{w) defined by (3.33) belongs to il"^. Hence the inclusion F{w) e 5^^ (—1 < a < +00) holds in view of the properties of the Blaschke type product. NOTES. One can find the general scheme of the proof of Theorem 2.3 in NevanHnna's hard accessible paper [80] and in the monograph of Boas [10], where some misprints occur.

76

CHAPTER 3

Qualitatively different from Theorem 3.2 factorizations for functions of any finite order, meromorphic in the closed half-plane, first were found by R.Nevanlinna [83]. Later N.V.Govorov [39] extent them to the case when the function is meromorphic in the open half-plane. In view of Lemma 3.4 the classes 01^ and 01™ considered in Theorems 3.5 and 3.6 for a=0 coincide, in essence, with the classes OT and O^m of V.LKrilov. This is precise for 0 ^ = ^^ while 0^0 coincides with the subset of functions from OT, having bounded sets of zeros.

CHAPTER 4 MEROMORPHIC FUNCTIONS WITH SUMMABLE TSUJI CHARACTERISTICS

1. LEVIN TYPE FORMULAS AND TSUJI TYPE CHARACTERISTICS All over this section, we assume that F{w) is a function meromorphic in the lower half-plane G~ = {w : Im w < 0}, having finite sets of zeros {a^} and poles {bj^} in any half-plane G~ = {w :lm w < p} {p <0). 1.1. Let p G (—00,0) and R > max^^^y {|Im a^|,|Im bj^]} be fixed numbers. Then the mapping z = [w ~\- i(R — p)]/R = {w — ip)/R-\-i (a parallel shift with a contraction) transforms the disc \w + i{R — p)\ < R into \z\ < 1. On the other hand, there is a finite set of zeros and poles of F in the initial disc, and they lie upper its center by our choice of R. Hence, the function f{z)^F{wl

z='^+i,

(1.1)

is meromorphic in the closed unit disc \z\ < 1, where / has a finite set of zeros a* = {afj, — ip)/R-{-i and poles 6* = {by — ip)/R-\-i lying in the upper half-plane. It is obvious that / admits the factorization (7) of Introduction for any a > — 1 (where A = 0, U / CK = 1 and CA = Co — /(O)). Replacing a by a — 1 in that formula, we get

iog/(.).^// (i-icpr-y^^^^<^(c) TT J J|C|<1

(1 - zC,Y+°'

+ log7r„_i (z,{a;}) - l o g 7 r „ _ i ( ^ , { 6 : } ) - l o g / ( 0 ) (1.2) {a > 0, \z\ < 1), where da{() is Lebesgue's area measure and log7r„_i(^,{^4) = - ^

/

./^

/ i l + a f ^ - E

*-(^,^fc),

(1.3)

where {zk} ^ {a* } or {zk} = {6^}. Using (1.1), one can write (1.2)-(1.3) as a representation for F. As for C ^ 0{R, p) = {w : \w -\- i{R - p)\ < R} 1-

R

' = 2^HL1ZPI_K^,(0,1],

(1.4)

78

CHAPTER 4

l _ , ! ^Ri ^ + i U 'R^ - ^ i{w — C ~ 2ip)

{w-ip){C R

+ ip)

(1.40

for 0 < a < +CXD and w € 0(i?, p)

a2^ / / 0{R,p) V ' TT J Jo

logF{w)

\C-ip\' 2i?|Im C - p| |ImC-pr-Mog|F(C)|

R

+ E *«

+ 1-

R

1+ a^^(C)

+ i

^+i^-^+i]-logF{-i{R-p)).

(1.5)

b^€0{R,p)

In view of the definition (1.3) of ^a, taking w = —i{R — p) we get T^dr \\lms-

- ( " • ^ - ) = (l)'^"i-»-'-i'^r3i = f-J

|Im5-p|^+^/a(5,p,i?),

P\T

0 < a < + o o , 5 G 0 ( i ? , p ) . (1.6)

Consequently, the same substitution in (1.5) gives

''iTrjJc 0(R,p)\ ^

llmC-pr-'loglFiOldaiC)

2i?|ImC-p|,

\Im a^-p\'+''Ia{a^,p,R)+J2

a^€0{R,p)

\lm K -

p\^+"Ua^,p,R)

b^eO{R,p)

-^r^log\F{-iiR-p))\,

0
(1.7)

1.2. The following two lemmas are necessary for the passage R -> +oo in (1.7), which will lead to a family of Levin type formulas. L e m m a 1.1. Let (p{t) >0 be a measurable function on (0, +oo). Then for any a G [l,+oo) / Jo

e-'^0 Js

it-e)°'-'^
FUNCTIONS W I T H SUMMABLE T S U J I CHARACTERISTICS

79

Proof. It is obvious that the right-hand side integral monotonely increases and tends to left-hand side integral for any sequence e = Sn -> -^0. L e m m a 1.2. Let ^(C) be a measurable function in the half-plane G~ (—oo 

for some a e [1, +00). Then lim

/ /

(l-

jl~'f^—rl

|ImC-pr"^*(C)cif^(C)

Proof. If a-l

PaAC,R)

= { V

'^ '^' ' 2i?|ImC-p|.

1 for -1

CeO(ii,p)

for

C^O{R,p),

then our assertion is equivalent to lirn

/ /

P„,^(C,i?)|ImC-pr'*(C)MO=0.

(1.8)

R-^+00 J JQ-

For proving the latter relation, observe that \Pa,p{C,,R)\ < 1 for ^ G G~. But _^lim

Pa,p(C,i?)=0

it—>-+oo

for any fixed C ^ G~. Therefore, (1.8) follows by Lebesgue's theorem. T h e o r e m 1.1. Let F be a meromorphic function in G~, having a finite set of zeros {a,j,} and poles {bu} in G~ C G~ for any p < 0. / / fj

_ |Im C r - i |log |F(C)|| da{0 < +0O

(1.9)

for a given a G [1, +00); then for any p < 0

+ TT7;

T. buGG-

| I m 6 ^ - p | i + ' ^ - 2 - ' * lim e+'^log\F{-it)\,

(1.10)

80

CHAPTER 4

where the limit exists and is finite. Proof. By Lemma 1.1, (1.9) implies the requirements of Lemma 1.2 for ^ = log |F|. Consequently, the integral in (1.7) tends to that in (1.10) as i? -> +oo. It is obvious from (1.6), that limR-^^OQ Ia{s, p, R) = (1 + a)~^ for any fixed p < 0 and s G G~. Therefore, the passage R —)- +oo transforms the (finite) sums of (1.7) into those in (1.10). The existence and the finiteness of the limit in (1.10) is provided by the existence and the finiteness of all other limits. 1.3. The below theorem, which easily follows from Theorem XX of [76] by a passage, contains a special case of Levin's formula which gives rise to Tsuji characteristics. T h e o r e m 1.2. Let F{w) be meromorphic in G~, and let {a^} and {bj^} be zeros and poles of F{w). If for any p < 0 in G~ lies a finite set of points b^ and for ^{w) = log \F{w + ip)\ the second condition of (4) in Introduction is fulfilled, then for any p <0

Yl K-p\-

Yl \^^-p\ 1 f^"^ 1 = — / log\F{u + ip)\du-Urn tlog\F{-it)l 27r J_^ 2 t^+oo

(1.11)

where the limit exists and is finite. Assuming that n(t, F ) is the number of poles bj^ of F{w) (counted according to their multiplicities) lying in the half-plane G^, consider the functions m{p, F)=

Y,

|Im b,-p\=

r

{p - t)dn{t, F), (1.12)

0^(p,F-i)= Y.

\i^a,-p\=

r

{p-t)dn{t,F-'),

1 /'"'"°° m(p, F) = —j log+ \F{u + ip)\du, (1.13) 1 f^°° 2-K J_^ C{p,F) = ra{p,F)+m{p,F),

1 \F{u + ip)\ C{p,F-')=

m{p,F-^)+

^{p,F-').

(1.14)

Formula (1-11) can be written as the equilibrium relation C{P,F)

= C{P,F-'^)

+ CF,

-OO
(1.15)

81

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

1.4' We will reduce a family of Tsuji type characteristics depending on a G [1,+CXD) from the Levin type formula (1.10). For this, observe that ^

I I

|Im C - pr'

where the functions

IIIQ

log \F{0\da{0

- a [xn^{p, F) - m<,(p,P-')] ,

are some integrals of Tsuji characteristics (1.13):

ma(p,F)=

f\p-tr-^ui{t,F)dt,

J —OO

It is obvious that ^

|Im6,-p|^+"= r

{p-tr+'dnit,F)=

J—OO

h^eG-

f

{p~trdm{t,F),

•/—OO

where 0^(t, F) is the Tsuji characteristic (1.12) for the poles of F. An integration by parts gives

bueG-

where

?io(p,F) = -]-

[

(p -

tr-^^{t,F)dt.

Similarly, au^Gp \-1T for zeros of F . Additionally, denoting {a2'^)~^\\mt^+^t^-^'^\og\F{-it)\ Coc{F), one can write (1.10) in the form

m(p, F) + ^{p, F) = m(p, F'^) + 0^(p, F ' ^ ) + Ca(F),

=

- o o < p < 0,

or, which is the same, in the form of an equilibrium relation: £ , ( p , F) - £a(p, i^-') + Ca{F),

-OO < p < 0,

(1.16)

where Z^a are the Tsuji type characteristics

c.{p,F)= r «/ — O O

.(p,F-i)= r

ip-tr-'

mit,F)

+

---m{t,F) 1 -f a

( p - i ) " - i [ m ( i , F - i ) + - ^ g i ( f , F - i ) dt.

(1.17)

82

CHAPTER 4

1.5. The equilibrium relation r ( r , / ) = r ( r , / - i ) + 0(l),

0
between Nevanlinna characteristics of a function / meromorphic in the unit disc D = {^ : 1^1 < 1} is well known. In view of this relation, NevanUnna's condition (5) (Introduction) defining his weighted class is equivalent to

L

1

(1 - rY [T(r, / ) + r ( r , f'^)] dr < +oo,

- 1 < a < +oo,

/o

and for 0 < a < +CXD it is equivalent to

sup [T«(r, / ) + Ta{r, f-^)]

< +oc,

0
where T^ is a Nevanlinna type characteristic of the form

Ta{rJ)=

Jo

f{r-trT{tJ)dt.

But one can be convinced that the equilibrium relation (1.15) between Tsuji characteristics is true not for all functions meromorphic in the half-plane G~. Therefore, the similarity of Nevanlinna's weighted class is to be defined as the set of meromorphic functions F in G~, for which 0

/

l^p-i [C{t, F) + C{t, F-^)] dt < +00,

0 < a < +oo.

(1.18)

By Lemma 1.1, for 1 < a < +oo the last condition is equivalent to sup

[Ca{t,F)-i-Ca{t,F-^)]

<+00.

-OO
One can verify that (1.18) is equivalent also to requirements of Theorem 1.1 which provide the validity of the Levin type formula (1.10) and the equilibrium relations (1.16) for the above Tsuji type characteristics. 2. CANONICAL FACTORIZATIONS In this section, successive passages i? -> +oo and p - > — 0 i n ( 1 . 5 ) will lead to canonical factorizations of the below classes of meromorphic functions. 2.1. Definition 2 . 1 . A function F meromorphic in the half-plane G~ is said to be of the class ^^^3 (0 < a < +00, 0 < P <1-^ a), if

f

J —c

'*'" [C{t,F) + C{t,F-^)]dt<+00. l + \t\0

(2.1)

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

83

The following inclusions are obvious: K/J.^^"/?.'

0<(3^
+ a,

"^,0^^+5,0+6,

0 < / ? < l + a,

(2.2)

0<6<+oo.

(2.2')

L e m m a 2 . 1 . If F £ 9T"^ {a > 0, 0 < (3
and

^

|Im 6^|^+" < + o o .

(2.3)

Besides,

Proof. (2.4) follows from (2.1). From (2.1) also follows that O

l/|a-l

/ Hence (2.3) holds. Indeed, for proving, for instance, the boundedness of the first sum in (2.3) suppose that for some po < 0 there is an infinite set of zeros a^ of F in G~^. Then for any t G (po/2,0) ^{t,F-')=

Y, >

(|Ima^|-|t|)> Yl

Y.

(|Ima^|-|po|/2)

(|Im a j - |po|/2) >

Yl

(|po|-|/>o|/2) = +oo

which contradicts (2.5). Thus, max^ |Im a^| = mo < H-oo, and O

|y-|Q:-l

/•|Im a^\

Ula-l

/ /»|Im a^\

>{l + m^)-^J2

\t\''-^{\lma^\-t)dt

= {a(l + a ) ( l + m ^ ) } " ' ^ | I m a j i + " . 2.2. Now the passage R -> +oo in (1.5) will lead to some similarities of the Jensen-Nevanlinna formula for functions from ^ ^ I + Q , - These formulas, as well as following them canonical representations of ^^^^ (a > 0, /? G [0,1 -h

84

CHAPTER 4

a]) contain some Blaschke type products which are considered in [69]. These products are similar to those considered in Ch. 1 and consist of factors of the form r /"Slim CI ^a^^ ^ aa{w,C)=exp\—— - 7 1 + ^ f. (2.6) I Jo [r + i{w - C)] J As the factor of Ch. 1, for any fixed C ^ G~ and a > — 1 the function aa{w, () is holomorphic in G~ and vanishes only at w = ( which is a first order zero. Besides, the product Ba{w,{wk})

= Y[^c^{'^^'^k),

{wk}cG~,

-l
(2.7)

k

is convergent when ^|Im^fc|^+^<+cx).

(2.8)

k

We start by some preliminary lemmas. Lemma 2.2. Let w and s are fixed points in G~ {p < 0). Then for any a> -1 ^ fw — ip . s — ip hm ^a[ —:^— +t,—^—+i]

K^^+oo

\

rC

it

\ J

, . . =-logaa{w-ip,s~ip).

. .

. . (2.9)

Proof. Let z, C ^ G~ be any fixed points and i? > 0 be chosen enough great to provide \z + iR\ < R and |C + iR\ < i?. Then by (1.3) (1 - 1 ) ^

dt

z±iRA

C

Hence

i?

'i?

T+i{z-C,)-irzR-'^ 1-iCR-^

1+a 1 _ X

by (1.4) and r = R{1 - t). Thus, 2|Im CI

hm ^ a - ; ^ 4 - ^ , — ^ - ^ = /

—

r'^dr

In view of (2.6), taking z = w — ip, ( = s — ip we arrive at (2.9).

85

FUNCTIONS W I T H SUMMABLE T S U J I CHARACTERISTICS

L e m m a 2.3. Let F G OTr,i+a (<^ > !)• Then for any p<0 lim / / -^+00 J J R-i'^ooJJ

a-l

IC - ^pP

y ~ 2R\lm C-p\

i{w - C - 2ip) - ^^-'P)^^^'P)

OiR,p)

= [[ \lmC-pr\ ^ JG-

|ImC-pr-Mog|F(C)|

^^!'^^^^' ^,tia(C), [i{w - C - 22/?)]

1+a

weG;.

da{0

(2.10)

P r o o f is similar to that of Lemma 1.2. Namely, introducing the function l+a

[i(ti; - C - 2ip)]

a-l

{^

2R\lmC-p\J

j ^

if

C^OiR.p)

if

<:^o{R,p)

(w-ip)(C-\-ip) R

= < \i(w - C ~ 2ip)

0 write (2.10) in the following equivalent form: lim

/ /

^,,,(^,C,i?)|ImC-pr'log|F(C)Ma(C)-0.

(2.11)

For proving this relation, evaluate ^a,p in assumption that w G G~ is a fixed point and R> 2\w — %p\^ j |Im w — p\. For ( G 0(i?, p) obviously -(l+a)

\^o.p{w, C,,R)\<\{w-

ip) - (C - zp) ,12

Xn+

1-

ic-^p|-

\ "-1

2i?ilm C - p|

{w - ip){C, - ip) ' - ^

-|-(l+a)>

{w - ip) - (C - ip)

Besides, -1

[w - ip){C^ - ip) {w - ip) - (C - ip)

"R w -ip

(C - ip)

< i?i \f ,„w_L^r=i j i z i ^ <' — ip) i?|Im It; — p| 2 since Im (w — ip) < 0 and Im {( — ip) < 0. But \{w - ip) - (C - ip)\ >\Imw-p\-}-

|Im ( - p\.

86

CHAPTER 4

Therefore by (2.4)

\^aA^X,R)\ yOV"^,S5^VI < ^

1 + 21+a ,,

I ,

IT

/-

l\l+«

for any fixed p < 0^ w E G~ and any R > 2\w — ip^ j |Im w — p\^ (^ E 0{R, p). In view of the definition of ^a,p, this estimate remains true for C, € G~\0{R, p). Further, the use of the inequality 1 TTT^ < -rr

7

1

rn- <

max|l,a-(^+"H , T-hrz (0 < a, 6 < 4-oc, a > 1)

gives

where Ca,p(tf;) = (1 + 2^+") m a x { l , |Im if; -/9|-(^+^)}. Now set g(x) = m—5 ^^ ^ l + ( x - a ) i + « '

0 < a < X < +oo.

Evidently ^(a;) < (l + x^+'') {x - a)-(i+") < 1 + (1 + a)i+^ for a; > 1 + a, and this is true also iov a < x < 1 -}- a. Consequently

l + (x-a)i+«

<—-^ :rr l + a;i+«'

,

0
1 < a <+oo.

Therefore, by (2.12) C . , » [ l + (l + |p|)i+-] |^a,p(^,Ci^)l <

l + |ImCP+^

From this estimate, from the convergence of (2.4) (with /3 = 1 + a) and from Lemma 1.1 it follows that the integrand in (2.11) has a summable majorant independent of i? > 2\w — 2pp|Im w — p\~^> On the other hand, it is obvious that limi?^+oo ^a,/)('^5 C^^) — ^ ^^^ ^^Y fixed p < 0 and wX ^ G~. Therefore, (2.11) follows by Lebesgue's theorem. T h e o r e m 2.1. Let F G ^ ^ 1 + ^ (a > 1). Then for any p < 0

+

^

logaa{w - ip, ay, -ip)

- l o g j lim F{-it)\,

weG-,

-

^

\ogaa{w - ip, K - ip) (2.13)

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

87

where {a^} and {bj^} are the zeros and poles of F and the limit exists, is finite and not equal to zero. P r o o f follows from formulas (1.5), (2.9) and (2.10). Lemma 2.4. Let {wk} C G~ satisfy (2.8) for a given a > — 1. Then lim

y^

log aoc{w -ip.Wk-

ip) = Y^ loga^{w,Wk)

(2.14)

p->

w^eGfor any w e G lmwk
which does not lie in an intercept Ik = {w : Re w =^ Re Wk, 0} {k > 1).

Proof. Assume Im. w < —4mo, where mo = max^jlm Wk\. Then

yZ

logaa{w -ip.Wk <

^ Im

-ip)

-

y]\ogaa{w,Wk)

\\ogaa{w -ip.Wk

-ip)

-\ogaoc{w,Wk)\

Wk
+

^

\\ogaa{w,Wk)\ = S1 + S2.

p < l m WkKO

One can verify that the series ^2 is convergent. Hence, for any s > 0 there exists pi <0 such that 52 < e/2

for

pi <

p<0.

(2.15)

Now observe that by (2.6) 2|Im Wk\

/

^ ' ^Im^w,

2{\Imw,,\-\p\)

r'^dr \T-^i{w-Wk)\^-^'^''

where \w — Wk\ — r > mo since \w — Wk\ > 3mo and 0 < r < 2mo. Therefore 2I+CX

Si<

j^^

(1 + a)mn ^ / u

J2 \lmwk\'^--

IT ^ ^^Xm Wk
53

T ^ Im Wk
(|Im^,|-H) l-\-a

One can verify that under the condition (2.8) lim

p-^'-O

(|Im Wk\ - \p\f'^'^ = X ! |Im Wk\l+a

^ Im

Wk
88

CHAPTER 4

and the first summand in tlie above figure brackets tends to the same limit. Consequently, there exists p2 < 0 such that ^i < 6/2 for p2 < p < 0. Hence by (2.15) the relation (2.14) holds for any w such that Im w < —4mo. Besides, if {wk} is a finite sequence then (2.14) is true for all w G G~\Uk {h}- Therefore, (2.14) is valid out of the set Uk{lk} also in the case of infinite sequence {wk}L e m m a 2.5. IfFe

hm / /

^ ^ i + ^ {a > 1), then

\lmC-pr'

-IL

^^^^^^^^MC)

[i{w - C - 2ip)\ ,,_! |ImCr\

G-

Iog|F(OI ^'-;ViU^'^(0, [i{w - 0 ]

weG-.

(2.16)

P r o o f is similar to that of Lemma 2.2. Lemma 2.6. Let {wk} C G~ satisfy (2.8) for a given a > —1 and let w be fixed and such that Im w < —4mo (mo = max{l,maxfc |Im t^fel}). Then logB, (^, {Wk}) ^ - E

/

, ^ /

,a+,

(2.17)

is a continuous function of ^ e [a, +00). Proof. All summands in (2.17) obviously have the required property. Therefore, it suffices to prove the absolute and uniform convergence of this series in any segment [a,/3] C [a,+00). Assuming that 7 G [a,^], we have \T + i(w - Wk)\^'^'^ > mj+^. Hence for 0 < |Im Wk\ < 1/2 2|Im Wk\

/h

ol+a

T^'dr [T ^-i{w

~Wk)]^^^

(1 +

a)ml^''

24. T h e o r e m 2.2. Let F G ^ ^ a + a (o^ > 0). Then p (w)

=CF-^—.

77-TT

Ba{w,{b^})

where Ba are convergent products constructed by zeros and poles of F, and CF =

lira \F{~it)r^ t->+oo

is a finite number.

^ 0

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

89

Proof. Let a > 1. Then (2.18) follows by the passage p -> - 0 in (2.13), in view of (2.16), (2.14) and the uniqueness of a holomorphic function. Now let 0 < a < 1. Then, by (2.2') F G ^ ^ 1 + 7 for any 7 > a, and hence F admits the representation (2.18) for any 7 > 1, i.e. F{w) '^B^iw,{h})

"^[nJjG-^

[iiw-0]

J

Setting rui = max^^i^ {|Im a^|, |Im bjy\}, fix any w G G~ such that Im w < —4mi. Then the right-hand side of (2.19) becomes identically constant. On the other hand, by Lemma 2.6 the factors in (2.19) are continuous functions of 7 G [a,+oo). Hence formula (2.18) holds. R e m a r k 2 . 1 . In view of inclusion (2.2), the factorization (2.18) is true for any function from 0^^^^ (a > 0, 0 < /3 < 1 + a). R e m a r k 2.2. The argument used to prove the case 0 < a < 1 in Theorem 2.2 permits to extend the assertion of Theorem 2.1 (i.e. formula (2.13)) to the case 0 < a < 1. 2.5. Below we prove a useful estimate for the exponential factor of (2.18). L e m m a 2.7. If F e 0^^^ (a > 0, /3 G [0,1 + a]), then for any w e G'

^

J J a-

[i{w-c.)y+'^

I

Proof. Obviously |w - CI > \^^ w\ + |Im C| for any t«, C S G~. Therefore, using the inequahty a + b> o(l + b)/ (1 + a) (a, 6 > 0) we get 1

^

1

(l + |Inm;|)^

l ^ - C I ' ' ~ |Im«;|^ (l + | I m C | ) ^ ' Further, (1 + xf/

(1 + x/^) e [1,2'^-!] {x > 0, P > 0). Thus, 1_ ^ ,|g_i| 1 + |Im ^/^ \w-C\0\lmw\0 l + |ImC|^

and ^ ^lg_il 1 + |Inm;|^ |Im w|i+" l + |ImC|''' y,_^|i+a-

90

CHAPTER 4

Using the last inequality we come to (2.20). satisfies (2.8) for an a; > —1, then one can show-

Remark 2.1. If {wk} C G that |logB„ {w,Wk})\ < ^

1 ^ |Im « ; , r + 4 |Im w\-^'+'^^

(2.21)

for Im w < —5maxfc |Im Wk\. By (2.20) and (2.21), (limt->+oo F{—it)) the factorization (2.18), i.e.

= 1 in

CF = lim [F{-it)]-^

(2.22)

= ±1

/;—>-+oo

for any F G OT^^ (o^ > 0, /3 G [0,1 + a)). 2.6. Definition 2.2. A function F holomorphic in G~ is of the class OT^^ (0 < a < +CXD) if for

any p

<0 +CXD

/

|log \F{u -{-iv)\\du<

4-00

(2.23)

-oo

and

lV|^|i+"^V-oc '°g"^l^(^ + ^*)l^"<+°°- (2.24) By Theorem 1.2, Levin formula (1.11) is true for any function F G 01^ (0 < a < 4-CXD). This implies the equilibrium relation (1.15). Consequently ^ a

C ^r,l+a'

0 < a < +00.

(2.25)

Hence, all assertions of Lemma 2.1, as well as the factorization (2.18) of Theorem 2.2 are true for these new classes 0^^. 3. DESCRIPTIVE REPRESENTATIONS We start by some assertions related to inclusion of multipliers of the factorization (2.18) in 9T^^ (a > 0, /? G [0,1 + a]). 3.1. We start by a Tsuji type estimate which is similar to that found by F.A.Shamoian [94] in \z\ < 1.

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

91

L e m m a 3.1. Let a sequence {wk} C G~ satisfy the condition ^|Im^fc|i+^<+oo

(3.1)

k

for a given a > — 1. Then log \B^ {w, {wk})\
L-^|i+« '

"'^^~'

^^-^^

where Ca > 0 is a constant depending solely on a. Proof. It suffices to prove (3.2) for a single factor of Ba- To this end, we consider two cases in assumption that ( = ^ + irj G G~ is fixed, (a) Let 2|r/|/|i(; — CI > 1/5, then we use the recurrent formula aa{w, C) = aa-p{w, C) exp \ ^

-——— ( — ^ ) \ ,

(3.3)

where p > 0 is the integer from p — 1 < a 0 is an integer. But in our case

vi«.-ci/

-

*i ^ " > f J 2 L | . i<„
Therefore log|ap(^,C)|
,

p = 0,l,2,...

(3.4)

If Q; > — 1 is not an integer, then from (3.3) one can derive logM^,C)l
^ S = a-pe{-l,0).

(3.5)

For evaluation of log \as{w, C)\ we change t = r /i{w — () in (2.6). Since i{w — C) = —Im ti; + 7/ > 0 for Im t(; < 7/ and Re tt; = ^, by uniqueness of holomorphic function we get ^og\as{w,0\

= - R e J^

{i-^t^-^S'

92

CHAPTER 4

If 2|ry|/|K;-C| < 3, then (3.6)

^1

Jo i-*-

And if 2|77| /\w - C| > 3, then ^U^^^ t^dt log|a,5('^,C)l < C ( 5 - R e / , ,77 + ty+^ 2\V\

= Cs-Re

1+5

[^^^ ^^ ( 1 - - ^ I

d\t[

again by uniqueness of holomorphic function. As 3 < \t\ < 21771/111; — C|, we conclude that |1 + 1 | > 2 and 1+.5

Re

1-

1+5

1-

1+ t

cos (1 + ^) arg

l+t

1 -

1+t

>

Vs

Consequently, —

2|r?|

log|a,KC)|
\V\

14-a

rr- > - ,

a > —1.

(b) Now let 2|77| /\w — C\ < 1/5. Then one can verify that 4r?Im w w~ ( = 1——=:z w—C \w-C\'

, and

2|Im w\ 2\rj\ T^ =^ < -,—^-^ \w-C\ \w-C\ + 2.

Therefore w—( \w — (

>

^

12\lmw\

1 /

2\rj\_

b\w-C\

^\\w-C\

^\ J

>

14 25

and consequently \w — ( w—(

w-C 2i\r]\x > w—( w—(

* '

H-CI

>i

2'

^e[0,l|.

(3.7)

FUNCTIONS WITH SUMMABLE TSUJI CHARACTERISTICS

93

Thus, for a > - 1

log|aa(w,C)ll <

<

/

[^(-Ofe-;f=c)]

4I+0C

x^dx

k-ci

w—C, _

2i\7]\x

1+a

1+^

<

\V\

l+a

1 + a V|^-CI

Hence by (3.7) we come to the desired estimate: \0g\aa{wX)\

\v\

< Ca

w,C G G ,

\w-C\

a >-1.

3.2. Lemma 3.2. Let a sequence {wk} C G satisfy (3.1) for a given a > 0. Then the following equilibrium relation is true for Ba{w, {wk}): C{p, Ba) = C{p, 5 - I ) ,

(3.8)

- 0 0 Np-\-l. Then we write 'No

\\og\Ba{w,{Wk})\\<

(Xl"^ \k=l

XI \ \^0g k=Np+l/

\aa{w,Wk)

(3.10)

li k > Np-\-l, then \u + iv -Wk\ > \v\ + |Im Wk\ > 2|Im Wk\ + |p|/2 for v < p. Consequently, by (2.6) i-oo

/

\ log \aa{u -^ iv,Wk)\\du -00

<

l + a

|Im Wk J —c

du [\u + i(\v\ + |Im Wk\)\ - 2|Im Wk\i l + a *

Observe that ^+00

du

-hoo

/ -00 fk + i(hl <

\\P\J

|Im Wk\)\ - 2 | I m Wk\] l+cx J\u\<2\p\

du J\u M^!>2|p| {\u\-\p\y+-

23+a 4 < -r-T7+ \p\a\p\-'

94

CHAPTER 4

Therefore +00

\\og\aa{u + iv,Wk)\\du /

-oo \

^

<^^H-"(-+2i+"J

o3+a

/I

Yl

/

|Im^A:r-^"<+oo.

(3.11)

k=No-\-l

Let V < p, let k {1 < k < Np) be fixed, and let rrik = max |Im Wkl- Then -t-oo

/

\ log \aa{u -\- iv,Wk)\\du -oo 2|Im Wk\

< /

T^dr

C?i/ / \u\>Amo

Jo

+ /

k + 1^1 + |Im Wk\ +

\log\aa{u + iv,ilmWk)\\du

i^P+^

= Ii-{-l2.

(3.12)

/|u|<4mo

Obviously r2mo / l <

/

du f

^a^^

J\u\>4:mo \u\>4mo

Jo Jo

[\u\ + i{\v\ H- |Im Wk\)\ - r]^"^"^

J\u\>Amo

Jo

(|u|-2mo)i+-

a ( l + a) '

^

^

and /2 ^ ^ock{i^) is a continuous function of t'(—oo < !» < 0). Besides, lim^^oo ^oi,k{v) = 0 in view of (2.21). Consequently I2 < max

max "^akM < +00.

l
' ^ ^

Hence (3.9) follows by (3.12)-(3.12'), (3.11), and (3.10). It remains to observe that by (3.9) Ba{w, {^k}) satisfies the requirements of Theorem 1.2, and hence (1.11) and (1.15) are true for this function. In view of (2.21) the relation (1.15) can be written in the form (3.9). L e m m a 3.3. Let {wk} C G~ be an arbitrary sequence. Then: 1°. / / (3.1) is true for an a > 0, then Bao{w, {wk}) G ^ ^ o /^^ ^'^V ^o > Oi. 2°. / / Y\ |Im ^fe|^+" log

r < 4-00

for an a > 0, then Ba{w, {wk}) G ^ ^ / ^ for any l3 > 0.

95

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

Proof, r . Let QQ > a be arbitrary. By (3.2), /

e-^dt

/

Jo

log+ \Bc,,{u - it, {wk})\dv

J-oo p+oo

< Cc V |Im ^fc|i+"« /

t"-i/«„^(t)dt,

(3.13)

^^^C;,(^+|Im^.|)--«.

(3.130

u

Jo

where + 00 •^otQ.k

dx

J-oo

[(t4-|Imi(;^|)2+x2]^^

= {t+\lmw,\)--^

r ^ J-oo

(1+^2^

2

Hence />+oo

/• 4-00

Jo

1 t t^"a--M

Jo r+oo

= C ; j l m w,r'"^

j^

a-1 7

( T T ^ ^ C(a,ao)|Im w;,!"-"".

Consequently /• + 00

/

/' + CXD

r-'C{-t,Ba,,)dt=

Jo

/» + 00

f'-Ht Jo

\og+\B^,{u-it,{wk})\du J—oo

< C'^C{a, ao) Y, Ilm Wk\^+'' < +00. k

Hence B„„(tz;, {w^}) G m^o by (3.8). 2°. Let /3 > 0 be arbitrary. Then by (3.2) and (3.13)-(3.13') p-\-oo j.a—1

/

/'+00 j.a—1

Y^^^-t^Ba)dt=

r+oo

YTJp^^ ^

log^\Bao{u-it,{wk})\du

,

p+00

j.a—1

jj.

One can verify that A fl

1+t/3(t+|Im«;,!)" ^ i i fO-l

^+

/.|Imto| 1

70 l + i ' 5 ( t + |Imwfc|)" ^ / o

t^~(3' x°'''^dx

(l + :r)^ 1

Jo /o

Im^fcl 2/(1 + 2/)'' Jllm

<- +

log

I m Wk\

96

CHAPTER 4

Consequently +00

.a-l

log

\lm Wk\

< +00.

G 0^r,/3 by (3.8).

R e m a r k 3 . 1 . T h e assertion 1° of t h e pervious lemma and L e m m a 2.1 prove t h a t t h e condition (2.3) presents a complete characterization of t h e density of zeros {a^} and poles {bj^} of functions from OT^^^ {0 < a < + 0 0 , 0 < (3 < l-\-a). L e m m a 3 . 4 . For any a > 0 there exists a sequence {wk}i^ C G~ satisfying (3.1) hut such that B^ {w, {wk}T) 0 ^ r , a + i P r o o f . Let t h e sequence {wk}T = {~'^'^k}T (0 < r^ < 1) be such t h a t

^ r^+" < +00 but

J2 ^fc^"" ^^g ~ = + ^ Tk

k=l

k=l

Consider t h e function Ga{w)

{w = re'^,

Ba+l{w,{Wk}) Ba{w,{wk})

-7r
By (3.3) aa^i{w,Wk)

ga{w,Wk)

aoc{w,Wk)

1

2rk l-\- a \iw -hrk

exp

l+a'

Consequently 2 l + a r^^

COS

log \ga{re''^, -irk)\ = 7 - — ' ' 14- a

[(l + a ) a r c t a n ( ; ^ i | ^ ) ] il+a

;.e<^+f)+;.,|

Hence 2l+a

/"O |t l + a

COS

(l + a ) a r c t a n ( ; : ^ i g ^ ) ]

log r e: < ' ' + t ) - f

G;n(t),

l+a

where n ( t ) is t h e number of those rk for which — r^ < t G (—1,0). that cos ( l + a ) arctan

rcosi? r\sm'd\ + |t|

> —^

V2

for

^+

Observe

< 4(1 + a ) "

F U N C T I O N S W I T H SUMMABLE T S U J I CHARACTERISTICS

97

Therefore ^^'^''^"^log+\GaiC)\da{C) /•f + 4(l+c)

> /

/"l

/

1

.

1

|rsin??|"-Mog+|Gc.(re^ )|rdrd^

where C^ = m i n | ^| ^^^ 4(i+a)]

'"'"}* ^ ^ ^ ^^ ^^ obvious that

> -^ / j t | - . . o g l < i u ( , ) - 4

= +oc.

Hence Ga{w) 0 ^ ^ i + a ? ^^d ^ a ('^? {'"^fc}?^) does not belong to ^ ^ i + c ^ since Ba-\-i{w^{wk}'i^) G ^r,i+Q! by the assertion 1° of Lemma 3.3. 3.3. The below theorem is the main result of this section. Theorem 3.1. ^^^^ (0 < a < +oo^ 0 < /3 < a) coincides with the set of those functions which for some aQ> a are representable in the form

where Bao{w,{a^}) and Bao{w,{bjy}) are convergent Blaschke type products with zeros {a^}; {bj,} C G~ satisfying

J2 |I«i ^MI^"^"" < +0C and Yl \^^ ^^l^"^"" < + ^ '

(3-1^)

and /i(C) = Mao(C) ^5 ^ function of bounded variation in any compact from G , such that llm Cl"^'^ ' ^' CI'' |^MC)l<+oo. (3.16) / / .G- 1 + |Im

98

CHAPTER 4

Proof. Let F{w) G ^^^p (a > 0, 0 < /3 < a). Then the sequences {a^} and {bi^} satisfy (3.15) by Lemma 2.1. Besides, the integral (2.4) is convergent. Further, m^^ C 91:^1^, C OT^oj+^o (ao > «) by (2.2)-(2.2'). Therefore, by Theorem 2.2 and (2.22) we come to the conclusion that F{w) is representable in the form (3.14)-(3.16), whatever be ao > a, and dii{() = log |F(C)|(icr(C). For proving the converse statement, note that Bc.,{w, {a^}), Ba^,{w, {K}) e ^ZP

(3-17)

by the assertion 1° of Lemma 3.3. Thus, it suffices to prove that also

To this end, observe that i|a —1

/ / .G- 1 + |Im w\l^

-^?~II

|ImCr-'/(a,ao,/3,OMMOI,

(3.18)

where ff |Imt/;|"-i da(w) J JQ- l-\-\lmwf \w-C\^-^'^''

,^ _ „

For a G (0,1] a suitable estimate of the last integral follows from the inequality

(which is true for any a > 0). If a > 1, then one can be convinced that

Jo (1 + t0)it + a)"o - "

Jo a-f^ + xP-""

'A-.

1 +t0 a—ao—fd

Besides,

A

(i+t^)(t+a)-o ^ y.

(l + ax)^o

~ ao - a + Z?'

Hence, for a > 1 J{a,cxo,l3,a)

<

a""'''-^ —^ [a- ^){a.o - a + /3)

= C-,a!^-^--K

FUNCTIONS W I T H SUMMABLE T S U J I CHARACTERISTICS

99

By the continuity of g{a) = J7(a, ao,/3, a) on (0,+oo), the last estimate and (3.19) imply ^a—ao

J{a,aQ,f3,a)
1 + a^

0 < a <+OO.

Returning to the boundedness of the right-hand side of (3.18), we conclude r+oo .a—1 71 /'+00

t"-^dt

r+°°

dx /•+°°

dx

oo x-in+oco a—ao

=

C4J{a,ao,(3,\r]\)
ioi C=^ + iV ( ^ < 0 ) . Thus ^

J

J

Ilm

Cr-'l{a,ao,/3,(:)W{0\

Hence the inclusion (3.17') follows by (3.18)-(3.18'). Thus, any function representable in the form (3.14)-(3.16), where a, P {a > 0, 0 <^ are any numbers, belongs to ^ ^ ^ . 3.4- Now we shall show that ioi a < f3 < l + a (3.14)-(3.16) is not a descriptive representation of the classes ^^^p (a > 0). To this end, for any a, {3 and ao > a consider the measure rfMao(C)-CaoN^(l+^')-'rf^c/r/

( C ^ ^ + iry), -1

where C.^ - r(ao) [2^'T (ao + ^ ) T ( l - ^ ) ] ' . This measure obviously satisfies (3.16). We shall prove that nevertheless

Previously we note that the integral in the exponent is absolutely and uniformly convergent in G~, and hence the function ao2"° r [

llmCr-'

,

,,.

100

CHAPTER 4

is holomorphic in G~. One can show that pao G{W)

/'+00

= ^2^Cao / T Jo

i " ° + ^ - ' / a o ( ^ - ^<)rf<,

(3.21)

where I.

The last integral can be calculated by residues. Namely, taking w — it = s(e G~) we get

'-^^^=loo [i(.-.)]^+-(i+.^)"y_ *^^^)'^^' where ^5(2:) = {[i(5 — >^)]-^"^°'°(l + 2:^)} is holomorphic in the z-plane, except the cut {z : Re z = Re 5, Im z < Im s) C G~, and ^s(z) has simple poles at z = ±i. Calculating the integral of this function along V R — {Re^^ : 0 < ^ < TT} U {[-i?,i?]} and letting R -> +00, we obtain la^^s) = 7r(l + is)~^-^+"°^. Inserting this expression into (3.21) and using the well-known formula for Euler's ^S-function we find G{w) = (1 -[- iw)~^~^. Consequently, F{w) = exp{(l +iK;)~2 ^j iov w = u-\-iv e G~ and \og\F{w)\ = \l-\-iw\

2

^cos1

(3-a\ ( u 2—y arctan VI+ 1^1

Besides, log \F{w)\ > 0 since 1/2 < 1 - (/? - a ) / 2 < 1. Therefore, 1

C{t,F)^—

/"-hoo

j cos < —

\og^ \F{u + w)\du

27r

/; _ , , [(i + | t | ) 2 + ^ 2 ] | -

= +00

J —c

for any t < 0, and hence (3.20) is true. The above counterexample shows that for a < /? < 1 -|- a (3.14)-(3.16) is not a descriptive representation for OT^^^ (0 < a < +00), in contrast to the case 0 < yS < Q; considered in Theorem 3.1. The case j3 — a remains open. NOTES. A similar method of mapping and successive passage has been used earUer by M.M. and A.E.Djrbashian [24] for proving integral representations of holomorphic functions in G from L^(G , y^dxdy). One can observe that the functions (1.3) are the inversions of a special, infinite case of Tsuji characteristics considered in [38] (Ch. I, Sec. 5). As F.A.Shamoian noticed, the passage of this chapter can be used for proving the representation (2.18) for holomorphic functions from somewhat larger classes (the suitable density condition for zeros must be presupposed). But there were some difficulties in finding descriptive representations. Later a version of the descriptive representation (3.14) was proved [5] for subharmonic extensions of the particular classes OT^Q (a > 0). Namely the approach of F.A.Shamoian [94] was used to prove some descriptive representations over the real axis, where d^ is replaced by (p(t)dt and (^(t) is from a definite O.V.Besov space.

CHAPTER 5 BOUNDARY VALUES 1. MAIN RESULTS 1.1. In view of results of Chapters 1 and 3 the below definition is natural. Definition 1.1. A function F{w) meromorphic in the lower half-plane G~ is of a-bounded type in G~ or, which is the same, of the class Ncc{G~} (—1 < a < +oo), if it can be represented in G~ in the form ^, , Ba{w,{an}) i /"+^ - ^ H = J. ) \ {{ exp <^ Co + ciw + /

da{t) .^ T^iT^+^CL

1.1

where CQ, ci (ci = 0 for a > 0) and C are any real numbers, a{t) is representable as the difference a = ai — a2 oi two nondecreasing functions satisfying

/ where p > 0 is the integer deduced hyp—l
|Im anl^"^"" < 4-00

n

and

^

|Im 6^|^+'' < +oo.

m

Note that for a G (—1,0) the classes Na{G~} are subsets of Nevanlinna's class N of functions of bounded type in G~ (i.e. representable in G~ as fractions of two bounded holomorphic functions). Definition 1.2. Let E C (—oo,+oo) be a Borel measurable set (-B-set) and let 0 < 7 < 1. Then we shall say that E is of positive 7-capacity {C^{E) > 0) if there exists a Borel measure (B-measure) /i supported in E {/JL ^ E) and such that '•^^ dfi{t) - 1 (1.4) (l + |t|)7

I

J —a

and Si=

sup

r°°-ML<+oo.

(1.5)

102

CHAPTER 5

If there is no such a measure, i.e. ^ i = +cx) for any nonnegative 5-measure H -< E satisfying (1.4), then we say that E is of zero 7-capacity {C.y{E) = 0). Definition 1.3. Let E C [-7r,7r] be a B-set, and let 0 < 7 < 1 be any number. It is said that E is of positive 7-capacity if there exists a nonnegative 5-measure 11 •< E such that dfi{^) = 1

(1.9)

/ •

and sup /

oii?|7

< +00.

(1.10)

If there is no such a measure, i.e. if ^2 = +CXD for any nonnegative ^-measure ji ^ E satisfying (1.9), then E is said to be of zero 7-capacity. Remark 1.1. The above definition of 7-capacity on the unit circle belongs to O.Frostman [33] and is well-known. One can verify that a conformal mapping of the disc onto the half-plane transforms it to a form which is equivalent to the Definition 1.2 for bounded sets of real axis. 1.2. The below theorems are the main results of this chapter. Theorem 1.1. Any function F € Na{G~} (—1 < a < 0) has non-zero, finite nontangential boundary values in all points u G (—00, +CXD)^ with possible exception of a set of zero 1 + a-capacity. The next theorem gives a more complete result for the Blaschke type products in G~ (Ch. 1), which are special representatives of classes NociG"}. Theorem 1.2. 1°. Let 7 G (0,1) and a G [7 — 1,7 + 1] be any numbers, and let {wk} C G~ be a sequence satisfying ]^|Im^fcP<+cx).

(1.11)

k

Then Bot{w^{wk}) has non-zero, finite nontangential boundary values in all points u G (—cx), +oc)^ with possible exception of a set of zero ^-capacity. 2°. Let a G [0,2] be any number, and let {wk} C G~ be a sequence satisfying Y^\lmwk\

< +00.

(1.12)

k

Then Ba{w, {wk}) has non-zero, finite nontangential boundary values in almost all points of the real axis. Also we shall prove a similar theorem in the disc.

BOUNDARY VALUES

103

Theorem 1.3. 1°. Let 7 G (0,1) and a G [7 — 1,7 + 1] be any numbers, and let {zk} clD) be a sequence satisfying

^(l-|z,|)^<+oo.

(1.13)

k

Then the Blaschke-M.M.Djrbashian product Ba{z, {^k}) ((10) in Introduction) has non-zero, finite nontangential boundary values in all points of the unit circle, with possible exception of a set of zero ^-capacity. 2°. Let a G [0, 2] be any number, and let {zk} C B 6e a sequence satisfying ^ ( 1 - \zk\) < +00.

(1.14)

k

Then Ba{z^ {zk}) has non-zero, finite nontangential boundary values in almost all points of the unit circle. We shall prove also the below inequality which is of independent interest: if — I < a i < a 2 < 0 are any numbers and {wk} C G~ is a sequence satisfying (1.11) with 7 — 1 + ai, then \Ba,{w,{Wk})\

< |5a2(^,{^A:})| ,

W £ G'.

(1.15)

Note that for a2 = 0 this inequality was proved in the end of Ch. 1. 2. PROOFS OF THEOREMS 1.1 AND 1.2 2.1. We start by some lemmas necessary for the proof of Theorem 1.1. Lemma 2.1. Let Ei, E2 C (—CXD,+OO) be any B-sets such that Cy{Ei) Cj{E2) = 0 for some 7 G (0,1). Then C^{Ei U E2) = 0.

=

Proof. Let /i -< Ei U £"2 be a nonnegative jB-measure satisfying (1.4). It is obvious that, if taken over Ei or J5'2, the integral (1.4) equals to a number M G (0,1]. For instance, let it be taken over Ei. Then also the measure ^;L = M~^iJi\^ supported in Ei will satisfy (1.4). But C^{Ei) = 0. Therefore sup / weG-J-00

, ^^ ^ , > M sup / w-ty weG-J-00

, ^^ , = +00. \w-ty

The next assertion easily follows by Fatou's lemma and a passage to a limit. Lemma 2.2. Let 7 G (0,1); let E C (-00, +00) be a B-set such that C^{E) > 0, and let fi ^ E be a measure from Definition 1.2. Then f^^

dfi{t)

^ ^

/^+^ dfi{t)

,

^

.o.x

104

CHAPTER 5

Lemma 2.3. Let (3 G (—1,0) be any number, and let a{t) be a nondecreasing function such that

Further, let f{w) be a holomorphic function given by the formula

/H=r°°,/"^'L,^ ^eC-. J-co

(2.3)

[l{w - t)\ ^

Then the set of those u G (—oc, 4-cx))^ where the nontangential boundary value f{u) exists as a finite limit, is of zero 1 -{- (3-capacity. Proof. From (2.2) and (2.3) it follows that the function exp{—/(^)} is holomorphic and bounded in G~. Therefore, by Lindelof's theorem the existence of a finite angular boundary value f{u) in a point u G (—oo, -\-OQ) is equivalent to the existence of the finite limit lim / ( ^ + i^) {=f{u))

(2.4)

V—>—0

in the same point. For proving the existence of this limit, write f{u + iv) = f{u + ivo) -f- i / f{u + iy)dy

(2.5)

Jvo

for any u G (—00, +00) and v, VQ (—00 < VQ < v < 0). One can verify that for any w = u-\-iv e G~, t G (—00, +cx)) and K > 0

\w-t\^ where (2.3)

CK,{W)

- {\u -1\ + \v\)'^ - ( l 4 - | t i ) « '

is a constant depending only on K and w. Therefore, by (2.2) and +00

\nu

+ iy) \dy <(1 + /3)2i+^/2 /

-00

p-\-oo

da{t) /

H

J—00

i-oo +^

= 2^+^/2

/

J\v

dx

{\u-t\+xf-^^

da{t) \l+/3 {\u-t\ + \v\y

-00 (\u -

r+00

< V2CI+R(U

+ iv) /

. . "", .V/i . r, < +00.

Hence, the passage VQ -^ —00 in (2.5) gives f{u + iv) = f{u - ioo) 4- i / f{u + iy)dy, «/—00

v <0, - 0 0
BOUNDARY VALUES

105

where the integral is absolutely convergent and f{u — zoo) is a finite limit. Now let EQ be the set of those u G (—00, +00) for which 0

/

\f\u 4- iv)\dv = +00

and let E be the set of those u G (—00, +00), where (2.4) is not a finite limit. Then E C EQ by (2.7). Hence, Ci^p{Eo) = 0 implies Ci+p{E) = 0. Indeed, if we had Ci^p{E) > 0, then there would exist a nonnegative 5-measure fi •< E for which (1.4)-(1.5) is true with 7 = 1 + /?. But E C EQ. Consequently, /i ^ EQ, and CI+(3{EQ) > 0 by Definition 1.2. Thus, it suffices to prove that Ci+/5(-Eo) = 0. We shall show that the > 0 leads to a contradiction. Indeed, if Ci-\-p{Eo) > 0, assumption CI^^{EQ) then there must exist a nonnegative 5-measure //o -< ^0 satisfying (1.4)-(1.5). By (1.4), one can fix an enough great A> 0 such that / dfio{t) = I diio{t) > 0. /[-A,A]nEo J-A J[-A,A]nEQ Then

A /

pO

-A

diio{u) / \f{u + iv)\ dv = +00. If \u\ > 2A and \t\ < A,-A then J—00 < 77—rr^TlTIT' - ( i + H ) ( i + |t|)'

\u-t\

C,{a) = {l + A-'){1

(2.8)

+ 2A).

Hence for Iwl > 2A ^1+/?/

J-A \U - t|l+^ - (1 + \v\Y^f J^A (1 + 1*1)^+'' {l + A-^)^+>^{l + 2Ay+^ On the other hand, if \u\ < 2A, then by (2.1) ^

diXQJt)

5 i ( l + 2A)i+^

/ Hence, particularly for any u € (—00, +00) ^ /

rfMo(^)

|u-t|i+/'

, [(1 + A-'y+0

+ 5 i ] (1 + 2^)1+/^

(i + iuDi+z^

C2(^)

{i + \u\y+0'

,...

106

CHAPTER 5

But for any u G (—00, -f-00)

r \fiu+iv)\ dv < 2^+^/2 r°° -^

da{t)

J—00

J—00

1^

1+/3'

where the integrals can be finite or infinite. Therefore, by (2.9)

< +00

which contradicts (2.8). Lemma 2.4. Let /3 G (—1,0) he any number, and let the sequence {wk} = {uk + ivk} C G~ he such that ^kl'+^<+oo.

(2.10)

Then Bp{wj{wk}) has non-zero, finite nontangential boundary values in all points u G (—00, -f 00); with possible exception of a set of zero 1 + (3-capacity. Proof. It was proved in Sec. 4 of Ch. 1, that B(^ = Boexp{—/}, where / is of the form (2.2)-(2.3). By Lemma 2.3, the exponential factor in (2.2)-(2.3) has the required boundary properties. Therefore, it suffices to prove that BQ has non-zero, finite angular boundary values at all real points, with possible exception of a set of zero 1 + ;9-capacity, and to use Lemma 2.1. To this end, observe that BQ diff'ers from the ordinary Blaschke product in the half-plane only by a constant multiplier. Consequently, by the Frostman's [33] result we can (after mapping conformally the disc onto the half-plane) only state that

for any nonnegative B-measure fi satisfying (1.4) (where 7 = 1 + ^ ) and supported on the set of those u G (—00, +00), at which Bo{u) ^0 does not exist. After we choose ^4 > 0 enough great to provide that the integral (1.4) (with 7 = 1 +/3) taken over [—A, A] equals a number M > 0, we consider the measure M~^fi\r_^ ^. = Jl ^ E. It is evident, that this measure satisfies (1.4) (with 7 = 1+/?). On the other hand, it is obvious that Ji satisfies also (2.11). But L \w

(2^+^ + A - ^ - ^ ) ( l + A)i+^

BOUNDARY VALUES

107

for IK;I > 2 ^ and \t\ < A. Consequently H-oo = M s u p \{l + \w\)^-^^ f oo

djl{t) r \w-t\

.A

< ( 2 i + ^ + A - i - ^ ) ( l + ^ ) i + ^ 4 - ( l + 2A)i+^ sup

/

\W\<2AJ-A \W\<2AJ-A

dii{t) !^-^P+^'

Hence Ci-^ 13(E) = 0 in the sense of Definition 1.2, which proves our lemma. Proof of Theorem 1.1 is now obvious. By Lemmas 2.3 and 2.4 both the exponential factor and the Blaschke type product in the representation (1.1) of F G Na{G~} (—1 < a < 0) have non-zero, finite nontangential boundary values out of some sets of zero 1 + a-capacity. Hence, F has the same property by Lemma 2.1. 2.2. Now we turn to lemmas which are necessary for proving Theorem 1.2. Lemma 2.5. 1°. Let a G (—1,-hoo) be any number, and let {wk} = {uk -f ivk} C G~ be a sequence satisfying (2.10) with (3 = a. Then Ba{w,

{Wk})

= Ga{w,

{Wk})Ha{w,

{Wk}),

(2.12)

where GUwdm})

= l[9aiw^i^^})^I[^MT" k

H^{w,{w,})

= llh^{w,{w,})^l[exJ-

[

r ^ 'W ^^ '-uk)-t]^ " " " ' " / ^ T i + JJ -^0

(2-^^)

/'^'IJh^i^^

(2.130

both are holomorphic functions in G~, and Haiw^ {'^k}) 7^ 0 (i<; G G~). 2°. If —1 < (3 < 0 and f3 < a < 1 -}- /3 and the sequence {wk} = {uk+ivk} C G~ satisfies (2.10), then the functions Baiw, {wk}), Gai'i^^i'Wk}) and Ha{w^{wk}) belong to Np{G~}. Proof. 1°. One can easily prove that B^, also the product (2.13), as well as the product (2.13') in (2.12) are absolutely and uniformly convergent inside G~. Consequently, (2.12) follows form formulas (1.1') and (3.4) of Ch. 1. Thus, it remains to observe that the integral in the exponent of the representation of each factor of Ha is holomorphic in G~. 2°. Assuming (" = ^ + iry G G~ an arbitrary point, we use the well-known formula F77T /

T-

r{6) JQ

(ZW H- a)^

\—^ =

-WTT-^TT-TT-T'

r(A)

(tw)^-^

0 < J < A < +00.

(2.14) ^

^

108

CHAPTER 5

Then for /? < 0 we obtain W~^\ogga{w,C)

= ^.^_^ . /

a^[logga{w-ia,C)]'da

T{l + a-(3) {\r]\ - trdt '-(3) ff r(l +•a)a) J [i(^_^)_t]i+"-^' Similarly

{H-trdt [i{w-i)^t] 1+a-

r f l ++aa) ) Jo r(i Jo

Besides, the same representations are true also for ^ = 0, since VF^ is identical. Consider the case a = /3. Clearly + 00

\W~^

/

log \Ha{u

+ ii;, {I(;A;})| Uii

-oo

1

^r-^

< T(l + a ) ^ . < W o

/^l^^'

/"^

""'

'

J-^

{u-utY

+

it^vf

One can verify that also the remaining conditions of the inclusion Ha ^ ^ ^ are satisfied (0^^ are investigated in Ch. 3). 9 1 ^ C Na{G~} by Theorem 3.6 of Ch. 3. Hence Ga G Na{G-}, since ^ ^ G iVa{G"}. Now let ^ < a < 1 + /3. Then, by (2.6) +00

\W-''\og\g^{u /

+

iv,0\\du

-OO

-

r(i + a-/3) (3+.-,)/2 /•'"Vi -f)"dt

r(i + a)

io ^'^'

Similarly /•i-oo + 00

/

\W-'^\og\K{u -OO

+ ivX)\\du

p _ _ _ ^ L _ _

^ y-oo(^ + IH-t|)^+"-''

BOUNDARY VALUES

109

Hence -hoo

\W-''\og\Ha{u +

/

iv,{wk})\\du
-CX)

^

and it is easy to verify that Ha G OTj C Np{G~}. For evaluation of / in the above estimate for ga, first assume \v\ > \r]\. Then obviously

=r(^f^-r<'"-'>^-^

1+/3

+0

If \v\ < \rj\ and a < 0, then H^l

/'(l^l+hl)/2

I
{\v\ - tfdt + /

Jo

J\v\

1 + /3

M^+^ + 2 ^ M z ^

r-hl

{t- \v\fdt + /

(IT^I -

tfdt

J{\v\+\ri\)/2 1+/3'

< jijH'«.

Now let \v\ < \r]\ and a > 0. Then integration by parts gives

Al^l+!^i)/2

J

(l+/3)(l + ^ - a )

From these estimates it follows that \W~^\og\Goi{u-[-iv^{wk})\\du

< H-oo.

-oo

Hence Got ^ ^ J ^ C Ar^{G~}, and it remains to see that 3^ G Np{G~} by the inclusions Go. G iV/3{G~} and H^ G iV/3{G~}. L e m m a 2.6. Let a G (0, +oo) and let the sequence {wk} = {uk^-ivk} that

he such

Then Ba{w, {wk}) = Ba-i{w, {wk}) [Ha-i{w, {wk})]-'

,

(2.15)

where Ha-i is a function holomorphic in G , having the form (2.13'). 0 < a < 1; then Hot-i is a hounded function in G~.

If

110

CHAPTER 5

If a G (l,+oo) and the sequence {wk} = {uk -hivk}

satisfies

k

then Ba{w, {Wk})=Ba-l{w,

where Ra-i

{Wk}) [Hc,-2{w, {Wk})f

[Ra-l{w,

{Wk})f

,

(2.16)

is holomorphic in G~ and hounded for 1 < o; < 2.

P r o o f follows from formulas (2.12), (2.13), (2.13') by successive integration by parts. Besides, it is clear that |i5/^a-i| < 1 in G~ for 0 < a < 1. In addition, one can show that

Roc-i{w,{wk}) ==exp^-——y^n—~—vi^^ r{

l-\-a^

[i{w - Uk)r

J

Using this, one can prove that |i?a-i| < 1 in G~ for 1 < a < 2. 2.3. P r o o f of T h e o r e m 1.2. 1°. Let (1.11) be valid with a fixed 7 G (0,1). First consider the case 7 — 1 < a < 7. If we denote /3 = 7 — 1, then — 1 < ^ < 0 and /3 < a < ^ + 1. Therefore B^c e Np{G-} = iV^_i{G-} by Lemma 2.5, and our assertion follows from Theorem 1.1. Next, let 7 < a < 7 + 1. Then the representation (2.15) of Lemma 2.6 is valid, and 7 — l < a — 1 < 7 . Therefore -B^-i, Ha-i G Nry_i{G~} by Lemma 2.5. Hence Ba is of the same class, and our assertion follows from Theorem 1.1. At last, let a = 7 + 1. Then, by (2.15) and (2.16) Ba = ^ 7 + 1 = ^7[-f^7]~ 5

^ 7 = [Hy-iRy-.i]~

,

(2.17)

where, by proven above, B^ has non-zero, finite boundary values at all u G (—cxD, +CXD), with possible exception of a set of zero 7-capacity. By Lemma 2.5, Hj-i G Nj-i{G~}. Therefore, by Theorem 1.1 also the set of those u, where H^-i has not non-zero, finite boundary values is of zero 7-capacity. Thus, if we also prove that lim HJu + iv,{wk})y^O (2.18) v—>—0

out of a set of zero 7-capacity, then, by Lindelof's theorem, non-zero, finite boundary values of i?^_i will exist out of such a set. By Lemma 2.1, H^ will have the same boundary property. And returning to the first equality in (2.17), we shall arrive at the desired assertion for -B7+1. Thus, for completion of the proof of assertion 1° it suffices to show that (2.18) exists as a finite limit outside a set of zero 7-capacity. One can prove this

BOUNDARY VALUES

111

quite similar to Lemma 2.3. The following estimates will be the only difference: r\'^k\

^ogH^{u + iv, {wk})\'\ < (7 + 1) 5 1 k

^ A

/

Jo

{\vk\-t)^dt [i{w - Uk) + t]2+7

{\u-uk\ + \v\y

pQ

dfJ.o{u) / -A

<

[logi/'^(w + i'?;, {t<;fc})] |d^

./—cx)' 2(l+7)/2 - 1 y

' | ^ ^ | 7 [^

dl'oiu)

^ 2(l+7)/2

-15^y

|^^|7 < ^ o o .

2°. Let (1.12) be satisfied. The desired assertion is well-known for a = 0. If 0 < a < 1, then (2.15) implies that Ba is a function of bounded type in G~, and hence our assertion follows. If 1 < a < 2, we use the representation (2.16), where Ha-2 and Ra-i are holomorphic and bounded in G" and Ba~i is of bounded type in G~ by proven above. Thus, also Ba is of bounded type in G~, and hence our assertion follows. The proof is complete. It remains only to prove the inequality (1.15) for the Blaschke type products in the half-plane. To this end, we use formula (2.14) one more time. In view of formulas (1.1')-(1.2) of Ch. 2 we get ip{w) = W-"^^ log

6a2('i/^+^,C)

r(l+ai-a2)

Re/Jo r(i+ai)

1 Re / " r(l+a2) Jo

M-tr

1 _ {iw —

t)f\l-\-ai-a2

+

1

dt

1 {\v\-tr iw — t + -iw -\-t dt

for any a i , a2 (—1 < a i < 0^2 < 0), K; € G~ and (^ = ^ -{- irj e G~. Hence it is evident that the function (^ is harmonic in G~ and continuous up to the real axis. But a simple calculation shows that for if; = li G (—00, +00) the integrand of one of the integrals in the representation of (^ is a positive function, while the integrand in the other integral is identically zero. Hence (^{u) > 0. Further, ^{w) > 0 (it; G G~) by the Phragmen-Lindelof principle, and applying Theorem 1.4 of Ch. 1 we find that Boc^{w,{wk})=Boc2{'^'>{^k})^y^v\-

/

dG{t) [z(^-t)]l+"2

weG-

where a{t) is a nondecreasing function satisfying (2.2) with /3 = 0^2. Hence (1.15) follows.

112

CHAPTER 5

3. PROOF OF THEOREM 1.3 3.1. First note that in our case there is no similarity of formula (2.14) in the disc. Therefore, it is not evident how to prove the similarity of the inequality (1.15) for Blaschke-M.M.Djrbashian products. Nevertheless, for our aims it is enough to replace (2.14) by the following L e m m a 3.1. For any (3 G (—1, +oo) and a> (3 — 1

f

^-^'6,= /o (l-tz)2+«

^ '-&^ ., (l_^)l+a-/3'

..C,

Jo

where Ja,p{z) is such that Ma,p{Ro)

=

s u p |Ja,^(2:)| < + 0 0 , \z\
0 < i?0 < + 0 0 .

Proof. The change of variable t = 1 — {1 — z)x/z where

JaAz) - ^ l0

gives our representation,

{1+X)^+-

is a function holomorphic in the z-plane cut along 1 < z < 4-oo (Im z = 0). Besides, here occurs the well-known arbitrariness in choice of integration contour. Since \z/{l — z) + 1| = 1/|1 — 2:| > (1 + Ro)~^ > 0, one can choose this contour to be distant from —1 not less than (1 + -Ro)""^- Concretely, we choose the integration contour in the following way. If the point w = z/{l — z) is out of Ai^o = {w : \ a,Tgw\ > n — arcsin(l +i?o)~'^, Re li; < —1 + (1 -\-Ro)'^}, then we choose the contour to be the intercept [0,w]. li w G ARQ and \w\ < 2, then we choose the union of the intercept [O, |t(;|e*^^^)] (where (p{w) = [TT — arcsin(l + i?o)~'^]sign(argt<;)) and the of a circle of radius \w\. Further, ii w e AR^ and \w\ > 2, then we unite the intercept [0,2e^^^^)], the arc [2e^^(^), 2e^^"s^] of a circle with the radius 2 and the intercept [2e^^'s^, tf;]. Let \z\ < 1/3. Then w ^ AR^ and |^| = \z/{l - z)\ < S\z\/2, Therefore

- « | < H F ^ P " ^ = ^ H ^ ( I ) ' - ' (^•'i If 1/3 < \z\ < Ro and w 0 AR^, \W\ < 2. Then \JaA^)\

< 3^+^(1 + i?o)'+" / JQ

x^dx = f — ( 1 + Ro) 2 + a i

H-p

If 1/3 < 1^1 < RQ and w ^ AR^ but \w\ > 2, then obviously

Jo

J2

\X — i~) "^

BOUNDARY VALUES

113

Thus, if z/[l — z) remains out of Ai^^, then sup

(3.10

\Jot,(5{z)\ < + 0 0 .

1/^<\Z\
Now let 1/3 < \z\ < Ro and let w G AR^, \W\ < 2. Then roa,rgw < 6^+^(1 + i?o)'+"[(l + / ? ) - ' + TT] < +00. Further, if 1/3 < \z\ < RQ and w G AR^ but \w\ > 2, then

\J^,l3{z)\
1 1

+ 7r + 7r + 31+/3

/. + 00

x^dx < +00. + x)2+«

Thus, the relation (3.1') is true also when the point w = z/{l — z) lies in ARQ. Together with (3.1), this completes the proof. 3.2. We shall use also the below similarities of Lemmas 2.5 and 2.6. Lemma 3.2. 1°. Let a G (—1, +00) be any number, and let {zk} C'D {zk y^ 0, k > 1) be a sequence satisfying (1.6). Then Boc{z, {Zk})

= CaGa{z,

{Zk})Ha{z,

(3.2)

{Zk}),

where Ca is a constant depending on a and {zk} and Gcc{z,{zk)) = W9cx{z,Zk) = f j e x p < -

/

Hc.{z,{zk})

j

= X[h^{z,Zk)^Xle^A-

(1 - x)""

dx \

(1 - xY

dx

, 7 - ^

1+a

(3.3)

(3.3')

are holomorphic functions in J}, and H^^z, {zk}) 7^ 0; z G B . 2°. / / -l 1) satisfies (1.13) with j = 1 + p, then the functions Ba{z, {zk}), Ga{z, {zk}) and Ha{z,{zk}) belong to the class Np of M.M.Djrbashian. Proof. 1°. It is easy to verify that the products (3.3) and (3.3') are absolutely and uniformly convergent inside B. Hence, (3.2) directly follows from

114

CHAPTER 5

(1.7)-(1.8). It remains to observe that the integral in the exponent of the representation of each factor of Ha is a holomorphic function in D. 2°. First observe that under our requirements there exists po ^ (0? 1) for which \zk\ > Po {k > I)' Further, assuming ( = pe^^ {po ^ p < 1) and z = re'^ (0 < r < 1) we use the following formula from [19] ((1.22), p. 573): if f{x) G Li(0, /) (0 < / < +oo), then for any a G (0,1)

Then for any /? G (—1,0] and a G (—1, +oo) we get r-f'D-l'

logg^iz, () = r-l'D-^'+0^^

+ I3)p r \1 (l + /3)p l+g

log g^{z, C) + ^^^^^

7Jp^/

' J oJo

\oggc,{z, Q

h-r^tei{'P-^)?+" (i-tftei-P-^))'

Z*^ ( 1 - a

T{1+I3)J,

7

Similarly, we find l + a

f^{l-x)

-dx

For a — P, one can show that

.-«i5-.l„g*.(re«,p.<^)|<ji±?^^'

2r(2 + a ) i p

—^—ax

|l_ree*(^-v)p

^

Using this estimate, we obtain that for 0 < r < 1 1 r \r--D--i^^\H^^,,i^^{,^}^\\ ^^J-7v

< "

\ V^(l - |z,|)i+- < +00, Pol (2 + a) ^

which provides i^a ^ -^a- Hence G^ G N^ in view of 5c^ G iV^ (see [19], Ch. IX). Now let l3 < a <1+p. Then |r-/^i^-/^log/..(re^^pe^-)| < ^ 1 ^ ^ P P oo l( l++ aa) rr ((ll+ / 3 )

+ 1+^ f\i-xrdx

C

(1 -

'

tfdt

BOUNDARY VALUES

115

and by Lemma 3.1

(1 - x)"daj

+ WW)"""''

(3.4)

i(^_c^)|l+«-^"

p \p — rxe

Observe that iov 'd — if = \ |A| < TT ^^1^ =.[xI a: — rpte

rptf

+ 4a:rptsin^ - > (a; - rpt)'^ +

4xrpt^

Therefore |A| X — rpt + 2y/xrpt

|^_^p^e^A|-(2+-)<2iW2

-(2+a)

and p /

\r-'^D-^loghaire''^,pe''^)\d^<

(i-p) y90(l+a)r(l+;9) dA 2+a

(a; — r/9f + | ^/xrptX)

The change of variable t = l — (x/p— 1)T gives

i

Ci (1 + T ) I + " - (a; - p)""

0 {x - pty+'- ^ {x- p)"-/3 Jo

Consequently, by a; = 1 — (1 — p)y we get ^

r

\r-'^D-l^loghc{re'^,pe''^)\d-d <

po{l+a)T{l+p)

nT{ '^7rT{l+(3)Jo

(1-2/)"-'^^

'^^

•

Hence, for 0 < r < 1 l-J^ 27r

\r-^D-^logHa{re'^,{zk})\d^ 22+a/2^^

po(l+a)r(l+/3)

|.l

+ 7rr(l+/3)yo / Jo

^a^^

(1 - 2/)""^

5^(1-1^,1)1+^ < + 0 0 .

116

CHAPTER 5

Thus, Ha G Np. For proving that also Ga ^ Np , observe that in view of (3.4) the repetition of the above argument gives d^

27r po(i + a ) r ( i where ^a,/3 =

^ (l-x)"da; /,p | p - r a ; | " - / 3 '

(3.6)

For evaluating of the last integral, first assume r < p. Then

Ia0<

/

1 {i-x)''dx {I-PY^I^ I -p k—^ < ^' „ ' ' ' ,

0
(3.7)

Next, assume p
One can verify that 1

Ji < J2 <

/•"/'•/

^ / 1

/JN/5^

a; - -

dx<

((i1- -p p) i) + ^ ^ / '

/•(i+p/'')/2. p\0 ^ / a; - - dx H

1

/-i 2 ( 1 -- Pn\^+0 ) -J / (1 - a;)'^da; < (1+PM/2'

P5+"(1+/3)

Therefore

•^a,/3

^

Po

+ Po

( 1 - p ) 1+iS-, 1+/3

p
a
(3.8)

Now let p < e < 1 and 0 < a < 1 + /3. Then integration by parts gives . l+/3-a

'".""TTjT^I-/, a-r^(^'^) .(l+p/r)/2

PN1+/3-"!

1

/•!

(1 - x)''da;

(i+p/r)/2 (a; - f) a-/3 <

1

1

1

(1-P)^+^

p^''

2i+/5pJ+"

pr"(l+;9)

1+,9-Q;'

BOUNDARY VALUES

117

Hence, by (3.5)-(3.8) ^

j ^ | r - ^ i ^ - ^ l o g G a ( r e ^ ^ {zk})\d^ < C2 ^ ( 1 - kA.|)'+^ < +00

for 0 < r < 1, where C2 is a constant depending only on a, P, and po- Thus, Ga € Np^ which provides Ba G Np by already proven Hoc ^ -^/3 and (3.2). Lemma 3.3. 1°. Let a G (0,+CXD) be any number, and let {zk} C ID) (ZA; 7^ 0, k > 1) be a sequence satisfying

j2{i-\zk\r<+oo. k

Then Ba{z,{zk})^C'^Ba-i{z,{zk}) where C'^ = exp {—a~^ ^ki^

[^^-i(^, { ^ a ) ] " ' ,

(3.9)

~ \^k\)^} is a constant and the function

^:-i(^,{^4) = n e ^ p | r

V

T — ^ S ^ ^ U o

(4.1 (1-^)

(3-10)

J

is holomorphic in D and bounded if 0 < a < 1. 2°. If a e (l,+oo) and the sequence {zk} satisfies

^(i-i^feir-i<+(x., A:

then Ba{z,

{Zk})

= C'^B^-1{Z,

{zk})-

[^a-2(^,{^fc})]

(3.^^)

[HT^2i^,{^k})]'[Ra-iiz,{z,}f where

i?a-i(^,{^fe}) = e x p ( - - ^ ^ | z f e | - ^ l ^ ^ M r i |

(3.12")

118

CHAPTER 5

are holomorphic, non-vanishing functions in B , hounded if 1 < a <2. Proof. 1°. The representation (3.9)-(3.10) is proved in [6]. The required properties of H^_i are obvious. 2°. One can easily show (see [6]) that for any

^a(^,C) = ^ a - l ( ^ , C ) e x p < 2 /

—-a•^^-

JKl (l On the other hand, if h^-i {z, () is a factor of the product (3.10), then obviously

J\i\ (l

('-?)'

Further, one can verify that <Pa-l{z,0 ^a-2{z,C)

exp<

1

(1-|(|)»

""M>-«r

l + 2 ( a - l ) /•' ( l - i ) " - '

^

°"' ^'i^-fT"

The representation (3.11) is a consequence of these equahties. For 1 < a < 2 the boundedness of functions (3.12), (3.12') and (3.12") in B is obvious. Proof of Theorem 1.3. 1°. Let (1.13) be true for some 7 G (0,1). First, assume that 7 — 1 < a < 7. Denoting /? = 7 — 1 we have —1 < /? < 0 and P < a < P + 1. Consequently, Ba G Np by Lemma 3.2, and our assertion holds in view of Theorem 2 (Introduction) of M.M.Djrbashian-V.S.Zakarian. Next, let 7 < a < 7 + 1. Then the representation (3.9)-(3.10) of Lemma 3.3 holds, and 7 — 1 < Q ; — 1 < 7 . Therefore, in this representation Ba-i G Np by Lemma 3.2. In essence, the proof of H^_i G Np coincides with that of the inclusion H^-i G Np. The desired statement follows from the above mentioned boundary property of functions of Np. Now consider the case a = 7 + 1. According to formulas (3.9) and (3.11), Ba = 5^+1 = C;+iB^[if;]-2, where H; = H;*_^[H;*_\]-^R^. AS it is proved above, the multiplier B^ has non-zero, finite boundary values at all points of the unit circle, with possible exception of a set of zero 7-capacity. As it was mentioned, H*'!Li ^ -^7-1? ^^d one can easily verify that also iJ*!*! G N^-i. Hence these functions possess the required boundary property in view of the common boundary property of functions from Nj-i. Therefore, if we prove that the limit lim H;_,{re'^,{zk})j^O (3.13) exists and is finite out of a set of zero 7-capacity, then by Lindelof's theorem Rj and successively if* and B^^i will have non-zero, finite boundary values

119

BOUNDARY VALUES

out of such a set. And this will complete the proof of 1°. The proof of (3.13) is quite similar to that of Lemma 2.3. The following estimates appear to be the only difference. Let /i* be a factor of the product (3.10), let z — re*^ (0 < r < 1) and let C = pe*'^ (0 < po ^ P < !)• Then one can verify that

|:log/^;(re^^O <

(1 - x)'*dx

22+7/2

Ip

{x-rp+lV^p\§-ip\)^^''

Jo

(x — •

22+7/2

1 + 7 ( i _ r - p + f v ^ p | ^ - y , | ) [ p ( l - r ) + fv^p|^-v^|]^+' <

{1-pr

22+7/2.

Consequently, for Zk = \zk\e^'^^^ \zk\'> PQ> p

L

i| ^

dr

dr

\ogH*M\{zk})

dr ^

-^0

\\Zk\{l [\zk\{l-r)

+

lr\zk\W-ipk\] dr

27

< 22+7/2 ^ ( 1 _ | ^ ^ |

7+1

^ ^ p r ^ r\ '/Ai / 2 ((li+ 7 r - i \'&-^^\-ry+^ 22+7/2

2T +

<^^+rE(i-l^^l „7+l Po

k

1

7 l^-^fel^J

At last, if ^2 is the number from (1.10), then

r dpoW f J-ir

d ^^\ogH;{re^'^,{zk})

dr

Jo

22+7/2

<-9TrE(i-l^^ir ^0

k

TT '

2^ + — 5 2 7

< +00,

since for any (p G (—7r,7r]

dM-&) oi'd | 7

= 52 < + 0 0 .

2°. If (1.14) is true, then our assertion is well-known for a = 0. If 0 < a < 1 then by (3.9) BQC is a function of bounded type in D, which provides the validity of our assertion. If 1 < a < 2, then we use the representation (3.11), where

120

CHAPTER 5

the functions H^*_2^ -fi^a-*2? ^^d Ra-i are holomorphic and bounded in B . And Ba-i is of bounded type in D by the previous case. Thus, also B^ is of bounded type in D. This completes the proof. One can be convinced that the requirements of Theorems 1.2 and 1.3 for a > 7 + 1 (in 1°) or for a > 2 (in 2°) do not provide the inclusion of the considered Blaschke type products in the suitable classes of functions of a-bounded type. Therefore, the method used in this chapter is not applicable for investigation of boundary properties of these products in the mentioned cases. NOTES. Theorem 1.1 is a distinctive similarity of a result proved by M.M.Djrbashian and V.S.Zakarian [28, 29, 30] for the classes N^ (- 1 < a < 0) in the disc. Similar to the case of unit disc, here also a contraction of Nevanlinna's class leads to delicate boundary properties. For - 1 < a < 0 and y = 1 4- a the assertion 1° of Theorem 1.3 was proved earlier by M.M.Djrbashian and V.S.Zakarian [28, 29]. In another particular case 0 < a < 1 and y = a the same assertion, in essence, was proved by D.T.Baghdasarian and I.V.Ohanyan [6]. For all - 1 < a < 1 the assertions 1 of Theorems 1.2 and 1.3 can be considered as generalizations (for the Blaschke type products in the half-plane and in the disc) of the well known result of Frostman [33], which coincides with the case a == 0 of assertions 1° of Theorems 1.2 and 1.3. Further, by assertions 2° of Theorems 1.2 and 1.3 a well-known property of the classical Blaschke product is extended to Blaschke type products. The author's united work with G.V.Mikaelyan [71] contains similar results on the boundary properties of the Blaschke-M.M.Djrbashian products of the version of 1945 [16, 17] (formulas (7), (8) in Introduction), as well as for the Blaschke type products considered in [69] (formulas (2.6), (2.7) in Ch. 4) and used in factorizations of Ch. 4. The structure of that products permits to prove in [71] the similarities of results of this chapter for all a G ( - 1 , + 0 0 ) .

CHAPTER 6 UNIFORM APPROXIMATIONS

1. MAIN RESULTS In this section we prove the following three theorems which are related to approximations and some other problems in the classes Na{G~} of functions of Qj-bounded type (see Definition 1.1 in Ch. 5). T h e o r e m 1.1. The class Na{G~} (—1 < a < +oo) coincides with the set of functions which are representable in G~ in the form Ba{w,{am})

{

,

^ ^ ) ^ o / , , /L IN ^^PS ^0 4- ciw

Bc,[w,{hn\) hn iTypi-" /

\ogba{w,t + i'n)dii{t)+ic\,

J —oo

(1.1) J

where the limit is uniform inside G", B^ are convergent Blaschke type products (of Ch. 2), and the numbers CQ, ci, C, the sequence {bn} and the function ii{t) are those in factorization (LI), (1.2), (1.3) of Ch. 5. T h e o r e m 1.2. Let F{w) be a holomorphic function from Na{G~} (—1 < a < +oo); representable by the formula (LI), (L2), (L3) of Ch. 5, where Co — ci = C — 0 and fi{t) is a nondecreasing function. Then for any natural numbers K, and N there exists a triangular matrix of pairwise different complex numbers {wi \K>,N)} C G~ {k = 1,2,..., I = 1,2,... ,/c) such that: 1°. For any k >1 Im ^ P ^ (/^, N) = rjihi, AT),

/ = 1,2,... , fc,

and N

sup

{k\M-,N)\'-''^}<'^^ V^+1

2°. From the associated with the mentioned matrix finite Blaschke type products k

B^{w,{wl^\K,N)}',)

=

Y[b^{w,w^,^\K,N)) 1=1

122

CHAPTER 6

one can choose a sequence

such that F{w) = 5„(w,{a™}) lim 5„(«;,{«;f^'^(K,-,iV,)}^) j->oo

uniformly inside G~. R e m a r k 1.1. This theorem is similar to a well-known result due to Schur [93], stating that any bounded holomorphic function in |2:| < 1 (or in a half-plane) can be uniformly approximated by a sequence of finite Blaschke products, although, for a = 0 Theorem 1.2 somewhat differs from Schur's theorem. R e m a r k 1.2. Any function oi Not{G~} (see (1.1) in Ch. 5) is representable as the product of exp{co + ciK;+2C} and a quotient of two functions satisfying the conditions of Theorem 1.2. Therefore, Theorem 1.2 implies a similar assertion on uniform approximation of arbitrary meromorphic functions from Not{G~} (—1 < a < -hoc) by means of finite Blaschke type products. Theorem 1.3. Let F{w) e Na{G-}

{-1<

a < +oo), and let F{w) ^ 0.

r. If liminf / '^-^-0 J_^

\W-''\og\F(u-^iv)\\-

^ = 0, ' 1-hu^

(1.2)

then F{w) = e^o+ci^+ic Ba{w, {am}) ^ ^^Q-^ Ba{w, {On})

(13)

where CQ, C\, and C are those in (1.1) of Ch. 5, and Ba are some convergent Blaschke type products. 2°. Conversely, if F{w) is representable in the form (1.3), then + 00

/

I W - " log \F{u -hiv)\\du = 0,

(1.4)

-oo

R e m a r k 1.3. Being a generalization of Akutowicz' theorem [3] for the classes Na{G~}, Theorem 1.3 becomes a somewhat different assertion for a = 0. This difference of cases a = 0 of Theorems 1.2 and 1.3 from the well known results is a consequence of the fact that No{G~} is a subclass of the set of all functions of bounded type in G~.

UNIFORM APPROXIMATIONS

123

2. PROOFS OF THEOREMS 1.1, 1.2, 1.3 2.1. The following lemma is to be used for proving Theorem 1.1. Lemma 2.1. For any a G (—1, +00) and t-i-ir] E G~ n^{w,t + in)= ^^^^^^^^^^_^^^^^^+Ra{w,t

+ iv),

w€G-,

(2.1)

where Ra is such that for any compact K C G~ and any w G K. and t G (-00,+00)

\Raiw,t + ir,)\<^f^^^l^^ll",

\r,\
(2.2)

where Ca(K) 25 a constant depending only on a and K. Proof. In view of the proof of Lemma 1.2 in Ch. 2 {w, t + irj) = r{[i{w

-t)-

a ] - ' - " - [i{w -t) + a ] - ' - " } {\r)\ -

af^'^dG.

/o

Hence (2.2) follows by the estimate , 2\i{w-t)±a\>\

f W for 1^1 >2/^ + P " J t p

tor

\t\ <2K-\-

. , X . (o
p

where K = max^^^K |Re it;| and p = mini^^K |Im ILJI. P r o o f of T h e o r e m 1.1. First we show that any F{w) G Na{G~} (a > - 1 ) is represent able in the form (1.1). To this end, observe that by (2.1) ^ [i{w-t)]^-^^

-

^\r]\-'-''logbo.{w,t

+

iri)-^\rj\-'--Ra{w,t^irj)

for any rj < 0. Inserting this into formula (1.1) of Ch. 5, one can write the exponential factor of that formula in the form

-^^^^\V\~'~''

r^Ra{w,t

+ ir])df,{t)+iCy

(2.3)

Consequently, by (2.2) and (1.2) of Ch. 5 +CXD

/

Ra{w,t-{-irj)dfi{t) -oo

=0

(2.4)

124

CHAPTER 6

uniformly in respect to it; G K, whatever be the compact K C G~. Thus, we proved the representation (1.1), where the passage is uniform by w inside G~. Conversely, if (1.1) is true, then F(w) G Not{G~} by (2.3) and (2.4). 2.2. Proof of Theorem 1.3. Our assertion 2° is already proved (see the relation (3.12) in Ch. 2). For proving 1°, note that from formula (1.1) of Ch. 5 one can easily derive that for any w = v -\- iv ^ G~

W^-« log |F(^)| = W-'^ log I ^"^'^' ^"""^^^'

\v\ / ^ -

dix{t)

Boc{w,\hn\)

Denoting r»+co

H<-) = e i

,^_t)?+„2. '«-v + ivea-,

(2.6)

by (1.2) and (1.4) we find +00

lim

/ ^—^ i-oc 1

^

= 0.

Observe that for any x G (—00, +00) and y > 0

y

C{x,y)

~7

1

/

\2~^ 2 - 7 " ; 2 '

-00

^,
where C{x,y) > 0 depends on x and y. Hence, for fixed x and y

Further, set ^l,2(^.) = ^ ^ ~ 7 ^ % ^ > ''

/i,2 = -

w-u + ivGG-,

(2.8)

J —C

/

.

TT y _ ^

NO .

{x -uf

odu,

V <0

(2.9)

+y^

(where /ii,2(^) is assumed to satisfy (1.2) of Ch. 5). Then one can see that

h;

y+\v\

±m ff-^^ {x-tY d/xi,2(t) + {y-{-\v\Y'

UNIFORM APPROXIMATIONS

125

Now observe that for any i; < 0 y-}-\v\

C'{x,y) 1 ^ ^ . ^ . ^ 7 4.\2 . / — r r h 2 - TV72 ' - c x ) < ^ < + o o , TT {x-ty ^-{y-\-\v\y 1+t^ where C{x, y) > 0 depends on x and y. Therefore, by Lebesgue's theorem

for any fixed a; and y. But in view of (2.8) and (2.9)

y r+ ^

^u + iv) ^d^Z>|/l-/2|. {x — uY + 2/2

Together with (2.7) and (2.10), this leads to the conclusion that the quantity y_ / - + d^{t) ^ y r ^ TT J_^ (X - ty +2/2 TT J_^

c//ii(t) y P^^ _ {X - ty 4- 2/2 TT J . ^ (x- t ) 2 + 2 / '

is equal to zero. Hence ii{t) = const by arbitrariness of a: G (—00, +00) and 2/ > 0, and consequently the representation (1.1) of Ch. 5 can be written in the form (1.3). 2.3. For proving Theorem 1.2 we shall use Lemma 2.2. Let a G (—1, +00)^ let N > 1 be a natural number and let

where fi{t) is a continuous, nondecreasing function on [—N,N]. Then there exists a triangular matrix ofpairwise different complex numbers {w^i^^{^, N)}^^^ CG- (A: = 1,2,...) such that: 1°. For any k >1 Im w^,'\fi,N)=rjk{fi,N)

( / = 1,2,... ,fc),

(2.11)

and

k(M,iV)|^+<^ = i ^ ^ i ^ V M . ^

27r

_^

(2.12)

2°. Uniformly inside G fo{w,N)

= lim B^iw,{wl''\fi,N)}t^). fc—>00

(2.13)

126

CHAPTER 6

Proof. For any fc > 1 we split [-N,N] tk^i = N in a, way to provide

as -N

< t[^^ < 4^^ < • • • < tf^ <

N

M(iS)-M4'^) = | V M '^

(/-l,2,...,fe).

-N

Also, we assume that

Vk ^Vkif^^N) „W^ wl^'{ti,N)

= -

•r(2 + a) ^ 27r

Ak) = t\''>+irik

VM

-jv

J

{k = l,2,...;l

=

l,2,...,k).

One can be convinced that in these assumptions 1° is true. Besides, by ( l - l ' ) ' (1.2) of Ch. 2 and (2.1)

logB„(t/;,{<'(M,iV)}ti) = 5 ] log 6„ ( « ; , < ) (;t*,iV)) 1=1 k

.j.{k)

^ _

,,(+{k)^

^^Ef-'':r'i2-E^(-.'!''+%). ' ti\i(„~tP) But, if K C G

is any compact, then by (2.2)

'Y^Roc{w,t^^'^ +ir]k)
as

k-> oo

1=1

uniformly in respect to it; G K. In addition, uniformly by it; G K

dn{t) Z=

l

[i(«;-4'=)y

1+a

t)]l+""

Hence (2.13) follows. 2.4- P r o o f of T h e o r e m 1.2. Assuming that a G (—1, +oo), set

/(.,iV).exp|-i:ii±^r

^^W

[i(w; - t)]i+" f '

UNIFORM APPROXIMATIONS

127

where fi{t) is a nondecreasing function satisfying the condition (1.2) of Ch. 5. Let {Kj}J° be a family of compacts exhausting G~, i.e. 00

K,CK,+i

(j = l,2,...)

and J J K . ^ G - .

Further, for any j > 1 choose Nj (AT^+i > Nj) enough great to provide \f{w)-fiw,Nj)\<^.,

weKj.

(2.15)

Consider the following sequence of functions, which are continuous and increasing on [-N,N] {N> 1):

Z^+i fj^^{t) = K

t if^{x)dx^-Yj;^^

/^ = 1 , 2 , . . . ,

where

r ii[N) (f^x) = < fj.{x) [ fi{-N)

a x>N if -N-^^<x
One can verify, that lim //«(t) = ii{t) for any t G [—N^N] and /c—>-oo

N

\^Ji.{t)\ < \ii{-N)\ +\Jii^i, -N

N

N

y ^.<\/ fi + i -N

-N

for any K, > 1. Therefore, by Kelly's theorem on the passage in the Stieltjes integral, lim r j-^t!M_^ = r , , '^^l^ , j_r^ [i{w - t)]i+" «^°o J-N [iiw - t)Y+"

. e G-.

(2.16) ^ '

One can be convinced that this passage is uniform by w inside G~. Now for any K>1 and j >1 set

[

-K J_ff. [l{w - i)]l+" J

Then, by (2.16) one can choose a sequence of natural numbers {KJ}J° to provide that for any j > 1 \f{w,Nj)-f.^{w,Nj)\<j,, weKj.

(2.17)

128

CHAPTER 6

By Lemma 2.2, for each j >1 there exists a triangular matrix of pairwise different numbers {wi (t^j^Nj)} (fc = 1, 2 , . . . ; Z = 1,2,... ,fc)satisfying (2.11) and (2.12) and such that (2.13) is uniform in respect to w inside G~. Consequently, one can extract a sequence of natural numbers {fej}i^ such that ,S^j)(

ife)

for any j > 1. Hence by (2.15) and (2.17) we come to the desired assertion.

CHAPTER 7 SUBHARMONIC FUNCTIONS WITH NONNEGATIVE HARMONIC MAJORANTS

1. MAIN RESULTS 1.1. It is well known, that Nevanlinna's factorization of the class N of meromorphic functions of bounded type in the unit disc [82] (see also [84], Ch. VII) leads to the following complete characterization of the growth of functions u{z) which are subharmonic in \z\ < 1 and have nonnegative harmonic majorants: P'ZTT

sup

/

P'2'K

u-^{re'^)dd ^limmi

tx+(re^^)d?? <-foo

(1.1)

(here and everywhere u'^ = max{n, 0}, u~ = u^ — u). In contrast with this, the problem of finding a complete characterization of the growth of functions which are of the same type in a half-plane was solved only partially, since the conditions l i m i n f i / w+(i?e^^)sin??d?? < + o o (1.2) R^-hoo i? Jo and

/

u-^{x)-

-\-x 2

< +00

(1.3)

J —(

arising from Nevanlinna factorization in the upper half-plane G'^ = {z : Im z > 0} [83] completely characterize the growth of such functions only in the particular case when they are subharmonic i n G + = {^ : l m z > 0 } (see also [10], Sec. 6.3-6.5, where it is assumed that u{z) can somehow be continuously extended to the real axis). 1.2. The main results of this chapter are the following three theorems particularly containing a complete characterization of the growth of functions subharmonic in G"^, having there nonnegative harmonic majorants. Besides, these theorems somehow improve Nevanlinna's uniqueness theorem ([84], Ch. Ill, Sec. 38) and also the Phragmen-Lindelof type theorem which follows from a result due to M.Heins and L.Ahlfors [1], by replacing the condition limsup z—>t, Im z > 0

u{z)<0,

—oo < t < + 0 0

(1.4)

130

CHAPTER 7

by a less restrictive one. Theorem 1.1. Let S{fl) be the class of functions u{z) ^ — oo subharmonic in G+ and satisfying (1.2) and nR

lim liminf /

u^{x + iy)Q.{x) dx <-^oo^

(1.5)

where Q.{x) is a continuous function such that f](x) > C{l-\-x'^)~^ (—oo < x < +CXD) for a constant (7 > 0. Then: 1^. S{Q) coincides with the set of functions representable in the form

{z == X + iy £ G~^); where u{() is a nonnegative Borel measure for which

h is a real number, and /i(t) is a function representable as the difference fi{t) = IJi+{t) — iJb-{t) of two nondecreasing functions such that n{t)dfi^{t)

and

< +00

/

-oo

-~—^ < +oo.

J — oo

2^. If the representation (1.6)-(1.8),

(1.8)

1 I ^

is valid, then fi±{t) can be deduced by

l^±{t) — lim / u^{x -\-iy)dx^

—oo < ^ < + o o .

(1.9)

With this choice of fi±{t) b

b

b

\ / / i - V ^ + + V^-' a

a

V[a,6]c(-oo,+oo),

(1.10)

a

and nb

pb

lim / u^(x-\-iy)g(x)dx

= / gix)dii±(x)

(l-H)

2^-^+0 J a Ja for any function g(x) continuous in [a, 6]. Besides, R

/

/•+00

u'^{x + iy)Q.{x) dx •= / -R

J-oo

Q.{x)dii^(x),

^hrnJ_M^+^.)^:^ = y _ ^ ,

(1-12)

(1.13)

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

131

and the relation +00

/

r+oo

u'^{x-\-iy)fl{x) -oo

dx = /

Q.{x) diJ.^{x)

is true, if n{x) = {!-{- \x\)-^ ( 1 < 7 < 2) or n{x) = (1 + \x\)-^ and h <0. 3^. If the representation (1.6)~(1.8) is valid, then h^ = -

(1-14)

J-oo

lim

4 /

( - K 7 < 1)

u^iRe'^)smMd,

(1.15)

for any 1} G (0, TT) /isin^ = lim sup R-^u{Re'^),

/i+siriT? = limsupi?-^^+(i?e^^),

i?^+oo

(1.16)

i?-)>+oo

and /or any 1? G (0, TT); except at most for a set of outer capacity zero, /i^sinT?-

lim R-^u'^iRe'^).

(1.17)

Remark 1.1. It is well known that the Nevanlinna class of functions (which are subharmonic in G+ and have there nonnegative harmonic majorants) coincides with the set of functions representable in the form (1.6)-(1.8) where ft{x) = (l + x^)~^, i.e. coincides with S{{1-{-x'^)~^). Thus, the pair of conditions (1.2) and (1.5) presents a complete characterization of the growth of such functions when Q>{x) = (1 + x'^)~^. Besides, if u{x) is subharmonic in G+, then it is obvious that the condition (1.5) with fl{x) = (1 + x^)~^ is equivalent to (1.3). The relations (1.5)-(1.7) are well known even in the most general case when fl{x) = (1 + x'^)~^ (these relations are true for h, h^ and /i~, since u{z) and u'^{z) have the same least nonnegative harmonic majorant). Remark 1.2. Using a result from [101] one can verify that the subset of functions of 5(1) {fl{x) = 1), for which /i < 0, coincides with the class of those functions subharmonic in G"^, for which +00

sup

y>o /

u'^{x + iy) dx < +00.

(1.18)

-00

T h e o r e m 1.2. Letu{z) he subharmonic in G^, and let there exists a sequence Rn t +00 such that for any n > 1 /»7r

I u^{Rne^'^)sm'dd'd Jo

pRn

<-\-OQ and

liminf / u'^{x-\-iy) dx < y^+^ J-Rr^

^-OQ.{1. 19)

132

CHAPTER 7

If for some Ro > 0 liminfi 4 / u(Re'^) sin M^ R^+oo \R Jo H-liminf /

u{x + iy)gRn^{x)dx>

=-oo,

(1.20)

where , ^

( 2-Ha;-2 - i?-2)

for

Ro < \x\ < R,

i?-2)

for

|a;| < Ro,

then u{z) = —oc. R e m a r k 1.3. Theorem 1.2 is an improvement of Nevanlinna's uniqueness theorem, since NevanUnna's conditions (1.4) and Hminfi^-^M(i?)--oo

{M{R) = sup

R^+oo

u{Re'^))

0
are more restrictive than the conditions (1.19)-(1.20). T h e o r e m 1.3. Let u{z) be subharmonic in G^, and let rR pR hminf / u-^ (x-^ iy) dx = 0 -R y-^+o J_j^

(1.21)

for any R> 0. Then: r.

hm R-^M{R)=

hm

R-^+oo

R-^[M{R)]-^

R-^+oo

= -

hm

^ [ u+{Re'^)smMi}

n R-^-hoo R JQ

Besides, if a = sup {Im z)~^u{z),

= pe[0,-hoo].

(1.22)

then

Im z>0

a+ =

sup (Im z)-^ix+(z) = /3.

(1.23)

Im z>0

2\ IfP = a-^< -foo, then (1.15), (1.16) and (1.17) hold for /i+ = ^ - a+. R e m a r k 1.4. The above theorem is an improvement of the Phragmen-Lindelof type theorem which follows from a result of M.Heins and L.Ahlfors [1] since their theorem implies the same assertions under the condition (1.4) which is more restrictive than (1.20).

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

133

1.3. The techniques used to prove Theorem 1.1 can be appHed to prove also a theorem on some weighted H^ classes in G+. Consider the class H'P{Q.{X) dx) (0 < p < +oo) of functions f{z) holomorphic in G'^, for which liminf4 / and

|/(i?e^^)|^sin^di?<+oo

(1.24)

nR

lim liminf /

\f{x-\-iy)\Pn{x)dx<-\-oo.

(1.25)

By Theorem 1.1, i7^((l + x'^)~^dx) coincides with the set of functions f{z) holomorphic in G"^, for which \f{z)\P has a harmonic majorant in G+, i.e. it coincides with the conformal mapping of Hardy's H^ class for the disc. Besides, it is obvious that, by Remark 1.2 H^{dx) coincides with the class H^ introduced by Hille and Tamarkin by means of the condition (1.18), where u{z) = \f{z)\P. The following theorem relates to a more general case. T h e o r e m 1.4. Let 0 C{1 -^ x'^)'^ for a constant C > 0 and such that for any R> 0, rt{x) is uniformly bounded almost everywhere in {—R,R). Then: 1^. The class H^{^{x)dx) coincides with the subset of those functions of the conformal mapping of Hardy's class, for which f{x) G Lp{fl{x)dx). 2°. The condition (1-24) can be replaced by liminf 4 / il-> + 00 i t

without changing H^[Vt{x)dx). lim

log" \f{Re'^)\smddd

= {)

(1.26)

JQ

Besides, if f{z) G H^{Q.{x)dx),

~ I

\f{Re'^)\PsmMi}

then

= 0.

(1.27)

i?-^ + 00 R JQ

3^. If Q{x) satisfies the additional condition

I

^+^ log+ n{x) -dx < + 0 0 , 1+^2

(1.28)

then HP{^{x) dx) = [VL{z)]-^^^H^{dx), where

n(.)^exp|-y_^ t - . 1 + tV^^h ^^^^-

(1.29)

134

CHAPTER 7

Particularly HP{{1 + \x\)-^dx)

= {z + iy^PH^idx),

- o o < 7 < 2.

(1.30)

2. REPRESENTATION IN THE SEMI-DISC The main tool used in this chapter is a theorem on necessary and sufficient growth conditions under which a function subharmonic in a semi-disc larger than G^ = {z : Im z > 0, \z\ < R} has a nonnegative harmonic majorant in G^. Before stating our theorem we consider the function T-»

,

.

\

TV/a'

Rp + z-ip^ Rp- z + ip

^piC^z)

Rp-z-ipJ

I

where 0 p, |(^| < R} onto the unit disc and ijjp{z^ z) = 0. Therefore gR^p{(^z) = — log |cjp(C, z)| (z, ( G G'^ ) is the Green's function of G J . If u{z) is a function subharmonic in a semi-disc G^* (i?* > R), then by Riesz' theorem in G^

[

u{Re'^)ipRAd,z)dd+

[ ' u{t-\-ip)xljRAt,z)dt,

(2.1)

//3

where i^(^) is a nonnegative Borel measure in G j * , finite in any domain D compactly contained in G^*, and (fR^p, i/^R^p are the expressions for the Poisson kernel of G^^^, written on the arc {( = Re^'^ : (3 < d < -K — f3} {(3 = di.vcsin{p/R) = -K12 — a) and on the interval {C^ = t -\- ip \ —Rp < t < Rp}. Using the well known formula

^^p(C,^)

dC ^p(C,^)'

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

135

where dn is the differentiation along the inner normal, one can calculate Tv/a^

-KIOL '^

{R, + Re'^ - ipyl-

- {R, - Re'^ + ipyl'^

i ^ ± ± - ^

^ f Rp -{- z-\-ip X J (i?p + Re'^ - ipyl'^ - {Rp - Re'^ + ip^^' Rp — I — ip

-K/a

-1

(2.2)

(2.2') 0 (/? < ?? < TT - /3) and il)R,p{t, z)>Q {-Rp Rp). The below lemmas relate to the behavior of these functions when p —>• +0. Lemma 2.1. If R> 0 and z G C^ are fixed numbers and p>0 small, then Ci<

(sixi^'^'^'f}^ 7r-2/3

/3
(2.3)

where Ci,2 > 0 are constants depending only on z and R. Besides, in (0,7r)

uniformly

/o

^

Vfl,p(i?,^)
is sufficiently

2Im^

E(i?2-|^|2)sini?

; n n ^ ^ . „ ( ^ , . ) = -;r\Rei^-zme-^^-z\^

,

,„ ^,

^ ^^^^^'^^

,„

^'''^

Proof. First we shall prove the inequalities (2.3) for iS close to the ends of [/?,TT — /?]. Next we shall prove the relation (2.4). Then, estimating (pR{'d,z), we shall extend (2.3) to all -j? € [/3, TT —/3]. So, we start from the obvious relation .. fRp + z-ip\ ^ 2Ry /^ '7r\ lim arg -~ — = arctan —x —pr = r? G 0, — ,

y = lm z,

using which one can verify that for sufficiently small p > 0 n/a

^R^Re'^m a

f^^ + ^'^P

Rp - z + ip

6i?2

f5R\^

136

CHAPTER 7

where A = sinry, if 0 < r/ < 7r/4, and A = sin(7r/2 — 77), if 7r/4 < 77 < 7r/2. To estimate the denominator in (2.2), first observe that

\R, + Re'^ - ipl''^'' + \R, - Re'^ + i/^r/"

Rp-\z-ip Rp- z + ip bR

<

-K/OC

3

-'TJ

for sufficiently small p > 0. Next, observe that Rp^z-ip \Rp + Re'^ - %p\''l'' - \Rp - R^^ + i p r / ^ Rp- z-\-%p > I i?p + i? cos (5r/<^ - (^47? sin ^ )

-KIOL

Z'—")

> i?^

for sufficiently small p, J > 0 (J > /3) and for any ?? € [/3, J]. Now, using (2.5) and the last two inequalities (where the quantities estimated are even functions of ?? — 7r/2), we obtain that for sufficiently small p, 5 > 0 (J > /3) and for any ^ e [/3,J]U[7r-(5,7r-/3] ax < \Rl - (i?e*^ - ipf\'-^/"ipR,p{^,z)

< a2,

where ai,2 > 0 are constants depending only on z and R. Since \Rl - {Re'^ - ip)^\ = 2i?2 sin [(i? - p)/2] sin [(TT - /3 - 0)/2], we conclude that the estimates (2.3) are true for sufficiently small p, 6 > 0 {6 > (3) and for any i? G [/3, J] U [TT - (J, TT - /?]. Now observe that (2.4) can be easily derived for any i? G (0, TT). To prove that this relation is true uniformly in [(5,7r — 6] for any S G (0,7r/2), it is enough to observe that the limits of both the numerator and denominator of (fR^p{'d^z) are uniformly separated from zero. Finally, it follows from (2.4) that for any i9 e [S^TT — S] {0 < S < 7r/2) we have 0 < al < (^R{d,z) < a^ < +oo, where the constant aj depends only on z, R and 5 J and the constant a2 depends only on z and R. Using the uniformity of (2.4) and the estimates (2.3) (which were proved for i? G [/3, J] U [TT — (5, TT — ^]), we conclude that these estimates hold for any 7? G [/?, TT — /9]. Lemma 2.2. If R > 0 and z G G^ are fixed numbers and p > 0 is sufficiently small, then Cl < {Rl - i')i-"/"Vii,p(i, z) < C | ,

-Rp
Rp,

(2.6)

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . . where C^ 2 > 0 ^^^ constants depending only on z and R. Besides, in {-R,R)

Imz f 1

lim i;R^p{t,z)

137 uniformly

K'

Z\^

|i?2_t2|2

= - ^ \t-zm^-tz\^

=^fi(<.^)-

(2.7)

P r o o f is similar to that of the previous lemma. T h e o r e m 2 . 1 . Let u{z) ^ —00 be a function subharmonic in the semi-disc G^* {0 < R* < +00). Then u{z) has a nonnegative harmonic majorant in G\ {0 < R < R*) if and only if the following conditions are satisfied: f

i/+(i?e^^)sin^d^<+oo,

(2.8)

liminf / \ R I - x^y/'^-^u-^(u^iy)dx

< +00,

(2.9)

where Ry = ^/R^ — y'^ and a — arccos (y/i?). / / it is so, then

"''''lit

log

+ ^R(R^

z-CR'^-Cz z-CR^-Cz u{R e'"^) sin ^

- 1^12

d^

Jo i?2

\R'^ - tz\

id/i(t),

z^x + iyeC^,

(2.10)

where i^(C) is a nonnegative Borel measure in G j , , such that Jj

^{R-\<:\)Im(:du{0<+oo,

(2.11)

and ii{t) is a function representable as the difference ii{t) = fJ^-\-{t) — fJ'-{t) of two nondecreasing functions such that / {R^ - t^)dii±{t) < + o o . J-R

(2.12)

The functions /x± {t) can be chosen to be those deduced from rt

li±it) = lim / u {x-\-iy)d'.X,

-R
(2.13)

138

CHAPTER 7

Under this choice b

b

b

\/M = V ^ + + V ^ - ' a

and

a

V[a,6]c(-i?,i?),

(2.14)

a

r

lim / u^{x-\-iy)g{x)dx = / g(x)dfj.±{x) (2.15) y-^-^O J a Ja for any function g{x) continuous in [a, 6]. Besides, for any RQ {0 < RQ < R) [

\u{Roe''^)\smi}d^<-\-oo.

(2.16)

Proof. Choose 0 < yo < R such that u{iyo) 7^—00, then take z = iyo and suppose p > 0 in (2.1) is small enough to ensure the validity of the estimates (2.3) and (2.6). Using these estimates we obtain

/X+

9RAC^^)di^{0 + C^I^

+ Ci* /

u-{Re'^)[sm'^^^^y^

dd

{Rl - t^Y^''-^u-{t + ip)dt

J-Rp

pRp

+ C2* / {Rl - t^Y^'^-^u^it + ip)dt. (2.17) J-Rp Further, assuming that A^ i 0 is any sequence such that the last integral is uniformly bounded for p = Am ( ^ > 1)? we conclude that the whole right-hand side of (2.17) is uniformly bounded for such p. Thus sup / / 9R,Xm{C^Wo)du{0 < +00, m>iJJG+^^

sup r~^^ m>l Jp^ sup /

u-{Re'^)

f s i n ' ' ^ ^ ~ f " ^ ^ V "^ d^ < +00, \ TT - 2fJm J

{Rl^ - ^2)^/^-^-i|x^(t + iXm)dt < +00,

(2.18)

where am = a,vccos {Xm/R), Pm = 7r/2 — am and Rx^ = ^/R^ — A^. It is not difficult to verify that the first of these relations is equivalent to (2.11)

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

139

while the second, together with (2.8), gives (2.16) for RQ = R. Now let Sn iO {0 < 6n < R) be any sequence. Then, by (2.18), the nondecreasing functions

Jo are uniformly bounded in any segment [—{R — Sn),{R — Sn)] {n > 1), if m > N{n) > 1. Consequently, by Kelly's theorem, there exists a subsequence {Am } £ {Am} such that the relations (2.13) are true when t G [—{R — 6n), {R — Sn)] and y = Am i 0. Therefore, the relations (2.13) are true for any t e {—R,R), if y takes values from the diagonal sequence {Pn} - {A^"^}. At the same time, Kelly's theorem on passage to a limit leads to relations (2.15) for y = Pn I 0. This implies the validity of the equality (2.14). Consider the nondecreasing functions Ai±)(t) = fiRl Jo

- a;2)-/"-id^(±)(x),

- i ? , „
(2.19)

and put AW(t)=Ai±)(i?pJ

{R,^
and

Ai±)(i) = Al±)(-i?,J

{-R
Then (2.18) shows that these functions are uniformly bounded in [—R,R]. Kence, by Kelly's theorem, there exists a subsequence of {pn} (which for convenience we again denote {pn}) such that ^^rt\t)

-^ A±(t)

{-R
as

n ^ oo,

where A± {t) are some nondecreasing and bounded functions. It is clear that A±(t) =

{R^ - x'^)dp±{x),

-R
(2.20)

Jo and so the relations (2.12) are true. On the other hand, Kelly's other theorem (on the passage to the limit in Stiltyes integrals) implies the relations (2.15) in any [a, 6] C [—R^R] for the measures dA^r^\t). Therefore, b

b

lim \ / A 1 ^ ' = V A ± , a

a

and, since the function i/jR{t, z)/{R^ ~'t^) is continuous in [—i?, i?],

(2.21)

140

CHAPTER 7

Let

C (/-';.r/i-.-'<*'">=C ^^'^"'<"+«?'• f^-^^) Then i^R,pAt,z) JR-

i5<|t|
•>pR{t,z)

(i?2^ - i 2 ) - / a „ - l

+

dAi±)(t)

i?2_i2

(R-S)

+

(R-S)

{m

-i2)V".-l

i?2_i2

dAi±)(i) = j ; + j ; '

(2.24)

for any 6 {0 < 6 < R). Assuming that 5 > 0 is arbitrary, choose 5 > 0 such that ' -(R-S)

R

V +VP^i'=^< 8C2*' where the constant Q is that of (2.6). Then, by (2.21) -{R-5)

V +V|A.<J^. -R

R-Sj

if n > 1 is sufficiently large. Therefore, J^ < s/2 by (2.6). On the other hand, it follows from (2.7) that the integrand of J^ tends to zero uniformly in [-{R - 6), {R - S)] as n -> oo. Therefore, by (2.21), J^ < e/2 for sufficiently large n > 1. Consequently, the relations (2.19)-(2.24) give lim /

u^{t + ipn)ijR,pAt^z)dt=

/

^R{t,z)dii±{t).

(2.25)

It follows from (2.3) that 0 < (pR,p{'d, z) < C2 sini? for sufficiently small p > 0. Hence, using the relation (2.16) (still proved only for RQ = R), we obtain lim /

u^{Re'^)^Rp{d,z)d^ ?1!B(JJ'-

W')'" 7o

\Re''

«"('''">™'' zmRe-i^-z

jdi}.

(2.26)

For the next passage, observe that the function ^{z)=gR{(,z) — gR,p{C^z) is harmonic in the closure of G'^ whatever be C ^ G^ . Besides, $(i?e*^) = 0 (/? < 79 < TT - /?) and $(t + ip) '= 9R{C, t + ip)>0 {-R^ < t < Rp). Therefore,

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

141

^{z) > 0 in G+^^, and gniCz) > gR,p{C,z) for any z, C ^ G^^^. It is easy to prove that the condition (2.11) is sufficient for the convergence of the integral

//^gniC,z)du{C) ^ ff J JG+

iog\'~^ll~^j\du{C).

J JG+

\Z-CR^

-CZ\

Using this, one can prove that

The relations (2.25)-(2.27) permit to obtain the representation (2.10)-(2.12) by letting p = pn iO in (2.1). On the other hand, the representation (2.10)-(2.12) is necessary and sufficient for the existence of a nonnegative harmonic majorant of u{z) in G^, since using a conformal mapping this representation is derived from the similar representation of subharmonic functions which are of the same type in the unit disc. Thus, u{z) has a nonnegative harmonic majorant in G^, and, as its majorant is the same in any half-disc G J {0 < Ro < R), the relation (2.16) is true for any Ro {0 < Ro < R). Finally, the relations (2.13) and (2.15) without the indices ± can be easily verified using directly the representation (2.10). But also u'^{z) is a subharmonic function having the same nonnegative harmonic majorant. This proves the relations (2.13) and (2.15) with the index +, hence follows their validity with the index —. 3. PROOFS OF THEOREMS 1.1 - 1.4 Proof of Theorem 1.1. Let u{z) ^ —oo be a function of 5(f^), and let i?^ t oo be a sequence on which the lower limit in (1.2) is attained. Then the hypotheses of Theorem 2.1 are true for any R = Rk, and hence the representation (2.10) is valid for any R = R^. From (2.10) we subtract the same representation written for a smaller half-disc G^ {Q < Ro < R)^ put z = iy, divide the obtained equality by 2y and let y -> +0. This gives the following Carleman type formula:

+ (if - i^) / I **'> - i : [ "'^°'"'^'"'''"'- *=*•" The right-hand side of this formula remains bounded from above as R = Rk -^ -hoo. This follows from (1.2), (2.16) and from the relation +00

/

pR

ft{x)dfi-^{x) =

lim liminf /

u'^(x-i-iy) ft(x) dx <-\-oo

(3.2)

142

CHAPTER 7

which is a consequence of (1.5) and (2.15). Hence, also the left-hand side of (3.1) is bounded. Using this we arrive at (1.7). The proof of convergence of the second integral of (1.8) as well as the proof of the representation (1.6) are similar to the proofs of the corresponding assertions of Nevanlinna's theorem [83] (see also [10], Sec. 6.3 - 6.5). Now let u{z) be a function representable in the form (1.6)-(1.8). Then, obviously

Hence (1.2) follows. Further, it is easy to show that for any R> 0 there exists a constant C > 0 such that

y f"" ^n(x)dx ^^

C <5-, t)2 + 2/2 - 1 + ^2 '

0<2/
-oo
Hence, using Lebesgue's theorem on dominated convergence we arrive at relation (1.12) for U{z). This implies (1.5), and so, u{z) G S{Ct). Consequently, the measures fi±{t) in (1.6) can be recovered by the relations (1.9), and (1.10), (1.11) follow from (2.14), (2.15) of Theorem 2.1. Further, R

/»+oo

/

u~^{x-{-iy)Q{x) dx < / Q{x)dfi^{x) -R J-00 since (1.2) is true for U{z). This inequality and (3.2) imply (1.12). On the other hand, one can prove that 1 dx C If -^"^ (a;-i)2+2/2 TTT"? (1 + 1^1)7 -< 7(1 + |«|)T"

""^^
for any Q < y < M < +00 and —1 < 7 < 2, where the constant C > 0 depends only on M and 7. Therefore, if f](t) = (1 + \t\)~^ (1 < 7 < 2) or 0(t) = (1 + \t\)-^ ( - K 7 < 1) and /i < 0, then by Lebesgue's theorem + CX)

/

/• + 0 0

u^{x-\-iy)Q.{x) -00

dx < I

Q{x)d/j.-^{x).

J—00

The converse inequality for the lower limit is true by (3.2). Hence (1.14) holds. To prove (1.13), observe that the function / / .G+

log

c z-C

du{C) - u{z)

is from 5((1 + \x\)~'^). Therefore, the relation (1.14) with n{x) = {1-\- \x\y is true for this function. Using (1.14) we come to (1.13).

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

143

Remark 3.1. By a similar argument, one can prove that the set S{ft 1,0.2) of functions which are subharmonic in G"^ and satisfy the conditions (1.2), (1.5) with f2 = f^i and also the condition lim liminf /

u (x -{- iy)ft2M

dx < +00

(where 01,2(2^) are continuous and such that fli^2 > ^1,2(1 4- x^)~-^ for some ^1,2 > 0) coincides with the set of functions representable in the form (1.6)(1.7), where ^±{t) are such that +00

/•+00

/

fli{x)diij^{x) < +00 and / ft2{x)dfi-{x) < +00. J—00 Besides, by-00the results of [103] the subclass of 5(1,1) (fii = fi2 = 1) for which h = 0 and / /

Im C duiC) < +00

coincides with the set of those functions u{z) subharmonic in G"^, for which ^+00 i-CXD

/

\u{x + iy)\dx < +00. -00

Proof of Theorem 1.2. Assuming u{z) ^ — cx), observe that the conditions of Theorem 2.1 are satisfied for any R = Rn- Therefore, they are satisfied for any i? > 0, and (2.15), (2.10)-(2.12) and (3.1) are true for any i?, Ro {0
/

u{Roe'^)smi}d^<^

^o Jo

[

u{Re'^)sinM^

^ Jo

+

lim /

gR^R^{x)u{x-\-iy)dx,

where the left-hand side integral is absolutely convergent in accordance with (2.16). So, the hypothesis u{z) ^ —00 contradicts to the condition (1.20). Proof of Theorem 1.3. 1^. Let lim sup i?-iM(i?) = +00 as i? -> +00. In this case, for any i^ > 0 we can find z = x -\- iy ^ G+ such that u{z) > K\z\ > Ky. Hence a = a'^ = +00. If we have also liminf i?~-^M(i?) < +CXD as R -^ -hoo, then by Theorem 1.1, u{z) is representable in the form (1.6)-(1.8), where /i+(t) = 0. Thus, u{z) satisfies the condition (1.4), and, by the result of M.Heins and L.Ahlfors, there exists lim. R~^M{R) a,s R -^ +00. This is a contradiction. Therefore, if lim sup i?~^M(i?) = +00 as i? -> +00, then there exists lim R~^M{R) = +00 as i? —)> +00, and a + == /3 = +00. In our case lim

^ /

R^-^00 R JQ

u-^(Re'^)sinM'i}

= +oo

144

CHAPTER 7

since otherwise u{z) would be representable in the form (1.6) - (1.8), where /x_i_(t) = 0, and obviously a < h < +oo which is a contradiction. Now consider the case when \iui sup R~^M{R) < +oo as i? -> H-cx). In this case u{z) is representable in the form (1.6)-(1.8) with /i+(t) = 0, besides, (1.4) is true and there exists lim R~^M{R) =(3 = a-^ as i? -^ +oo, according to the result of M.Heins and L.Ahlfors. Besides, from the representation (1.6)-(1.8) immediately follows that a < h. On the other hand, a > \ixnsupy~^u{iy) a.s y ^ +oo by the first of relations (1.16). Thus, a"*" = /i"^ = /3 and, additionally, (1.15) is true. 2^. U (3 = a^ < -foo, then u(z) is representable in the form (1.6)-(1.8). For such functions the relations (1.15)-(1.17) with h = fS = a'^ hold. Proof of Theorem 1.4. P . Let f{z) G HP{n{x)dx). Then \f{z)\P has a harmonic majorant in G"^ by Theorem 1.1. Thus, f{z) belongs to the conformal mapping of Hardy's class, and it has nontangential boundary values f{x) almost everywhere on (—OO,+CXD). Hence R

pR

X < +00, i? > 0,

/

\f{x)\Pn{x)dx

< liminf /

-R

2/-^+0

(3.3)

\f{x + iy)\Pn{x)d:

J-R

by Fatou's lemma, and f{x) € Lp{Q,{x)dx). Now let f{z) be from the conformal mapping of Hardy's class H^ {\z\ < 1), and let f{x) G Lp{fl{x)dx). Then, using the factorization of f{z) one can obtain

On the other hand, it can be easily verified that for almost all t G (—oo, +00) Q,{x)dx

-1

< C*nit),

0
R>0,

for a constant C* > 0 depending only on R and Q,{x). Consequently, R

/

pR

\f{x + iy)\Pn{x)dx< / \f{t)\Pn{t)dt (3.5) -R J-R by Lebesgue's theorem. The relation (1.27) follows from (3.4). 2^. If / ( z ) G HP{Q{x)dx), then f(z) is from the conformal mapping of Hardy's class. Using the factorization of this function we obtain (1.26). Now let (1.24) be replaced by (1.26). Then evidently log"^ \f{z)\ has a harmonic

SUBHARMONIC FUNCTIONS WITH NONNEGATIVE . . .

145

majorant in G+, i.e. f{z) is of bounded type in G^. Therefore, |/(i?e*'^)|^ is continuous in 0 < i? < TT for almost all i? > 0. Besides, liminf / ?/->+0 J-R

dx \f{x + iy)\P- —^ 1

<+oo

for any R> 0. Therefore, by Theorem 2.1, \f{z)\^ has a harmonic majorant in any C^ (JR > 0), i.e. in any G^ the function f{z) is from the conformal mapping of Hardy's class. The transformation of the corresponding factorization by means of the conformal mapping of |z| < 1 onto G^ gives

log|/WI= E l°g z,eG+

z — Zk R — zzk z - ZkB? - zzk I d^

Here ZA; are the zeros of f{z) and dii{t) = \og\f{t)\dt — duj{t), where uj{t) is a nondecreasing function such that (jo'{t) = 0 almost everywhere in {—R^R). But we have already proved that log"^ \fi^)\ has a harmonic majorant in G^. Therefore it is obvious that the passage R -^ +oo in (3.6) leads to the factorization

(2: e G+), where £i/i(i) = log |/(t)|di - duj{t), Im C = 0 and /i=-

lim

4 /

log|/(i?e*'')|sin^di?<0

according to (1.15) and (1.26). Consequently, f{z) is from the conformal mapping of Hardy's class, and (3.4) is true. Hence (1.27) follows. 3^. Let f{z) e HP{Q{x)dx), where fl{x) satisfies the additional condition (1.28). Then it is clear that Q{z) and

both are non-vanishing holomorphic functions in G'^. Using (3.7) one can easily obtain

146

CHAPTER 7

where |/(i)|Pfi(t) € Li{dt). Hence f{z)[n{z)]^/P S HP{dx). Thus

On the other hand, the assumption f{z) Lp{Q,{x)dx). Therefore, fiz)[n{z)]-'^/P

G HP{dx) gives f{x)['[l{x)]~^/P £

G HP{n{x)dx)

by assertion 1°, and HP{dx)[n{z)]-^^P C fl-J'(J^(a;)da;). The equahty (1.30) is for the particular case of (1.29), when il(x) = (1 +

CHAPTER 8 WEIGHTED CLASSES OF SUBHARMONIC FUNCTIONS

1. GREEN TYPE POTENTIALS 1.1. We start by the following auxiliary Lemma 1.1. Let U{w) he subharmonic in G~, and let U{w) G Mp {= Mp{G~}) for a natural p. Then for any a G (0,p]; 1°. W~^ u{z) is continuous and subharmonic in G~, 2°. —-W-^P-'^^W-'^u(z)

= U(w), w =

u-\-iveG-.

Proof. 1°. Since Uiw) G Mp, the integral W~^ u{z) is absolutely convergent for any w G G~. By the same reason, for any fixed t/;o = ixo + ifo ^ G~ the integral />+oo

/

I

I

a^-^ \U{w - iG)\dG,

\w - w^\ < - 4 ^ , JM 2 can be made arbitrary small by taking M > 0 large enough. On the other hand, one can verify that for any finite M > 0 the integral rM

a'^-'^\U{w-ia)\da /o is a function continuous at w = WQ. TO this end, one can use, for example, the Riesz representation of U{w) in any disc in G~, which contains the rectangle {w -ia : \w - wo\ < bo|/2}, 0 < a < M. Hence it follows that W'^^uiz) is continuous in G~. The inequality between W~^u{z) and its mean value is easily obtained by changing the order of integration. 2° follows from Lemma 1.2of Ch. 1. 1.2. The Green type potentials considered below are constructed by means of the Blaschke type products considered in Ch. 2. Lemma 1.2. Let u{() be a nonnegative Borel measure in G~, such that

IL

\il\'+"duiO<+oo

iC^^

IG-

for some a > 0. Then the Green type potential

+ iv)

(1.1)

148

CHAPTER 8

lo.{w) = -jj

log|6„KC)MKC)

(1.2)

is a superharmonic function in G . Proof. Let po < 0 be a fixed number, and let w G G~^ = {w : Im w < po}- Using the representation (2.2) of Ch. 2, one can verify that for any C e Q- \ G^^/2 w^ ^^^^ \^og\ba{w,C)\\ < Ca,po \v\^'^'^, where Ca,po > 0 is a constant depending only on a and po- Hence

li

^c.,poM

\G PO/2

logK{w,0\dHO

is a harmonic function in G^^. For consideration of the remaining part of Iot{'^)', recall from Ch. 2 that the function Uc,{w,0 =

\og\ha{w,0\-\og\ho{w,C)\

is harmonic in G~ and representable in the form + CXD r" C/„(w,C)=C„+Re / \_{iw — iC, + TY^^

Re

(1.3)

1 dr iw — iC-\-T

T^dr ^ /"l^' dr + Re iw — i( — r)^+^ iw — i( — r / Jo + 00 "^ da >0

/

where

(1.4)

rH^)\

Co.

l+ay J 1+a

is a constant while the other terms are functions harmonic in G~. One can see that for any w G G^Q and ( G G~ ,2

G. - r

"^^"

+^'^t1 iw —driQ —

< c'

|r/r+^

T\ Jo {iw-iC-ry-^^ Jo where C'^p^ > 0 is a constant depending only on a and po- To estimate the remaining term of (1.4), denote

J

{iw — iC,-\-TY'^^ + 00

1 dr iw — i(-{-T

^+l?l \v\

da

+ 1 + 0-

w = u-\-iv,

(1.6)

i ^ + i+a

and consider separately the cases \T]\ > 2\v\ and \r]\ < 2\v\ {( = ^ + ir] £ G - j , w = u + iv £ C^o). If \ri\ > 2\v\, then J =

Jo '^M-2) =

Jl+J2,

da \v\

+ 1 + 0(1.7)

149

W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S

where obviously ^"^"^

^^

.1

\'^\

^ riff

I

|i+«

(1.8)

On the other hand, putting

^ + l?l |t,| + l + a

where

1

A

^1^ +1 i ^ + l +a

observe that | A| < 1 if and only if cr > \r]/v\ — 2. When |A| < 1 we have |(1 - A ) " - 1|/|A| < M^ < +oo. Therefore J2 < 2Ma

u-C

+

1 V

1+a

+ 00

0/

U-^

da

+ l + (7

^1-2

(1.9)

< 2M« < C7-,„ |7?r

If |po|/2 < |7?| < 2|w|, then |A| < 1 for any a > 0. Hence J ^ <2M, ^-Lyj-a

4-CO

U-^

+

1

u-^

-2

+ 1 + 0-1

da

il+a

< 2M„ < c:,j7?r

Using formulas (1.4)-(1.9) and the last estimate we obtain that \Ua{zC)\^ ^a,po l^l^"^*^ for '^ ^ G^po5 C ^ ^ W 2 ' where the constant C* ^^ > 0 depends only on a and po- Therefore, by (1.3) and (1.1)

Po/2

= -y"y_ PO/2

\og\bo{w,i:)\duiO-JI_

Ua{w,Odu{0, Po/2

where the last integral converges in G~^ absolutely and uniformly, while the whole Ia,po ('^) ^^ superharmonic in G~^, as a sum of an ordinary Green potential and a function harmonic in G~^. The desired assertion follows, since /«('«;) = la^po {w) + 1*^^^ (w) for any po < 0. L e m m a 1.3. Let u{() be a nonnegative Borel measure in G~, satisfying (1.1) for some a > 0. Then the Green type potential (1.2) has the following properties: 1°. Ioc{w) G M/3 for any /? G (0,1 + a). 2°.

W~^Iot(w) is a continuous superharmonic function in G",

W-^Ia.{w) = - j j

W-"log|6.(^,C)IMC)>0,

weG-,

(1.10)

where the integral is uniformly convergent in any half-plane G~ (p < 0) and

150

CHAPTER 8 7,

+00

W-^Ia{-it) J < +00.

(1.11)

/ is representable as an ordinary Green potential:

3°. W~^Ia{w) W-''Ia{w)

=- J j

log\bo{wX)\diya{C).

weG-,

(1.12)

where I'aiC) ^^ ^ nonnegative Borel measure satisfying (1.1) for a = 0. Proof. 1°. Let po < 0 and w = u-{-iv e C^Q. Assuming a < ^ < 1 + o; set I - / Jo

a^-' da

I log \bc,iw - ia, QWdi^iO J JG-

^ \IL G-^IIG-I

(/^°°<^''"'|i°gl^"("'-^<^'Ollrf^) MO

= h+h

(1.13)

and estimate / i and I2 separately. By formula (1.1') and (1.2) of Chapter 2

-Jo

(1 + c.)^+« JJG-\G,

h=

tr^-^

Ivl'^'^duiO, J

where C{a,P,po)

' ^ ' ^ ^ ^ 7-1 {\v/r,\ -

(1.14)

JG-\GZ

> 0 is a constant depending only on a, f3, and po- Putting

du{C) / (T^-' \\og\ba{w-ia,C)\\ J JGJ\rj\ r r r\l\ + / / dv{Q / a^-^\\og\ba{w-ia,0\\da J JGy

da = Ji+J2

(1.15)

Jo

and again using formulas (1.1') and (1.2) of Ch.2, we obtain

Ji< / /

duio/

J JG-

(|7,i-\t\rdt /

J-\n\

,,/./^V»

J\v\ K\v\-t + (xY+'^

(1-16)

To estimate J2 we use the representation P-I

^

/

I I

\ ct-k

logMt«,C)|=X:-^f7-^) -^^

fl'nl / 7: Jo

-Re / ' " -

^ •PdT

r^ dr 7= Tj— ( a < p < l + a)

[IW - ZC — T)^^^

W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S

151

which is derived from the representation (1.1')-(1.2) of Ch.2 by integration by parts. Then we obtain P-^

JJG-

1

/

1^1

\«-^

JO L[^"-^Vl^l+^y ,=o- --• • -

+ Jo0 JM (|t;| +

|7?|-T + (7)'+"_

r\r}\

11^ -io '•-•

1^1+'^ + ^ 1 ' + " " ^

<7^-^da = Ki+K2

+ Kz.

(1.17)

For any 7 e (0, a]

Therefore

f:;^

a - A;

k=0

J JG-

On the other hand

K, < C"'{a,p,po) jj

H'-^'' dKO-

(1-19)

Now observe that where ^ ^

io

io | | i ; / ^ | - l + cT + T|^+"-^

and \\v/rj\ — 1 + cr| < 1. One can easily show that L{rj) < C^{a,P) < +00. Therefore, for K2 an estimate similar to (1.18)-(1.19) holds and by (1.17)

h
Thus, by (1.1) and (1.13)-(1.16) I
|r?|^+" (izy(C) < + 0 0 ,

where C*(a,/3, po) > 0 is a constant depending only on a, /3, and po- This obviously proves the inclusion / a ( ^ ) ^ Mp {a P2- The inequality in (1.10) is valid since the integral in the formula (2.2) of Ch. 2 is nonnegative. By Lemma 1.1, W~^Ia{w) is a continuous superharmonic function in G~ since Ia{w) G Mp {a < p < 1 -\- a). To prove representation (1.12), observe that by (1.10)

152

CHAPTER 8 + 00

IW-'^Iaiu + iv)] du /

-CXI

=n ff di^iC) [ ' J JGJ-\r,\

{\v\-\t\rsign{\v\-t)dt

< I T S //G-I i'l'" •*"«'•=+"= for any v <0. On the other hand, ip{v) -^ 0 as v -^ —0. Therefore, the results of [103] and Ch. 7 lead to (1.12), the latter implies (1.11). R e m a r k 1.1. Using the formula (2.2) of Ch. 2, one can verify that for any w,( e G~ and a > 0 T ^ - log \b^{w, 01 - - ^

/ ' r"-^ log \bo{w, C + iT)| dr.

r(a) Jo This formula permits to write explicitly the measure z^a(C) of representation (1.12). Namely, one can prove that Ua{E) = W-'^uiE) = —— / ^[^j Jo

a''-'^u{E - ia) da

for any Borel set E such that E C G~ (see [Q&\). 2. SUBHARMONIC FUNCTIONS WITH NONPOSITIVE LIOUVILLE PRIMITIVES Definition 2.1. By Sa {0 < a < +CXD) we denote the class of functions U{w) which are subharmonic in G~ and satisfy the following conditions: (i)

U{w) G Mp for the integer p E [a,l -\- a),

(ii)

W-'^uiz)

<0,weG-,

(iii) the associated measure of U{w) is supported in a neighborhood of the origin, (iv) if a = p>0

is an integer, then

I

+°°

'1

df

W-^Ui-it)

-t > -00.

2.1. Our aim is to find Riesz type representations for functions from SaL e m m a 2 . 1 . / / U{w) e Sa {c^>0), then its associated measure u{Q satisfies (1.1) and U{w) = l j

log\ba{w,0\du{0

+ U4w),

weG-,

(2.1)

W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S

153

where U^{w) is a harmonic function of SaProof. Let a > 0, and let J9 be a bounded domain such that D C G~. Then Uoiw) = U{w) - 11

logK{w,0\

dv{Q

is a function subharmonic in G~ and harmonic in D, and by Lemma 1.3 UD{W) e Mp. Therefore W-'^UDiw) = W-^'Uiw) - II

PF-^log|6a(^,C)l ^KC), ^ e G",

(2.2)

by Lemma 1.2, and consequently W^Uoiw)

<- j j

M/-"log|6a(u;,C)| dv{0 =

VD{W),

where VD{W) is a nonnegative superharmonic function, as proved in Lemma 1.3. The representation (1.12) yields sup / v
[W-''UD{u^iv)]'^

du < 27r / / |r/| dv^iQ < +oo. J Jo-

in the same way we obtain ^hm^ /

[W-'^UDiu + iv)y

du = 0.

Consequently, W~^UD{W) < 0 in G~ by Theorem 1.1 of Ch. 7 (see also Remark 1.2 after that theorem). By (2.2) VD{W) < -VF-"C/(^) {w G G"). But the measure z/(C) vanishes when \(\ is large enough, for example when ICI > R^. Therefore the representation (2.2) of Ch. 2 gives

>-^Rjh^) IIy*° ""•(>• Exhausting G~ by means of finite domains D we arrive at (1.1). Further, defining U^{w) by (2.1) and using Lemma 1.3 we obtain U^{w) G 5^. This completes the proof since our assertions are obvious when a = 0. 2.2. We are ready to prove our main theorem on descriptive representations of classes Sa-

154

CHAPTER 8

Theorem 2.1. The class Sa {0 < a < +oo) coincides with the set of functions representable in the form

_ R e n i ± i a e - . f O « ) r ^ ^ ,

„,G-,

(2.3)

where u{Q is a nonnegative Borel measure in G~, supported in a neighborhood of the origin and such that

IL I

|Im Cl^^° du{0 < +00,

(2.4)

'G-

while fj.{t) is a nondecreasing function for which '•+°°

J

^

dii{t) , ,,^

< +00.

(2.5)

—<

Proof. Let u{z) G Sa (0 < a < +oc). By Lemma 2.1 the measure associated with U{w) satisfies (2.4) and U{w) is representable in the form (2.1), where C/*(i(;) G 5a is a harmonic function. Using Theorem 3.5 of Ch. 3, we conclude that U^{w) can be written in the form of the last integral of (2.3), where /x(t) satisfies the mentioned conditions. Conversely, let U{w) be representable in the form (2.3)-(2.5). Then using Theorem 3.5 of Ch. 3 we conclude that the last integral in (2.3) is a harmonic function of Sa- At the same time, by Lemma 1.3 also the first term on the right-hand side of (2.3) is a function of ««• Hence U{w) G SaRemark 2.1. Using Lemma 3.3 and formula (2.10) of Ch. 1, we find that the inclusion U{w) e Sa (0 < a < +oo) implies that iov w = u-hiv E G~

W--u{z) = f[

W--logK{w,C)\

duiO + l r V ^ j f e l '

(2-^)

J JG^ J-oo (^ - t) + ^ where u{() and fi{t) are the same as in representation (2.3)-(2.5). By (1.12) ioT w = u-\-iv £ G~ W-- u{z) ^ jj^

log |6o(«;, 01 du^iO + I f ^

(u-t)*\v^'

^^'^^

where i^aiC) is some nonnegative Borel measure in G~, satisfying (2.4) for a = 0. Remark 2.2. Consider the class Sa {0 < a < +oo) of those functions subharmonic in G~, which are representable as differences of two functions of s^Evidently Sa coincides with the set of those functions U{w) subharmonic in G~, whose Liouville a-primitives (in imaginary variable) are representable in the form (2.7), with measures Ua{C) from certain class, while fi{t) is the difference of any two nondecreasing functions satisfying (2.5). This representation describes a subclass of functions which are subharmonic and have nonnegative harmonic major ants in G~.

W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S

155

3. THE GROWTH OF FUNCTIONS FROM S^ 3.1. We aim at proving T h e o r e m 3.1. 1°. The class 5a (0 < a < +oo) coincides with the set of those functions u{z) G Mp {a < p < 1 -\- a) subharmonic in G~, which satisfy the conditions liminf^ /

\W-'^U(Re-'^)\sm^d^ \W-'^U(u + iv)\

= 0, T-nTT

< +00,

(3.1) (3.2)

and associated measures of which are supported in the neighborhood of the origin. 2°. / / U{w) e Sa {0
-^ /

+oo), then \W-'^U(Re-'^)\sin^d^

i?->+oo R JQ ^

The function iJL{i) in the representations (2.3)-(2.4) found from the relations fi±{t)=

lim /

\W-''U{u

= 0.

(3.3)

o,nd (2.6)-(2.7)

can be

'

+ iv)]'^ du,

-/i(t) = / i + ( t ) - / i _ ( t ) ,

(3.4)

where the functions fi±{t) are nondecreasing and satisfy (2.5). If lJ±{t) are recovered by (3.4), then for any [a, 6] C (—oo,+oo) and for any function g{x) continuous in [a, b] \mi I

[W-'^Uiu + iv)]^ g{x)dx=

f g{u)dfi±{u).

(3.5)

t/(2

•/ a.

Besides, lim P

[W--U{u-,ivt

.

, t .

-

r

T-%T?^-

(3.6)

R e m a r k 2.3. Evidently, U{w) G 5a (0 < a < +oo) implies U{w) G Sa if and only if one of relations (3.4)-(3.6) gives /J^-{t) = 0 (—oo < t < +oo). If both /j.±{t) = 0 then U{w) is precisely a Green type potential. 3.2. To prove Theorem 3.1, we need the following lemma. Lemima 3.1. Let a>0 and let U{w) G Mp {a
R^, then v{Q satisfies (2.4)-

156

CHAPTER 8

Proof. Let p < 0 be any number. Then the function Up{w) = U{w) - j j

\og\ho^{w,s)\ dv{s)

is harmonic in G~ and is of Mp by Lemma 1.3. By the same lemma, W-'^Upiw) = W-"" u{z) - 1 1

W-""log\bc,(w,s)\

dv{s).

is harmonic in G~. Writing the difference of two Poisson representations of : \w\ < RQ} and in G-{R) {R>Ro> i?*), W-'^Upiw), in G-{RQ) ^{weQwe arrive at the equality

r[

^-a^(^)^^P,i^o(C^)|^^|^ weG;{Ro), (3.7)

'^^ JdG-{Ro)

where

ipA^^s) = ^f

w--\ogK{C,s)\ ^^^'^^^-^Vci {R = Ro.R)

and QP^RQ, Qp^R are the Green functions of G~{Ro) and G~{R). We conclude that IP^R{W,S) — IP^RQ{W,S) > 0 when w G G~{RQ) and 5 G G~, since W~^\og\ha{C,^s)\ is subharmonic in G~ for any fixed s G G~. Now assume that i? and RQ {R> RQ > R^) are such that / \W-''U{Re-'^)\smddd<^oo Jo

{R =

RQ,R).

By Lemmas 2.1 and 2.2 of Ch. 7, the right-hand side of (3.7) remains bounded for a sequence p = pnt ^' Hence, by Fatou's lemma / / liminf [/p,ii(if;,5) J J^-^ p ^ - o

IP^RQ{W,S)]

dv{s) < M{w) < +oo,

where M{w) > 0 is a constant depending only on W^RQ and i?, while po < 0 has sufficiently small modulus. Letting here po -> —0, we obtain / /

liminf [/p,i?(if;,5) - Ip^R^{w,s)\ dv{s) < M{w) < +cx).

(3.8)

157

W E I G H T E D CLASSES O F SUBHARMONIC F U N C T I O N S

But for any R> 0 there exists

p->-0

C—w

^T^'^JdG-(R) fdG-{R)

C—w

W

W

dC

+

/"llm s\

-|

r ( l + a) J_\ims\ where (see (2.2) of Ch. 2) jR{t,w)

-f

27ri JdG-{R) C - (Re s + it) C ~w

C—w w

+

w

dC

Calculating this integral and returning to (3.8), we arrive at the estimate

+t

L(fi+*)' + (Re.)^ ML + t

dt > du{s) < M{-ivo)

\vo\

< 4-cx),

(S+*) +(^^«)'J where M{—ivo) > 0 depends only on VQ (—i?* < i;o < 0) and i?, i?o- Taking i? sufficiently large, we obtain that u{() satisfies (2.4). 3.3. P r o o f of T h e o r e m 3.1. Let U{w) G Mp be a function subharmonic in G~ and satisfying conditions (3.1), (3.2) for an a G (p — l,p], and let the associated measure u{Q of U{w) vanishes in a neighborhood of infinity. Then u{() satisfies (2.4) according to Lemma 3.1. Therefore the function U.{w) = U{w) - I I is harmonic in G'. W-''U.{w)

log Kiw, 01 di^iO

By Lemma 1.3, Ut{w) E Mp and = W-" u{z) - I I

W"' log \ho.(w, 01 dzy(0

is harmonic in G~. Besides, this function satisfies (3.1) and (3.2) since W~''lo.{w) = ' 1 1

W " " log K{w, 01 du{0

(3.9)

158

CHAPTER 8

satisfies (3.1) and (3.2). Moreover, the representation (1.12) implies that Km

^f

W-'^lJRe-'^)sm^di} = 0

(3.10)

and + 00

W-''Ia(u-\-iv)du = 0.

/

(3.11)

-oo

Hence, by Theorem 1.1 of Ch. 7 W--U4w) = - f ^ ,

'^^}^^ , ,

w = u^-iveG-,

(3.12)

where ii{t) is the measure determined from relations (3.4) and satisfying (3.5), (3.6). In addition, (3.3) is obviously true. Applying ^ V F - ^ ^ - " ) to both sides of (3.12) and using the results of Ch. 1, we obtain a representation of the form (2.3). Therefore, Theorem 1.1 leads to the conclusion that U{w) G SoL' Conversely, by Remark 2.1, for any U{w) G Sa the function W~^ u{z) is representable in the form (2.6). Now relations (3.3) and (3.6) follow from (3.10), (3.11) and Theorem 1.1 of Ch. 7. NOTES. The inversion z = w" ^ transforms our requirement on boundedness of the support of associated measures in classes Sa and Sa to the requirement that the supports of measures are disjoint from the origin. This is a natural requirement for the classical Blaschke product with factors of the form (1 -z/(!^)/ (1 - z/Q in G"*" and its generalizations.

CHAPTER 9 FUNCTIONS OF a-BOUNDED TYPE IN SPECTRAL THEORY OF NON-WEAK CONTRACTIONS

1. FACTORIZATION OF REGULARIZED DETERMINANTS 1.1. For any p > 1 we denote by Cp the class of continuously invertible contractions T in a separable Hilbert space 53 for which the operator Dj^=I — T*T belongs to the Neuman-Schatten ideal 6p. The set Ci coincides with the class of all invertible weak contractions [104]. We define the characteristic function WT of the operator T as in [12]: WT{Z)WT{0)

WT{0)

= [/ - DT{I

- ZT)-'^DT]

= (r*T)i/2 I S T ,

I ST,

VT = DTS).

It is easy to verify that the operator-function Wj^{'z) differs from the characteristic function @T{Z) of B.Sz.-Nagy and C.Foias [104] by a constant isometric factor. Let us recall from [104, 12] that WT{Z) is holomorphic in \z\ < 1, where its values are two-sided contractions in 2 ) T , i.e. W>J^{Z)WT{Z) < I and WT{z)Wf{z) < / in \z\ < 1. Since / - W^\0)

= W^\0)

[(r*T)i/2 - /

\^i

= W f i ( 0 ) [ / + ( T * T ) i / 2 ] ~ \ r * T - / ) |S)T and

I - WT{Z)

= I-

W^\0)

+ DT{I

- zT)-^

DTW^\0)

operator / — WT{Z) belongs to &p for any z ^ a{T~^). the regularized determinant driz) = detpWriz)

= Y[Xk{z)expl k

^

, DT

-:[1 - Xk{z)y

^ 3=1 ^

e 62p,

the

Hence for each p > 1

(1.1) J

is holomorphic wherever the operator-function WT is holomorphic [37, Ch. IV]. In (1.1), {Xk{z)} is the set of eigenvalues of the operator WT{Z). Further, the functions VT = detpWT{z)Wf{z) (1.2)

160

CHAPTER 9

will play an important role, and one can state that formulas (1.1) and (1.2) give a correspondence between the operators T e Cp (where p is a natural number) and the functions (IT and 7>T which are holomorphic in |z| < 1. 1.2. The main result of this section states that (IT and J>T both belong to M.M.Djrbashian's class Na [19, Ch. IX]. As it is well known, the functions (IT and J>T are bounded in |2:| < 1 if p = 1. For the general case we have T h e o r e m 1.1. If p > 2 is an integer and T G Cp, then the holomorphic functions (IT and T>T belong to Np-i^^ for any e > 0. For proving this theorem we need L e m m a 1.1, If p > 2 is an integer, then for 0 < r < 1 i- / ^^

\\DT{I-zT)-'DTrp\dz\

< (l-r)-(^-i)p|.||^,

(1.3)

J\z\=r

where \\ • ||p is the norm in &p. Proof. First, suppose p = 2^ {k > 1). By the elementary properties of eigenvalues (A^) and singular values {sj) of compact operators [37, Ch. II] \\DT{I - ZT)-'DT\\1

= Y. s f '

{DT{I - ZT)-'DUI

-

ZT*)-'DT)

3

= J2 ^T i^Til - ZT)-'DUI - ZT*)-^DT) 3

= Y, Af' {Dl{I - zT)-^Dl{I

-

zT*)-')

3

< Y ^T i^Tii - zTr'DUi - zT*r') 3

3

<{l-rrP/'\\DUl-zT)-'Dm/;^

{\z\=r).

Using these relations k — 1 times we get \DT{I

- ZT)-'DT\\1

< ( 1 - r ) - ^ n ; . ' 2"' \\Dl'-' (/ -

=

fc-1

zXy^D^ 2

il-r)-^P-^)\\Q{I-zTr'Q\\l, 2

•

||2

2

FUNCTIONS O F Q-BOUNDED T Y P E IN ...

161

where Q = DZ! . Further,

/

\\DTiI-zT)-'DTt\dz\

J\z\=r

P

oo

= 27r(l - r)-(^-2) ^Sp(QT^'Q2(T*)^Q)r2^' 3=0

since

|2 = Sp(0(7 - zT)-'Q\l ||Q(J - zT)-iQ||2'

-

zT*r'Q)

= J2 Sp{QT^Q^{TyQ)z^z''. j,k>0

Recalling that Q = D^' , we get

Hence (1.3) holds for p = 2^ {k > 1). Note that the above argument remains vahd if we replace DT by any normal operator G with the spectral decomposition i

i

We shall use this later. Returning to the proof of (1.3), for any p > 2 we shall use the well known Hadamard theorem on three lines and some techniques similar to one worked out in the proof of the Riesz-Thorin theorem on interpolation of operators [7]. First note that if the values of a function F{ip) are in &p, then TV

sup G

/

\

(

f^

1 ^^^

^ sp{F{,f)G{^)}dcp]^ = |y_^ m^w^d^j

,

where the supremum is taken over all ©^-valued functions G{(p) for which

\\G{^)\\ld^ = l,

(1.4)

1 + 1 = '^-

J —t

Of course, we presupposed that F and G satisfy the standard requirements providing the existence of the integrals in (1.4). It is easy to verify that (1.4) is an immediate consequence of the precise estimate |Sp{FG}| < ||i^||p||G||q (see [37]) and Holder's inequality.

162

CHAPTER 9

For interpolating the inequality (1.3) (already proved for p = 2^~^, 2^) for all p G [2^~^,2'^], denote po = 2^"^, pi = 2^ and similar to the proof of the Riesz-Thorin theorem set 1 1— z z 1 1— z z 1 1 . / . ^ . x P{z) Po P i ' q{z) qo Qi' Pj Qj and for any n > 1 consider the operator-function ri

_p_

_ ^

n

k=l

j=l

where l/p-\- 1/q = 1 and {ek}, {%(v^)}> {'^i(^)} ^^^ some weak measurable orthonormal sets of vectors, 5^ > 0 and aj{(p), [djiy^)]"^ are nonnegative, bounded, measurable functions. In addition, we suppose that

X:sf = l

and

rJ2a%v)d^

fc=i

=l

•'-'^j=i

for any fixed n > 1. Now introduce the entire function fr{z)

= ^

f

Sp{^niz){I-re"'T)-^^n{z)Gn{
k,m,j=l

wheie Pk,m,j{^) = {ek,Vj{(f)){uj{(p),em){{I - re^'^T) ^6^,6^). The inequality (1.3) is true for p = po and for the normal operator $n(^)- Consequently 1/PO

\Miy)\ <

(27^)^-'!^^Ij\^n{iy)il-re'^T)-'^niiy)\Zld^'^ r

\

p-K

l/qo

<(27r)-i||$^(i2/)||^^(l_^)-i+i/Po|^j|Gr„(^,iy)||^orf^ 2p

=(27r)-i|X:

Po >j 1/po

go

(l-r)-

af^\^) 1 1/90

^27r)-^^.f

(1-r)-^

i/go

/

0

>j 1/go

y

_ ^

5 ] a ((/:?)d(yi? ^ = [27r(l - r)] ^o.

Takings = pi we similarly come to the inequality \fr{l-{-iy)\ < [27r(l—r)] ^/^^. Consequently, \frW\

< [27r(l-r)]""^~^,

0 < ? ? < 1.

163

FUNCTIONS O F Q:-BOUNDED T Y P E IN ...

Now let p e (2^ "^,2^) be arbitrary and let the corresponding T? G (0,1) be chosen from 1

1-1?

p

po

- =

^

+-, Pi

1

1-7?

q

go

- =

7?

/I

1

Qi

\p

q

+-

\

- +- =1 . J

Then obviously n

^nW

= Y^Sk{'

n

,ek)ek,

Gn(^,^) = ^ a j M ( . ,Vj{cp))uj{ip)

k=i

(1.6)

j=i

and therefore ^)]d^ < [27r(l-r)]

•^ j ^ Sp{$n(7?)(/ - re'^Tr^^nmOni^,

-i/q

for any n>\ and any functions 0n('^), Gn{^,'&) satisfying (1.5). Using this inequality and (1.4) instead of the first condition of (1.5), we get ^

fj^nWil

~ r e ^ ^ T ) - ^ ^ 4 ^ ) | | > < 11^^(^)11^ (1 - r) - ( p - i )

At last, taking as $n(^) the n-th sum of the Schmidt series of the operator DT and letting n —> oo we come to (1.3). P r o o f of T h e o r e m 1.1. Using a simple inequality for regularized determinants [31, Ch. XI, Sec. 22], from (1.1) we get log+ M T ( ^ ) | < mp\\I-WT{z)\\P {\z\ < 1). Therefore log+ \dT{z)\ < 2^M,{\\I

- Wi;\0)r,

+ \\Wi;\OW\\DT{I

-

ZT)-'DT\\I],

Consequently, by (1.3) /

D-^log+ |^T(re^'^)|^^ = Z ) - ^ | /
log+ \dT{re'^)W^^

Ci+C2(l-r)-^+i

for any a > p - 1. Hence dr G A"^ {a > p - 1). To prove that T>T G ^4° for any a > p — 1, observe that log+

\VT{Z)\

< Mp\\I -

WTiz)W}{z)rp

< 2PMp{||/ - WrizWp + < Ci + C2{\\DT{I

- zTr'Drr^

\\I-WS{z)\\p} + \\DT{I

-

zT^r'DTFp}.

164

CHAPTER 9

Application of (1.3) completes the proof. Corollary. If p > 3 is an integer, T e Cp and I — T'^T G Got for some a ^\p — l,p); then both (IT and J>T belong to ^ p _ i . Proof. As Lemma 1.1 is true for any p>2, using the above notation we obtain

- zTr'Dr\\;=^s^,

\\DT{I

which even may be not an integer,

= E4'"«i 3

3

<\\DT{I-zT)-^DTr-''Y.''i 3

-

<2^-"\\DT{I

zT)-^DTn.

Hence i- /

- zT)-'DT\\l\dz\

\\DT{I

< ^

^TT J\z\=r

/ ^^

\\DT{I

-

zT)-'DTUdz\

J\z\=r

<2P-"||D2^||^(l-r)-("-^).

Recalling the proof of Theorem 1.1 and using the fact that the class A° enlarges as a increases, we complete the proof. 1.3. The following assertion is another corollary of Theorem 1.1. Proposition 1.1. Let {zk} be a sequence of numbers from \z\ < 1, such that for a given integerp>2 ^ ( l - | z , | f <+oo.

(1.7)

k

Then the products

k

I fe=l •'

fp-l J J2 ^[l - Bk{z)]

Vo{z) = l[Bk{z)exp{

(1.8)

k^i^

k

where Zk — Z \Zk\

bzA^) = --, =1-7-

1 — YkZ Zk

^^^

^k{z) = bzj,{z)bj^{z),

are holomorphic functions in \z\ < 1, which belong to A^_i^^ for any e > 0. Proof. Consider the normal operator T = ^

Zk{'

,ek)ek.

FUNCTIONS OF a-BOUNDED TYPE IN ...

165

in our Hilbert space S^. Obviously, T eCphy (1.7). Further, for z ^ cr(T~^) U a{T*~^) the operators WT{Z) and WT{Z)W>}^{'Z) are normal and the sequences {bzk{z)} and {Bk{z)} are their eigenvalues. Hence ^0(^2^) = driz), ^o{z) = J)T{Z), and our assertion follows from Theorem 1.1. Now we shall check the precision of Theorem 1.1. First, it is easy to observe that the discrete spectrum of an operator T ^ Cp satisfies (1.7). Thus, the functions (IT and T^T in general do not belong to A^ when a < p — 1. For checking this, observe that the set of zeros of dx is the discrete spectrum of T*. Indeed, it follows from (1.1) that driz) — 0 if and only if ;2: = 0 is an eigenvalue of the operator WT[Z)^ and this is equivalent to noninvertability of WT{Z^ since / — WT{Z) G 6 p . On the other hand, applying the equalities T^r(^)W?(^"^) - W^(^-i)Wr(z) - / ,

z^ ai.T''),

\z\ < 1,

(see [104, 12]) we conclude that the operator WT{Z) {\Z\ < 1) is invertible if and only if z ^ a{T*). Similarly, {z^} U {zk} is the set of zeros of 2>T, where {zk} is the set of eigenvalues of T. Thus, if dx (or 7)T) belongs to A"^ for some a < p — 1, then

^(l-|z,|)i+«<+oo k

according to the general assertion on the density of zeros in ^ ° , and these series obviously does not converge for any T G Cp. It is not known weather the functions dr and X>T belong to Ap_i or even to M.M.Djrbashian's more wider class Np-i for any T E Cp. Also, it is not known weather the products do and I^o? constructed by a sequence satisfying (1.7), belong to Np-i. But we can state that there exists a sequence {2^fc}i° satisfying (1.7) and ^(l-|z,|)^-^<+oo k

for any £: > 0, such that do, ^ 0 G Ap_i for the corresponding products. However, we mention that Theorem 1.1 is as much precise as it is necessary for the further assertions. 1.4' Finding factorizations for C?T and J>T, we shall pay the main attention to the function T^T since its factorization will play more important role in our further considerations. First we note that 0<

VT{X)

which immediately follows from

< 1,

- 1 < a: < 1,

WT{X)W'}{X)

(1.9)

< I.

T h e o r e m 1.2. Ifp > 2 is an integer, T E Cp and {zk} is the discrete spectrum of T, then for any £ > 0

V^{z)=i:)o{z)e^^[-^j^

5p_i+,(e-^^z)d^,(i?)|,

\z\ < 1,

(1.10)

166

CHAPTER 9

where ips is a real, continuous function of bounded variation in defined by (1.8) and

[—TT^TT], 2>O

is

5p-i+.(C) = r(p + e ) { ^ 3 - A _ _ i | , Each factor of this representation satisfies (1.9) and if it is assumed that i/jsi—Tr) = 0, then the factorization (1.10) is unique for any s > 0. For proving this theorem we need the following Lemma 1.2. Let A and B be nonnegative contractions and let A = I — XP, where A G (0,1) and P is a one-dimensional orthogonal projector. Ifl—B G &p for some integer p>2, then

sp I E ^ [(^ - ^ ) ' + (^ - ^ ) ' - (^ - ^ ^ ) 1 1 ^ 0-

(1-11)

Proof. First, consider the case when A and B are acting in a finite-dimensional space and B is continuously invertible. Let {A^li', {l^k}^-, and {Z^A;}? be the sets of eigenvalues of the operators A, 5 , and AB correspondingly, numerated in decreasing order (note that the spectrum of AB is positive). It is well known [35, Addition] that TTl

771

n

n ^J ^ n ^^N 3= 1

(1 < m < n - 1) and

3= 1

71

H ^^' "" 11 ^^N3= 1

3= 1

In terms of these sequences, the inequality (1.11) takes the form n

p—1 ^

3=1

k=l

Observing that

3 = 1 k=l n

"^ \

= ^ ( l o g A,- + log iij - log I/,) - - log H - ^ 3=1

3=1

- 0,

^3

we conclude that (1.11) is equivalent to

EEfc{(i-^^)' + (i-^i)'-(i-'^i)'}^o3 = 1 k=p

(1-12)

FUNCTIONS O F Q-BOUNDED T Y P E IN ...

167

Now set ^(t) = f {I- e-'^y-^dx, Jo

0
+CX),

Thus, denoting — log Aj = aj, — log/ij = bj and — loguj = Cj {I < j < n) one can write (1.12) in the form

3=1

i=i

As A = I — XP, where P is a one-dimensional orthogonal projector, we have ai = a2 = ''' = ctn-i = 0, an = — log(l — A) and also m

>.

/ m

^bj

= -log( YlfiA

y m

\

<-log(Y[uj]

m

=J2^J^

1 <m
and an + Zlj=:i ^j = S j = i ^J- Consequently

f]{$(c,) - ^bj)}=j2 r ^^ - ^"')'"'^^' 3=1

3=1 -^^^

where Cj > bj (1 < j < n) and Cn > an since for each fc(l < fc < n) we have i^k < Xk and Uk < /J^k [35, Addition]. Note that at least the first n — 1 intervals {^3^ ^j) (1 ^ J < ^ — 1) are disjoint. Indeed, let C be the Hnear hull of those eigenvectors of AB, which correspond to the eigenvalues ^'l, z/2,.. • , ^j-i, and let P = (• , e)e. Then, using the minimaximal properties of eigenvalues [37, Ch. II] we obtain that

(ABx.x)

Uj — max —

(Bx,Ax)

r— > max —

(Bx.x)

r— = max —

r^ > //j+i

for any j ( l < j < n — 1). Thus Cj < &j+i. Further,

3=1

73=1"^^^ = 1 ^^3

> r\l-e-^r-^dx Jo

= ^{an)

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since {bj, Cj) (1 < j < n) are disjoint and the sum of their lengths is a^. Thus, we proved (1.11) under the assumption that A and B are acting in a finite-dimensional space. It is clear that (1.11) is valid also when B is not invertible. For proving (1.11) in the general case, observe that in virtue of the formula detpAB

={detpA){detpB) x e x p | - S p (j2l[{I-A)'-^{I-B)'-{I-AB)']^^

(1.13)

and other simple properties of regularized determinants [37, Ch. IV] the lefthand side of (1.11) depends on the operators I — A and I — B continuously in the ©p-metric. Consider a monotonely increasing sequence of orthogonal projectors {Pn}T which strongly tend to / and PnA = APn (n > 1). As (1.11) is already proved for An = PnAPn and Bn = PnBPn, letting n -> oo we come to (1.11) in the general case. 1.5. Proof of Theorem 1.2. As we have proved, {zk} U {zk} is the zeroset of the function X>T which belongs to Ap_-^_^^ for any e > 0. In virtue of Proposition 1.1, also the product !Do constructed by the sequence {zk} belongs to Ap_i_^^. Hence the function 'DT{Z)/'DQ{Z)^ which has no zeros in \z\ < 1, belongs to A'^_ij^^ for any 5 > 0, and consequently it allows the representation

VT{Z)/V^{Z)

= e'^^ e ^ p { - ^ f

Sp-i+e{e-'^z)dM^)V

|z|
where Im 7^ = 0 and i/^e is a real-valued function of bounded variation in [-7r,7r] [19, Ch. IX]. In this formula 7^ = 0 since X>T(0),X>O(0) > 0. Thus (1.10) is true. For proving that -0^ is continuous, observe that log ( P T / ^ O ) is holomorphic in |2:| < 1 and hence it can be expanded in a power series: 00

log {VT{Z)/VO{Z))

= ^ 4 ^ ^

\z\ < 1.

A;=0

On the other hand, using the power expansion of the kernel 5p_i+£ we find

Hence

169

FUNCTIONS OF Q-BOUNDED TYPE IN ...

As the same equality is true for any ei G (0,6:), we have \dk\

Tjl + k)

^1

-iM

r ( p + 5 + fe) TT

#.iW

Tip + £i + fc) T{p + e + k)

0(1)

as fc -> cxD, i.e. the Fourier coefficients of V^e tend to zero and hence '^^ is continuous on [—7r,7r] [122, Ch. Ill] (note that more might be said on the differential properties of t/^e). The uniqueness of the function ijje follows from the results of [19, Ch. IX]. Moreover, the following inversion formulas are true:

V^(±) . (^2)

- ^ f ^(^i) = ^Hm J j ^ rD"" . - a .log .J^T(re^^)

(1.14)

M,

'0^(^) :='0(+)(^)-'0(-)(^) (-7r<7?i<^2<7r, - 7 r < ' i ? < 7 r ,

a=p-l-\-e),

where ijje {'&) are nonnegative, continuous, increasing functions defined as the positive and the negative variations of ijje on [—TT,-??]. Finally, we shall show that each term of the factorization (1.10) satisfies (1.9). It is well known [12] that to each invariant subspace S)i of the operator T, on which T is invertible, corresponds a factorization of the characteristic function: WT{Z) = WI{Z)W2{Z), (1.15) where Wi and W2 are holomorphic contractions in |z| < 1. Now, let zi be the first eigenvalue of T, Tei = ziei, ||ei|| = 1 and let ^ 1 be the one-dimensional invariant subspace born by ei. Then by (1.15) and (1.13) VT{z)^detp[Wi{z)W2{z)W^{z)W^{z)] =

= detj,[W^ {z)Wi{z)W2{z)W^

{z)]

det^[W^{-z)Wi{z)]det^[W2{z)W^{-z)]

xexp|-Sp(X^i[(/-A)^ + ( / - 5 ) ^ - ( / - A B ) ^ ] ) | ,

(1.16)

where A = Wi{j)W\{z) and B = W2(-2^)1^2*(^)- For finding the elementary factor corresponding to zi, note that detpW^{z)Wi{z)

=

detpWi{z)W^{z)

and use the well known formula [12] Wi{z)W^{z)

= 7 - (1 - Z^)DTPI{I

-

zTi)-\l-ZT^)-^PIDT,

where Pi is the orthogonal projection on i j i and Ti =T\S)i. (1 - \zif)h (7 - zTiy^h

= (7 - Ti*Ti)/i = Pi (7 - T*T)Pih = = (7 - zzi)-'^h

and

As

PiD^Pih,

(7 - zT;;)-'^h = (1 -

zziy^h,

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CHAPTER 9

for any h e S^i, we conclude that the spectrum of the operator Wi{z)Wi {"z) coincides with the spectrum of the operator / - (1 - z'^){I - zTi)-\l = I-{1-

-

-zT^y^PiDlPi

z^){I - zz,)-\l

- zzi)-\l

- |zi|2)Pi

having only one eigenvalue Tyi{z) that differs from 1: vx{z) = \ - {l-z^)il-\z^\^) {\ - zz{){\ - zz{)

_{z-z^){z-z^) (}. - zz{){\ - zzx)

= h.,{z)h-,,{z) = B^{z). Therefore, obviously r pp--ii - .

detpT^i(z)l^i*(^) = ^ i W e x p ^ E ^ [ l - ^ i ( ' ^ ) ] ' \3 = ^ 3 is the desired elementary factor. Further, denoting as ^/c(>3) the elementary factor corresponding to z^^ by (1.16) we obtain VT{X)I^X{X)

< detpW2{x)W^{x),

- 1 < X < 1.

(1.17)

Here we used also (1.11) with A = W^{x)Wi{x), B = W2{x)W^{x), which, as one can easily verify, satisfy the conditions of Lemma 1.2. Since W2 is also a characteristic function of a contraction from Cp [12], the above argument is applicable for X>2(^) = detpW2{z)W^(z). By (1.17) this gives 0 < 'DT{x)/[^i{x)^2ix)]

< detpW3{x)W^{x)

< 1,

- 1 < a: < 1,

since W3 is a holomorphic contraction. Taking away the zeros oiT>T in this way, we come to the desired statement, and the proof is complete. The following question naturally arises: in which case the exponential factor in (1.10) is absent? The answer is given by Proposition 1.2. Let T he a completely non-unitary contraction from Cp (p > 2). Then its regularized determinant T>T is exactly the product T^o of (1.10) if and only ifT is complete and normal. We omit the proof of this statement since it will not be used in the future. Note that, as it was stated in the proof of Theorem 1.2, each discrete factor of the factorization (1.10) is a regularized determinant constructed by a divisor of the characteristic operator-function WT^ corresponding to an eigenvalue of T. So, the considered Blaschke-type factors have a definite spectral interpretation and their application is reasonable in operator theory.

FUNCTIONS O F Q;-BOUNDED T Y P E IN ...

171

In [19] M.M.Djrbashian used Blaschke-type factors of different form, which permit to obtain inversion formulas (1.14) without division by I>o- Note that products of nearly same type as in (1.10) were introduced in the early works of M.M.Djrbashian [16, 17].

2. BOUNDARY BEHAVIOR OF REGULARIZED DETERMINANTS In this section we find a connection between the behavior of the function J>T near the boundary point z = 1 and some spectral properties of operators T G Cp. Namely, our aim is to prove Theorem 2.1. Let p > 2 be an integer, let 1 be not an eigenvalue for an operator T E Cp and let 7>T be the regularized determinant (1-2). Then the following statements are equivalent: (i)

DTS^ C (/ - T^)9)

(ii)

sup

and

{I - T*)-^DT

G 62^;

{l-r)-^\\I-WT{r)W^{r)\\p<+oo;

0
(iii)

sup (1 - r)-P[l - Drir)] < +00; 0
(iv) if {zk} and i/js are the parameters of the factorization (1.10), then E f h l ' ^ ' I ' V < + ° ° '^'^d r

5 p _ i + , ( e - ^ ' ' r ) # e W = 0 ( ( 1 - rf)

(2.1)

as r -^ 1 — 0; (v) the following limits exist lim (1 - r)-n\I

- WT{r)W^{r)\\P,

r—>-l—0

^

hm ( 1 - r ) - ^ r r-^l-O

lim (1 - r)-^[l - D r ( r ) ] , 7—>-l—0

Sp.i^e{e-'^r)dM^)'

J_^

Moreover, if any of statements (i) - (v) is true, then 11(7 - TT'DTWZ

= 2 - ^ lim (1 - r)-^I

- T^T(r-)W?(r)||^

= p2-P lim (1 - r)-P[l - VT(r)]

= T.{]^)\W

- '^"2^ £ Vi+.(e-*^)#. W- (2.2)

2.1. The following lemmas are to be used for proving Theorem 2.1. L e m m a 2.1. //(ii) is true, then I \ogVT{r)\

< Mp\\I - WT{r)W^{r)\\l,

0 < ro < r < 1.

(2.3)

172

CHAPTER 9

Proof. From (1.1) it follows that oo

^

oo ^

I logI?T(r)| = E E 7(1 - ^'^W]' = E [ l - ^^'WF E 7[1 - ^A,(r)F-^ /e j = p "

J=p"^

where {Afe(r)} is the set of eigenvalues of the operator WT{r)Wj^{r). if r is sufficiently close to 1, then E [ l - Mr)r

Further,

= 11/ - Wr(r)Wf (r)ll^ < M ; ( 1 - r f < 5 < 1.

Thus 1 — Afc(r) < VS for each fe, and we get I logI>T(r)| < b ( l - ^ ) ] " ' E [ l - ^fc(^)]'' = ^pll-^ - ^T(r)W?(r-) \'pL e m m a 2.2. T/ie following assertions are equivalent: (a)

(b)

sup(l-r)-^[l-l^o(r)]<+oo, 0
^ ( i l l M y ^V|l-^A:r/

<+oc.

Proof. If (a) is true and r is sufficiently close to 1, then clearly 1

I

"^ 1

' k

j=p

|logX>o(r)|

o(r)]
n

Y^{l-r)-P[l-Bk{r)r
for any n > 1. As dr

Bk{r]

letting r ^ 1 — 0 we get

t{}0tr.. Zk

which implies (b) by arbitrariness of n > 1. Conversely, if (b) is true, then observe that l-B,{r)=^-^—p^^—p^ |1 - Zkr]''

and

\1 - z^rl >2-'\l

- Zk\

FUNCTIONS O F a-BouNDED T Y P E IN ...

173

for I^A;! < 1 and 0 < r < 1. Hence

(l-r)-.|l-B.(r)l=(l + r ) l ^ < 8 l ^ and by (b) J2[l~Bk{r)r
Consequently \\ogVo{r)\<M^Y}^-Bk{r)r, k

and finally we come to 1 - X>o(r) < | logX>o(r')| < (74Mp(l - r)^. Lemma 2.3. If the statement (i) of Theorem 2.1 is true, then s4im (/ - T ) ( / - rT)-^ = I

s4im (/ - T*)(/ - rT*)'^

and

1—)>1—0

= I,

r-^l—0

Proof. One can easily verify that the left relation holds if s4im (1 - r)(J - rT)-^ = 0.

(2.5)

1—)>1—0

To prove (2.5), note that for every h G 9) \\{I-rt)-^DTh-{I-T)-^DTh\\

< \\{1 - r)T{I - rT)-^{I <

Consequently, lim (l-r){I-rT)-^DTh

-

T)-^DTh\\

\\{I-T)-^DTh\\. = 0 and lim {l-r){I~rT)-'^f

1—>-l—0

=0

r—>-l—0

for any vector / of the form n

f=Y.

T'^Drhk,

n > 0, {kkjlr. C Sj.

k=—n

As 11(1 - r){I - rT)-iII < 1 (0 < r < 1), (2.5) holds if T is completely nonunitary [104, Ch. 1]. The relation (2.5) is true also in the case when T has a unitary component U. Indeed, 1 is not an eigenvalue for f7, as it is not an eigenvalue for T. Therefore, by the spectral decomposition of U s4im (1 - r)(I - rU)-^ = 0 . 2.2. P r o o f of T h e o r e m 2 . 1 . For proving (i) =^ (ii), we use the following well-known equalities for characteristic operator-functions [104, 12]: Wr(r/)W?(0 = / - ( ! - vODril - vT)-\l W^iOWriv) _ _ = / - {l-'n£,)W:^^{Q)DTT*{I - iT*)-^{I

-

1,T*)-^DT,

(2.6) - »7T)-iT£>TWf 1(0).

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CHAPTER 9

Hence / - WT{r)W^{r) and consequently

- (1 - r'^)DT{I ~ rT)-\l

(1 - r2)-i||7 - WT{r)W^{r)\\r, =

- rT'^y^Dr

- rT)-\l

\\DT{I

=

-

(0 < r < 1) rT^'DrW^

\\iI-rTn-'DT\\l.

On the other hand, II(/ - rT^r^Dr

- (J -

T^)-^DT\\^^

<(i-r)||(/-rr*)-i||.||(/-rT'i^T||2,<||(/-n-^i^T||2,. Thus, ||(/ - rT*)-iJ9T||2p < C < +00 (0 < r < 1), and (ii) holds. (ii) => (hi). By Lemma 2.1, for ro < r < 1 1 - -DTir) T(r)| < M^\\I - WT(r)W?(r)||^ < M'^{1 -

rf.

(hi) ^ (ii). If {\k{T)} is the sequence of eigenvalues of the nonnegative operator WT{T)Wj^{r)^ then 1 ^

I k

\ k

"^ 1

- I log X>T(r) I

j=p'^

< M[l - Vrir)]

< M i ( l - rf,

and it suffices to see that ||/ - WT{r)W^{r)\\P = J ^ J l (ii) => (i). From (ii) it follows that for ro < r < 1 (1 - r)-^\\I - WT{r)W^{r)rp

(1 - r)-^I

0 < r < 1, M^W-

- W^{T)WT{r)\\l < K.

Therefore, using the formulas (2.6) we obtain ||(/ - vT^r^Drhp

< Ku

\\T{I - rT)-'DTW^\0)hp

< Ki

and moreover sup | | ( / - r r * ) - i D T / i | | 2 p < H - o o

and

0
sup \\{I - rTy'^Drh\\

<+oo

0
for any vector h E'DT- Since 1 is not an eigenvalue for T, we have DT9)C{I-T*)S)

and

DT^C{I-T)S)

(2.8)

by the well-known theorem of B.Sz.-Nagy and C.Foias [104, Ch. IV]. Besides, by the same theorem the following strong limits exist: s4im (/ - rT*)-^DT r—>1—0

= {I - ry^Dr,

s4im (I r—>1—0

VTY^DT

= {I -

T)-^DT,

FUNCTIONS OF a-BOUNDED TYPE IN ...

175

and using (2.7) we come to the inclusions [37, Ch. Ill] {I-T*)-^DTee2p

and

{I-T)-^DT

eG2p

(2.9)

which mean that (i) is true. For proving the equivalence (iii) <^ (iv), we denote the exponential factor in (1.10) as G{z) and observe that (iii) is equivalent to the following pair of conditions: sup (1 - r)-^[l - X>o(r)] < +oo,

sup (1 - r)-^[l - G(r)] < +oo. (2.10)

0
0
The last condition obviously is equivalent to the second condition of (2.1). Therefore, application of Lemma 2.2 proves the equivalence (iii) 4^ (iv). (i) ^ (v). By Lemma 2.3, s4im (/ - T*)(/ - rT'')-^! s4im (J - T){I - rT)-\l

- T*)-^DT

= {I -

T'^y^Dr,

- T)-^DT

= (I -

T)-^DT-

r—>1—0

Consequently [37, Ch. Ill] ^hmJ{I-rT)-^DT-{I-T)-'DT\\,^

=

^i4Y'J^I-rTT'DT-iI-T*r'Dr\\,^=^0,

and by the equalities (2.6) lim \\{I-rT*)-^DT\\l

= Hm (1 =

r^y^ll-WT(r)W}{r)\\

\\{I-T*)-'Dj

\2p'

Further, if {Xk} is the set of eigenvalues of the operator WT{t)Wj^{r), using the already proven equivalence (i) <^ (ii) we obtain

then

oo

lp+1 k

j=p-hl

•"

^

' ^

k

= C'\\I - WT{r)W}{r)\\

. \\I -

WT{r)W}{r)\\l.

176

CHAPTER 9

Thus, by (ii)

^(l-r)-4logA,(r) + X:7[l-A.(r)FJ

{l-r)-P\logVT{r)\ oo

.

= E E 7(1 - '•)"''[i - ^-^ wi' = E ^(1 - ^)"'ti - ^*= w]' k j=p-'

k p

00

^

+ E E 7(l-0-"[l-A.(r)F = i ( l - r ) - P | | / - W r ( r ) W ^ ( r ) | | ^ + o(l)

as r - > 1 - 0.

Now (2.11) implies the existence of the hmits lini

^-^TJr)

r->l-0

{l-r)P

^

j . ^ r->l-0

llog'DTir)] ^ ^.^ 11^ - ^ T ( r ) W ^ ( r ) ||g (l-r)P

r ^ l - 0

(1 - r ) ^

Similar to the above argument (but Bk{r) taken instead of A^), we prove that p lim ( l - r ) - P [ l - X > o ( r - ) ] r—^-l—0

lim V ( l - r)-^[l - 5 , ( r ) f , r->l—0-^—' A;

and using the factorization (1.10) and Theorem 2.1 we obtain ,. 1-I?r(r) ,. I-Voir) p lim -^r— = p lim -7- r r-^1-0

(1—r)P

r-»l-0

[l—r)P

, ,. 1 - G{r) h p lim -7- r—. r-^l-0[l—r)P

As the equality (2.4) is true,

.]!?LoE(i-)-^[i - ^^Wf = 2^ E ( y i ^ ) ' Consequently

and the implication (i) =^ (v) is proved. The converse implication is an obvious consequence of previous results. Further, formula (2.2) follows from (2.11)(2.13), and the proof of Theorem 2.1 is complete.

FUNCTIONS O F ^ - B O U N D E D T Y P E IN ...

177

We close this section by two remarks related to Theorem 2.1. R e m a r k 2 . 1 . Theorem 2.1 remains valid if we add to (i) - (v), for instance, (vi)

sup(l - \z\)-^I zer

-

(vii)

sup(l - |2:|)~^|1 - VT\ < +00, zer

WT{Z)W}{Z)\\P

< +00,

where F is an angle of opening < TT in |z| < 1, symmetric with respect to the real axis and with vertex at z = 1. Moreover, all the limits in Theorem 2.1 exist as z —> 1 non-tangentially. The proof of this extension of Theorem 2.1 is simple and it needs no new idea. R e m a r k 2.2. Formula (2.2) will be discussed also in the next section. One has to note that it is not known weather (2.2) can be observed as a regularized trace formula for operators of Cp (p > 2). The previous considerations may be used to obtain essentially more general relations of (2.2)-type, which are called trace formulas in the case p = 1 [75]. The existence of e in (2.2) brings some dissatisfaction. The way in which e appears was explained in Sec. 1. One may get rid of it by letting e —> 0 but the properties of the limit function ipo (which may be even a distribution) would be hard to analyze. At last. Theorem 2.1 has been proved earlier for the case p = 1. 3. 6p-PERTURBATI0NS OF SELF-ADJOINT OPERATORS The meaning of the condition (i) of Theorem 2.1 becomes more clear for dissipative unbounded operators A whose Cayley transforms belong to Cp. Namely, it appears that (i) is equivalent to the representability of A as a sum of a selfadjoint operator and an operator from &p. 3.1. Denote by Qp the set of those operators A whose Cayley transforms T={AiI){A -\-iI)-^ belong to Cp. Then A e Qp ii and only if:

1)

±i^a{A),

2)

Im ( A / , / ) > 0 for a l l / G i ^ ( A ) ,

3)

The operator iR^i - iR*_. - 2R-iR'Li belongs to ©p.

(where Rx = {A-

A/)"^)

Besides, an operator T G Cp appears to be the Cayley transform of another operator from Qp if and only if 1 is not an eigenvalue for T. In addition, the following statement is true. P r o p o s i t i o n 3.1. Let A G Qp be an arbitrary operator. transform satisfies (i) of Theorem 2.1 if and only if A = AR+iAi

Then its Cayley

(3.1)

178

CHAPTER 9

where AR = A'^ and Aj is nonnegative and belongs to ©p. Proof. Let T satisfies (i). Then by (2.8) (/-T*r)i3C(/-T*)^

and

{I-T^T)^

(l{I-T)^,

(3.2)

Since / - T'^T = 2{iR.i - iR:_. - 2i21^i?_i), D{A) = {I - T)S) and Z}(^*) (/ - T'')S^, the inclusions (3.2) mean that {iR-i - iR'Li - 2R1^R-i)^ C D{A) and {iR-i - iRL^ - 2R'^_-R-i)S) C D{A*). By the first of these inclusions, R'LiTh e D{A) for any h e S), and D{A') C D{A) since T is invertible. Similarly, the second relation gives D{A) C D{A*), and so D{A) = D{A*). Consequently, we can write i{A* - iI)R-i

- i l - 2R-i = 2-\A*

and if we take h = {A + il)f 2 - \ A * - iI)Dl{A

+ il)f

- il){l - T^'T) = 2-\A*

-

iI)D^,

(/ G D{A), then = i(A* - il)f - i{A -I- il)f -2f

= i{A* -

A)f.

Thus Aj = ^ ~ ^ * ={I-

T'^y^Dlil

- T)-\

(3.3)

By the left relation in (2.9), Aj G &p. In other words, the closure of Aj (which initially was defined on D{A)) is a nonnegative operator from (5p. On the other hand, AR = {A -{- A*)/2 is a self-adjoint operator. Indeed, since A is a closed symmetric operator, it suffices to show that its defect index is (0,0). If we contrarily suppose that this defect index is {n,m) {n -\- m > 0), then introducing the operator A - ^ © {-A)

=AR

+ ZAI,

AR = ARe

{-AR),

AI

=

AT

e (-^/),

acting in the space M. = S) ^ f) we conclude that the defect index of AR is {n + m,n-]- m). Thus, AR has an extension in H. Since A/ is bounded, also A has such an extension. The latter is impossible since A and therefore also A have ±z as regular points. Suppose now that A is represent able in the form (3.1). Then similarly we obtain that Ajf = {I - T*)-^D^I - T)-^f for any vector / G D{A) ( - D{A'')). Besides, for any / G D{A) \\DT{I

- T ) - V | r = ((/ - Tn-'DUl

- T)-'f, f) = {Ajf, f) < M l l / p

since Aj is bounded. Consequently, the operator S = DT{{I — T)~^ can be extended by taking its closure to a continuous operator in i^. As A / G &p and Ai = 5*5, one can easily deduce that DTS) C (/ - T * ) ^ and 5* = (/-r*)-iL>TG62p.

FUNCTIONS O F Q;-BOUNDED T Y P E IN ...

179

3.2. Now we shall convert the statement of Theorem 2.1 to an assertion for operators A £ Qp. It is convenient to connect such operators with the functions DA{W) = DT (l^^^ , \i — w J

T={A-

iI){A + iI)-\

lmw<0,

(3.4)

which are holomorphic in the lower half-plane G~ = {w : Im. w < 0} and for which the factorization (1.10) takes the form VA{w)=Vo{w)g{w),

(3.5)

where ^ is a non-vanishing function in G~, corresponding to the exponential factor in (1.10), and X>o is the product jj—i

V"

^f^i^

J

1-Xkwl

+ XkW

(Im \k > 0) constructed by the discrete spectrum {A^;} G G~ of A. Note that a formal representation of the function g can be obtained from the formulas g(w) = G f ^ i ^ ) ,

Giz) = exp { - ^

r

Vi+,(e-^''z)dV.w|

I —W

which are true for w E G~. Nevertheless, g{w) has another, more natural representation given in the below similarity of Theorem 1.2 for unbounded operators. T h e o r e m 3 . 1 . Let p > 2 be an integer, and let A G Qp be an arbitrary continuously invertible operator. Then for any e G (0,1/2) 2..M=I>oHexp{4/;j^^^^^^},

lm.<0,

(3.6)

where //^(t) is a real-valued function of locally bounded variation, satisfying

I

(l + |t|)2e

< +00.

3.3. For proving Theorem 3.1, we need the following purely technical lemma which we give without a proof. L e m m a 3.1. Let G{z) ^Q be a holomorphic function in \z\ < 1, and let G{z) be holomorphic also in a neighborhood of z = —1, where it may be written as oo

G(z) = l + ^ a f t ( 2 + l)'= k=p

{p>2,

|z + l | < r o < l ) .

(3.7)

180

CHAPTER 9

If for some a e {p — l,p) I=ff

il-\z\^r-'\\og\Giz)\\daiz)<+oo J

(3.8)

J\z\
{where d(j{z) is the area element), then for each 7 G (a—_p-f-l,l) the function g{w) = G ( f z ^ l ; which is holomorphic in G~, satisfies

P r o o f of T h e o r e m 3.1. Evidently, - 1 0 a{T) (where T {A-iI){A-\-iI)-^) since A is continuously invertible. Using the similarity of Theorem 2.1 for the point z — — 1 (i.e. for the operator —T for which all remains true with the change z -^ —z), we obtain lim

(1 + x)~^[l - DT{X)]

=bp

lim

(1 + x)~P[l - G{x)] =ap

{0 < bp < +00),

and moreover (0 < a^ < +00),

where G is the exponential factor in (1.10). In our case G{z) is holomorphic in a neighborhood of 2: == —1. Hence, G{z) is representable in the form (3.7). Besides, for any a > p— 1 and any r G (0,1) 1

f\r

- i)"-i [ r

r(a) Jo

log+ \G{te'^)\d^] tdt

U~n < f

i D-" log+ |G(re^'')|di? <M < +00,

J —TV

where D~^ is the Riemann-Liouville integro-differentiation and M is a constant independent of r (the latter was seen in Proof of Theorem 1.1). Thus, sup 0
/

/

{r~t)'^-'^\og-^\G{te'^)\tdtd^<M.

J-TV

Further, the equilibrium relation between Nevanlinna characteristics of G{z) leads to (3.11) since G{z) 7^ 0 in \z\ < 1. Thus, G{z) satisfies the requirements of Lemma 3.1. From (3.7) it follows that there exists a neighborhood of it; = 00, where I log |^(if;)|| < C\w\~P. Assuming a=p—l-{-€{0<€< 1/2) and using Lemma 1.9 of Ch. 1, we conclude that for w G G" with enough great modulus \W-"\og\g{w)\\
(3.10)

FUNCTIONS O F a-BouNDED T Y P E IN ...

181

where W~^ is Liouville's integro-differentiation and Ci is a constant. Hence by (3.9) (where 7 = 2e) we find that for any v <0 +°°

du

L

-j

< sup—-

nv

n-\-oo

+°° I log \g{u +

{v-tr-'dt

it)\\-

du

+ KP^

J

r(a)

< +00.

On the other hand, by (3.10) Hm ^ /

\W-''log\g(Re'^)\\sinMi}

= 0.

R-^+00 i t Jo

But the function W~^ log \g{w)\ is harmonic in the lower half-plane (see Lemmas 1.5 and 1.8 in Ch. 1). Consequently, by the results of Ch. 7

„,-.,„,i,,„+,„,i=_i_£|;j_i^, „<„,

,3.„)

where ji^ is a measure with the required properties. Using these properties and the corresponding assertion in Ch. 1, one can verify that dP 1 V /*+^ dii,{t) _1 7; r W~^^~^^ dvP T{p) 4- 5) TT J_, J_^ {u - t)2 + i;2 = - R e i r , / ^ - ( ; )

•

(3.12)

On the other hand, the properties of G{z) provide that the holomorphic function Log g{w) is expandable in a neighborhood of ty = oo by a power series: L o g 5 H = ^/lfe(^u;)-^

(3.13)

k=p

As W-^P-''^^W-''{iw)-^

= {iw)-^

{k > p),

successively applying the operators W~^, d^/dv^ and W~^^~^^ to both sides of (3.13) (the termwise application of these operators does not change the uniform convergence of the sum) and then taking real parts, we get W-^f-''^—W-''log\giw)\

= log|5(w;)|,

Im w = z; < 0.

182

CHAPTER 9

Thus, g{w)=exp\--

I

r . / ^ ' S z ^ + e +^^f^

lmit;<0,

where (7 is a real number. It remains to observe that C = 0, since g{w) as If; -> oo and ^™

/

= 0.

TTT

Hence, our assertion follows from (3.5), and the proof is complete. 3.4- The next theorem is a consequence of above results. Theorem 3.2. If p > 2 is an integer, then for any operator A G Qp the following statements are equivalent: (i) A is representable in the form A = AR + iAi, where AR = A}^ and Aj >0 (ii)

sup

belongs to

6p;

\v\~P[l - DA{iv)] < +oo;

-l
(iii) The limit

lim |t'|~^[l — DA{iv)]

exists.

V—>—0

Besides, if A is continuously invertible, then (i) - (iii) are equivalent to (iv)

V^(Im Xk)^ < -foo and the following limit exists:

d,s)

lim - ^ r v^-0 4P7r J-oo

\v\P{\v\-it)P+^'

where {Xk} cind /i^ are the parameters of the factorization (3.5). In addition, if A is invertible and at least one of the statements (i) - (iv) is true, then

= V ( I m A.)- + lim / -

r ^

, , , y ^ , .

(3.14)

Proof directly follows from Theorem 2.1, Proposition 3.1, and Theorem 3.1, if one uses the formulas (1 _ ^)-P[i _ Zk =

DT{X)] T——:,

Xk+i

= (1 - ^)^(2|7;|)-^[1 - DA{iv)l 71

To = I m Xk

\^-Zkr

^ < 0,

FUNCTIONS O F a-BouNDED T Y P E IN ...

and the equality (3.3) by which | | ^ j | | ^ = ||(/ -

183

T*)-^DT||2P-

Note that formula (3.14) is an improvement of the inequality

j2{irciXkr<\\Ai\\i, k

which is well known at least for bounded dissipative operators. Further, note that for p = 1 the quantity Sp {Aj) was calculated by the parameters of the factorization of bounded holomorphic functions in a half-plane in [42]. 4. COMPLETENESS OF OPERATORS IN Cp The aim of this section is to show how the properties of the regularized determinant X>T may be applied to the completeness problem of non-weak contractions. Recall that an operator is called complete if the closed linear hull of its root subspaces corresponding to the eigenvalues of that operator coincides with the whole space. 4.1. In the case of dissipative operators, the following theorem improves a well-known result due to M.V.Keldisch (see [74] or [37, Ch. V]). T h e o r e m 4 . 1 . Let p > 1 be an integer, and let T G Cp be any operator for which 1 is not an eigenvalue. Further, let the spectrum of T be a sequence {zk} having 1 as its unique limit point, and let Vrir) liminf l o g - ^ ^ = 0

(4.1)

for the functions J>T{r) and X>o(r) defined by (1.2) and (1.8). Then the operator T is complete. Before proving this theorem, we shall show that it really is a generalization of the mentioned result of M.V.Keldisch. Indeed, (4.1) is satisfied "with considerable reserve" if sup (1 - r)-P[l - G{r)] < +00,

where

G{z) =

VT{Z)/VQ{Z).

0
If we additionally demand supo<^
< +00,

0
and the requirements of Theorem 2.1 will be fulfilled. Thus, by Theorem 3.2, the corresponding operator A (which is the inverse Cayley transform of T) will be representable as A = AR + iAj, where AR = A^j^ and Aj > 0, A/ € &p. Further, A and AR will have the same continuous spectrum since a compact

184

CHAPTER 9

perturbation does not change that. By the conditions of Theorem 4.1, the continuous spectrum of AR will be located in oc, i.e. AR will have only discrete spectrum. In this particular case Theorem 4.1 states: a 6p-perturbation of a self-adjoint operator with discrete spectrum is a complete operator. Namely this statement was established by M.V.Keldisch [74]. If we put p = lin the formulas for VT and 2>o, then Tyri^) will become the usual determinant of WT{Z)W'}{'Z) and 2^o(^) will become a Blaschke product. Thus, for p = 1 the condition (4.1) is also necessary for the completeness of T if this operator is a weak contraction. The latter fact immediately follows from the well-known completeness criterion for weak contractions [37, Ch. V]. 4.2. Proof of Theorem 4.1. Suppose our statement is not true. Then it is clear that S)i = Clos span{/^ : \zk\ < 1} ^ S), and denoting S)2 = ^ Q ^1 we conclude that the following triangulation given by the invariant subspace S) 1 is true:

Here T2 is an operator whose spectrum consists of the single point z = 1 which is not an eigenvalue. Thus, the characteristic function WT{Z) admits the factorization [12] WT{Z)

= WI{Z)W2{Z)

(4.2)

the factors of which are defined by the formulas Wi{z)W^{0)

=1-

DTPI{I

-

ZTI)-^PIDT,

W2{z)W^{0) =1-

Wr\0)DTP2{I

W2{0)

W{-\0)WT{0),

=

-

ZT)-^P2DTW^~\0),

where Wi{0) is any invertible solution of the equation Wi{0)Wi{0) = I — DTPIDT [12]. In the above formulas Pi and P2 are the orthogonal projections onto S)i and 9)2 correspondingly. As the operator Ti is complete, its characteristic function Wi can be represented by the product Wi(z) = W,,{z)W,,(z).-.

W,„(^) • • • = n W , , ( z ) ,

where Wzj^{z) are operatorial Blaschke factors. This can be checked by the standard methods introduced into operator theory by M.S.Livsic [78]. Note that this infinite product is convergent in the sense that the operator-function -1

An{z)=(flW.,{z))

Wi{z)

FUNCTIONS O F Q-BOUNDED T Y P E IN ...

185

uniformly converges to / inside \z\ < 1 in ©p-norm (i.e. ||/ — An(2r)||p —> ) 0 as n -^ oo). Thus, from (4.2) it follows that as n -> oo

W^''\z)^(flW,,iz))

WT{Z) = Y[

W.AZ)W2{Z)-^W2{Z)

in 6p-norm, uniformly in \z\ < 1. Consequently, by the properties of regularized determinants [37, Ch. IV] detpW^"'\z)W^''^'"{z)-^detpW2{z)W;{z)

as

n - > oo.

(4.3)

On the other hand, < detpW2{x)W^{x),

VT{X)/VO{X)

- 1 < x < 1.

(4.4)

Indeed, by Lemma 1.2 and relations in the end of the proof of Theorem 1.2 VT{X)

= detpWT{x)W^{x)

=

detpW^,W^^\x)W^^^*{x)W*^{x)

= detpW^^{x)W,,{x)W^^\x)W^^^*{x)

<

^idetpW^^\x)W^^^''{x),

where $ i , as before, is that elementary factor of the product I^o? which corresponds to the eigenvalue Zi. Evidently, one can apply the same argument to detpWj. \x)Wj^ {x) and so on. Thus for any n > 1 VT{X)

< n$fe(x)detpV4"^(a;)W^"^*(x).

Letting here n —> oo and taking into account (4.3) we come to (4.4). Now denote V2{z) = detpW2{z)W^{z). If {Xk} are the eigenvalues of the operator W2{x)W2(x), oo

then one can see that

oo

|logD2(a;)|=EE7[l-^'^(^)l' k=lj=p^ OO

^

oo

^

= E7ll(^-^2(x)l^2W)ii>E7ll^-^2(a;)T^;(^)ir'j=p ^

3=P "^

Further, by (4.4) and the second condition of our theorem oo

^hmJlogl>2(r)H^hm^X^^||/-T^2(r)W2*Mir-^0, J=P

(4.5)

186

CHAPTER 9

and consequently lim 11/ - W2(r)W:{r)\\

= 0.

(4.6)

Let r be the angle with vertex at z = 1, defined by the inequality 11-^1 < a

{\z\ < 1,

1-N

a> 1).

li z eT, then by formulas (2.6) ((7 - W2{x)W;{x))h, where Q2 =

P2DTWI~^{0).

(7 - zT^)-^Q2h

h) = {l-

\z\^)\\{I-zTi)-'Q2h\f,

h G 2)T,

Besides, it is obvious that

- (7 - rT*)-^Q2h

= {z - r)T;{I - zT^)-\l

-

rT^)-^Q2h

- r\\\{I -

rT^)-'Q2h\\

for \z\ — r and \l — z\ \z — r\ ^ 1 — r <+ 1—r 1— r 1 -
+a = K

for 2: G r . Consequently 11(7 - rr^)-^Q2h\\

< 11(7 - vT;)-^Q2h\\

+ (l - rr^\z

< K{{I - W2{r)Wi{r))h,h)

\im\\l-W2iz)W^{z)\\=0,

for any h € S T ,

zer.

Thus, in any angle F of the mentioned type 00

\\{W2{z)WUz)r'\\

< Ylp-W2iz)WUz)\\''

< C„ < +00,

where Ca is a constant depending solely on a. Henceforth, we shall suppose that a is enough great to provide that the opening of F is greater than 7r(l — 2/p). Now we may state that for any /ii,/i2 ^ 2 ) T the function f{z) = {W^^{z)hi,h2) is holomorphic in the whole complex plane, except the point z = 1, and is bounded in F. Besides, the entire function F{w) = f{z),

z = {w-i)/{w

+ i)

is bounded in the upper half-plane, except two angles adjoining the real axis, with openings less than n/p (since W2{z) is a unitary operator when |z| = 1

FUNCTIONS O F Q;-BOUNDED T Y P E IN ...

187

[12]). Aimed to apply the Phragmen-Lindelof principle in that angles, we shall show that F is of order p. To this end, we first note that if \z\ < 1, then ii^2~H^)ii = ii^2*(^-')ii < ^ 1 + C 2 i i ( / - z - ^ T ^ r ' i i = Ci+C2\z\-\\{T2-zI)-% Now introduce the operator A2 G Qp which is connected with T2 by the equality T2 = {A2 — iI){A2 -{-il)~^. li z = {w — i)/{w + i), then a simple calculation gives {T2 - zl)-^

= {2i)-\w

+ i)I 4- {2i)-\w

+ i){i - w){A2 +

wI)-\

On the other hand, if B2 = A ^ \ then B2 = -i{{I - T2){I + T2)~'^ and B2 - B* e 6 p since B2 - B^ = -2i{I + r 2 ) - i ( / - T2T^){I + T^)-\ As the spectrum of the operator A2 is an empty set, B2 is quasinilpotent and hence it is a compact operator [37, Ch. I]. Besides B2 G &p according to a theorem due to V.I.Macaev and, as it is well known, the order of the resolvent {A2 — wl)~^ is equal to p [37, Ch. IV]. Thus, the function F is of order p. Therefore, applying the Phragmen-Lindelof principle we conclude that F is bounded in the whole upper half-plane. By the elementary properties of characteristic operator-functions [12], F is bounded also in the lower half-plane. Consequently, the operator-function W2 is constant in the unit disc, i.e. T2 is a unitary operator. As cr(T2) = 1, we come to the equality T2 = I which contradicts the conditions of our theorem. Thus, Theorem 4.1 is proved. 4.3. Examining the proof of Theorem 4.1, one can see that a more general result could be proved. Namely, instead of the condition (4.1) one could require the function | \og{7yT{x)/7>o{x))\ to be bounded by a constant (which can be calculated) depending solely on p. Of course, stronger criterions of completeness given in terms of the regularized determinant T>T are possible. These can be obtained using more delicate Phragmen-Lindelof type theorems. For instance, the following result is true for the case p = 2 being of special interest. T h e o r e m 4.2. Let T be a contraction from C2, for which 1 is not an eigenvalue. Further, let the spectrum of T be a sequence {zk} with the unique limit point at 1, and let liminf (1 - r) log § ^

= 0.

(4.7)

Then T is complete. Moreover, for any given a < 0 there exists a non-complete operator in C2, for which the lower limit (4.1) is equal to a. Proof. It follows from (4.4) that liminf (1 - r)T>2(r) = 0 , r—>1—0

188

CHAPTER 9

where 1^2 is the function considered in the proof of Theorem 4.1. Suppose T>2{r) = exp{—(f{r)} {(p{r) > 0). Then {l — r)(p{r) -^ 0 as r -> 1 —0, according to Theorem 1.2. Further, (4.5) can be written in the form

^{r)>-log{l-\\I-W2{r)W^{r)\\)-Y^-:\\I-W2{r)WUrW. Using this we get (l — ||/ — W2(r)W2*(^)ll))~'^ ^ Cexp{(f{r)}

which impHes

oo

\\{W2{r)Wi{r)-^

3=0

Turning as ear her to the entire function F , one can see that hm sup y-'^ log \F{iv)\ < 0

(4.8)

and F is of second order and minimal type [37, Ch. V]. Application of the Pragmen-Lindelof principle to the function Fi{w) = F{w)exp{iw} separately in both quadrants of the upper half-plane gives that F is of first order and, consequently, it belongs to M.Cartwright's class [37, Ch. V]. Thus, by (4.8) the indicator of F is non-positive and hence F = const. The end of the proof of completeness is quite the same as that of Theorem 4.1 and we omit it. Now introduce an obviously non-complete operator T € C2 as the orthogonal sum T = Ti 0T2, where Ti is a complete operator from C2, for which (4.7) is true, and T2 is a contraction with one-dimensional defect, the characteristic function of which is of the form

T^2W = exp{|i±i}, a<0. Then

T>T{Z)

=

'DTI{Z)W2{Z)

exp{l — W2{z)} and consequently

Vrir)

limsup ^ , . = lim (1 — r) logl^2(^) = a < 0. r-,1.0 Voir) r->i-o^ J s zy J

NOTES. As it became clear from the later work [5], the measure dlp^ appearing in the similar to (1.10) factorization of the regularized determinant dx(z), is absolutely continuous and belongs to definite O.V.Besov class.

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INDEX

a-bounded type, xv, 101 Qf-characteristics in \z\ < 1, xiii a-kernels in \z\ < 1, xiii (1 + Q:)-capacity, xiv a-mean type, 68 ^ a { G + } , 62 ^ g (in \z\ < 1), xii Cp, 159 dT, 159, 160, 165 X>T, 158, 165, 168 dQ, 164 X>o, 164, 165, 168 G+(i?), 45 r*[^,oo), 46 7-capacity, 101, 102 HP (in \z\ < 1), ix iJ^ (in Imz > 0), X, xvi, 133 HP{n{x)dx), 133 iC^((5o,-D)-class, 7 i^^((5o,^)-class, 13 La(—oo,d)-class, 1 £c.(p,F^\81 M{R), 132 M/^(L)), Mp, 7, 147 Mp{D), 13 Blaschke-Djrbashian products, xi, xiii, XV, 103, 112 Blaschke type products, xiv, 21, 25, 33, 38, 84, 120, 121 M.M.Djrbashian factorization, xi, xiii

01, 9^"", X, 76, 131 AT, Nevanlinna class, 101, 120, 131 iVc-class, xiii, 113, 120, 160 91«, 0 1 ^ , 70, 71, 72, 75 Na{G-}, 101, 121 ?n^^, 82, 85, 86, 97 DA\W),

179

Qp, 111, 182 0t(p,F±),80 -R, Herglotz-Riesz class, 14 i?°, Nevanlinna class, 14, 23 i?T, I.S.Kats class, 14 Ra, 15 i?°, 15 Sa, 152, 154 Soc, 154, 155 6 p , Neuman-Scetten ideal, 159 it, il+, 69 il+,69 oo-starshaped domain, 6 oo-starshaped continuation, 45 W - " , 1, 6, 7, 20 W-^, 12 WT{Z), characteristic function, 159 na(^,C), 21 Ahlfors-Heins theorem, 132 Nevanhnna factorization in Im z > 0, ix, XV, 65, 129 Nevanhnna type characteristic, 61,82 Nevanhnna uniqueness theorem, 132

196

Eequilibrium relation, 61, 81, 82 Carleman type formula, 54,58 Cay ley transform, 177 Green type potential, 147, 149 Levin formula, 80 Levin type formula, 55, 59, 61, 62, 82

INDEX

Nevanlinna weighted class, xi, 82 R. and F.Nevanlinna type formula, 49,51 Phragmen-Lindelof Principle, x Riesz type representation, xvi Tsuji characteristic, xiv, 80, 100 Tsuji type estimate, 90 Tsuji type characteristic, 61, 62, 82

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