128 35 10MB
English Pages 380 [366] Year 2024
Frontiers in Mathematics
Armen M. Jerbashian Joel E. Restrepo
Functions of Omega-Bounded Type Basic Theory
Frontiers in Mathematics Advisory Editors William Y. C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Sprößig, TU Bergakademie Freiberg, Freiberg, Germany
This series is designed to be a repository for up-to-date research results which have been prepared for a wider audience. Graduates and postgraduates as well as scientists will benefit from the latest developments at the research frontiers in mathematics and at the “frontiers” between mathematics and other fields like computer science, physics, biology, economics, finance, etc. All volumes are online available at SpringerLink.
Armen M. Jerbashian • Joel E. Restrepo
Functions of Omega-Bounded Type Basic Theory
Armen M. Jerbashian Institute of Mathematics National Academy of Sciences of Armenia Yerevan, Armenia
Joel E. Restrepo Department of Mathematics Ghent University Ghent, Belgium
ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-3-031-49884-8 ISBN 978-3-031-49885-5 (eBook) https://doi.org/10.1007/978-3-031-49885-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.
Devoted to the memory of Mkhitar Djrbashian
Preface
p
In view of the contemporary development of the theory of .Aα spaces and its applications, it is natural and interesting to come out of the frames of the weights .(1 − r)α and consider p .Aω spaces with functional parameters .ω, which are associated with the M.M. Djrbashian integral operator .Lω . In some particular cases, this operator becomes the classical integral operators of Riemann–Liouville, Hadamard, Erdélyi–Kober, and many other ones. p The book gives the basic results of the theory of the spaces .Aω of functions holomorphic in the unit disc, halfplane, and in the finite complex plane, which depend on functional weights .ω permitting any rate of growth of functions near the boundary of the domain. p This continues and essentially improves M.M. Djrbashian’s theory of spaces .Aα (1945) of functions holomorphic in the unit disc, the English translation of the detailed and complemented version of which (1948) is given in Addendum to the book. Besides, the book gives the .ω-extensions of M.M. Djrbashian’s two factorization theories of functions meromorphic in the unit disc of 1945–1948 and 1966–1975 to classes of functions deltasubharmonic in the unit disc and in the halfplane. The book can be useful for a wide range of readers. It can be a good handbook for Master and PhD students and Postdoctoral Researchers for enlarging their knowledge and analytical methods, and also it can be very useful for scientists to extend their investigation fields. Yerevan, Armenia Ghent, Belgium August, 2023
Armen M. Jerbashian Joel E. Restrepo
vii
Acknowledgements
Joel E. Restrepo was supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and the Methusalem program of the Ghent University Special Research Fund (BOF) (Grant number 01M01021).
ix
Introduction
The origins of investigations related to the spaces of holomorphic functions, the squares of modules of which are summable over the area of the unit disc .D ≡ {z : |z| < 1}, can be found in the paper of L. Biberbach [5] and in other references of the classical book of J.L. Walsh [85]. While L. Biberbach studied approximations by rational functions in the space of holomorphic functions, the derivatives of which satisfy the mentioned summability condition, W. Wirtinger [87] studied approximations in the space .H2' of holomorphic in the unit disc .D functions f which themselves satisfy the summability condition. This space is being denoted by .A20 in the modern literature: ff
A20 (≡ H2' ) :
||f ||2 =
.
D
|f (z)|2 dS < +∞,
where dS is for the Lebesgue area measure. In the same work, W. Wirtinger, in particular, proved the representation formula for the functions .f ∈ A20 and found the orthogonal projection from the same type Lebesgue space .L20 to .A20 . Due to numerous misunderstandings on the issue in the contemporary literature, below we present the mentioned results of W. Wirtinger as they are given in J.L. Walsh’s book [85] (pp. 150– 151), where .C ' means the unit disc. .. . .Theorem
' .H such that 2
20. Let .F (z) be of class .L2 in .C ' . The essentially unique function .f (z) of class ff .
C'
| | |F (z) − f (z)|2 dS
is least is given by 1 .f (z) ≡ π
ff C'
F (ξ )
dS (1 − ξ z)2
,
|z| < 1.
(58)
xi
xii
Introduction The formal development of .F (z) on .C ' in terms of the functions .zk is ∞ Σ .
ak zk ,
k=0
ak =
k+1 π
ff C'
k
(59)
F (ξ )ξ dS;
this series converges to .f (z) of class .H2' in the mean on .C ' , hence (§5.8, Theorem 17) converges to .f (z) uniformly on any closed set interior to .C ' . Interior to .C ' , the function represented by (59) is .f (z)
≡
1 π
ff C'
] [ F (ξ ) 1 + 2ξ z + 3ξ z2 + · · · dS,
|z| < 1,
for the series in square brackets converges uniformly for .|ξ | ≤ 1 when z is fixed. This equation for .f (z) can be rewritten in form (58). Of course if .F (z) is an arbitrary function of class .H2' , then (58) is valid with .f (z) ≡ F (z). Theorem 20 is due to Wirtinger [1932], by a quite different method.. . . .. . . Theorems 20 and 21 and the remark just made extend to more general regions by the use of conformal mapping; compare §11.4. The study of extremal problems and their solution by methods of approximation is to be resumed in §11.3 and A 3. Of course one may study approximation in a multiply connected region (compare §1.6 and 1.7) in the sense of least squares, by orthogonalizing a suitable set of rational functions; see Ghika [1936] and Bergman [2] .. . .
Note that the real novelty on the .A20 space in the mentioned work of A. Ghika (1936) (no publication data in [85]), also in the monograph of S. Bergman [4], where his results were summarized, was the consideration of the unweighted space .A20 in some multiply connected domains. The complicated nature of the considered domains permitted S. Bergman only to prove the existence of the corresponding reproducing kernels and establish an analog of W. Wirtinger’s Theorem 20. Later, W. Wirtinger’s projection Theorem 20 was extended by V.P. Zakaryuta and p V.I. Yudovich [90] to the unweighted spaces .A0 (1 < p < +∞) in .D, the form of p bounded linear functionals was revealed, and it was proved that the dual space of .A0 is q .A .(1/p + 1/q = 1) in the sense of isomorphism. In W. Rudin’s books [72] and [73], 0 the same was done in the polydisc and the unit ball of .Cn , where the extension has some explicit forms of kernels, evident in view of W. Wirtinger’s Theorem 20 and the result of V.P. Zakaryuta—V.I. Yudovich. In W. Rudin’s books, the extension of W. Wirtinger’s Theorem 20 was called “Bergman projection,” and after that many investigators are attributing the terms “Bergman projection,” “Bergman space,” “Bergma kernel,” and even “Bergman-Nevanlinna classes” to any result on regular functions summable over the area of a complex domain. In fact, this cuts off the original information sources for numerous specialists and causes the above-mentioned misunderstandings on the origins in the contemporary literature. Coming to weighted spaces, the earliest paper of M.M. Djrbashian [8] is to be referred, the English translation of the detailed and complemented version [9] of which is presented
Introduction
xiii
in Addendum of this book. The mentioned papers were aimed mainly at improving R. Nevanlinna’s result of 1936 (see [66], page 216) on the density of zeros and poles of functions f meromorphic in .D, for which the Riemann–Liouville fractional integral of the growth characteristic .T (r, f ) is bounded, i.e., 1 . r(1 + α)
f
1
(1 − r)α T (r, f )dr < +∞
0
for a given .α > −1. This improvement results in a complete factorization formula for meromorphic in .D functions satisfying the above condition. The factorization formula contains some special Blaschke type product and a surface integral with the degree .2+α of the Cauchy kernel in the exponent and becomes the well-known Nevanlinna factorization of functions of bounded type in .D as .α → −1 + 0. The same works [8, 9] contain a large investigation of the similar Hardy type spaces .H p (α) of holomorphic in .|z| < 1 functions p for which the notation .Aα is used nowadays. M.M. Djrbashian [8] introduced these spaces by the boundedness of the Riemann-Liouville fractional primitive of the integral means .Mp (r, f ), i.e., by the condition f H p (α) ≡ Apα :
.
1 ≡ 2π
ff
1
(1 − r)α Mp (r, f )rdr
0
(1 − |ζ |)α |f (ζ )|p dσ (ζ ) < +∞,
|ζ | 0}, and in the whole finite complex plane .C are investigated. These spaces depend on a functional parameter .ω which compensate any growth of several integral means of functions near the p boundaries of the considered domains. Thanks to this, the spaces .Aω in the unit disc and 2 .Aω in the whole complex plane cover the whole sets of functions holomorphic in these
xiv
Introduction
domains. Besides, the factorization result of [8] is extended to some .ω-weighted classes of functions delta-subharmonic in the unit disc and similar classes in the upper halfplane, and the classes in the unit disc cover all functions delta-subharmonic in that domain. In the period of 1966–1975, an application of the Riemann—Liouville fractional integrodifferentiation and a more general operator depending on a functional parameter .ω directly to the considered functions led M.M. Djrbashian (see [11], Ch. IX and [10, 12, 13, 16, 20–22]) to the factorization theory of his Nevanlinna type .N{ω} classes, the union of which coincides with the whole set of functions meromorphic in the unit disc. Because of this comprehensiveness of the theory, M.M. Djrbashian first designated the last letter .ω of the Greek alphabet for the functional parameter. The new theory in a sense was more elegant, since it contains the Blaschke product and the classical formulas of Nevanlinna factorization, Jensen-Nevanlinna, Poisson–Jensen, and the Jensen inversion formula in the particular case .ω ≡ 1. The classes .N{ω} were introduced in [13] by application of a general Riemann–Liouville type operator .Lω [12]; see also in the monograph of S.G. Samko et al. [74] and many contemporary investigations. In a particular case, this operator takes the simple form f
1
Lω log |f (z)| = −
.
log |f (tz)|dω(t),
|z| < 1,
0
when applied to the logarithm of the modulus of a meromorphic in .D function. In the later extension of this theory [39] to the set of all functions delta-subharmonic in the unit disc .D, a growth condition was posed on .Lω u with an arbitrary delta-subharmonic in .D function u which replaced .log |f |. This led to a very explicit understanding of the Nevanlinna type .Tω characteristic and .Nω classes. Part II of the present book gives an extension of the M.M. Djrbashian factorization theory to a Riesz type representation theory of functions delta-subharmonic in the unit disc .D and the construction of a similar theory in the upper halfplane. Also, it contains some results on Banach spaces of functions delta-subharmonic in the unit disc and in the upper halfplane. August, 2023
Armen M. Jerbashian Joel E. Restrepo
Contents
Part I Omega-Weighted Classes of Area Integrable Regular Functions 1
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 M.M. Djrbashian Operators Lω and His Omega-Kernels . . . . . . . . . . . . . . . . . 1.2 Evaluation of M.M. Djrbashian Cω -Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 ω Decreases Not More Rapidly than a Power Function . . . . . . . . . . . 1.2.2 ω Decreases Exponentially . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Volterra Equation and Hausdorff Moment Problem . . . . . . . . . . . . . . . . . . . . . . . 1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Spaces .Aω (D) in the Unit Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 2.1 The Spaces Aω (D), Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 2.1.1 Definition and Elementary Properties of Aω (D) Spaces . . . . . . . . . . p 2.1.2 Representations of Functions from Aω (D) . . . . . . . . . . . . . . . . . . . . . . . . p 2.2 Projection Theorems and the Conjugate Space of Aω (D) . . . . . . . . . . . . . . . . . 2.2.1 Orthogonal Projection and Isometry for A2ω (D) . . . . . . . . . . . . . . . . . . . p 2.2.2 Projection Theorems from Lω (D) to p Aω (D) (1 ≤ p < +∞, p /= 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 2.3 Dirichlet Type Spaces Aω (D) ⊂ H p (D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Omega-Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 2.3.2 The Spaces Aω (D) ⊂ H p (D), Boundary Properties. . . . . . . . . . . . . . 2.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
p
p
3 3 13 14 23 27 39 41 41 41 44 46 46 49 56 56 61 62
3
Spaces .Aω (C) of Entire Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 3.1 Spaces Aω (C) of Entire Functions, Representations . . . . . . . . . . . . . . . . . . . . . . 3.2 Orthogonal Projection and Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Biorthogonal Systems of Functions in A2ω (D) and A2ω (C) . . . . . . . . . . . . . . . . 3.4 Unitary Operators in A2ω (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 65 68 71 73 78
4
Nevanlinna–Djrbashian Classes of Functions Delta-Subharmonic in the Unit Disc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Blaschke Type Factors and Green -Type Potentials . . . . . . . . . . . . . . . . . . . . . . .
79 80 xv
xvi
Contents
4.2 4.3 4.4
The Main Representation Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Universal Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86 87 92
5
Spaces .Aω,γ (G+ ) in the Halfplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p 5.1 The Spaces Aω,γ (G+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Representation by an Integral Over a Strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 General Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Orthogonal Projection and Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p p p 5.5 The Projection Lω,0 (G+ ) to Aω,0 (G+ ), the Space (Aω,0 (G+ ))∗ . . . . . . . . 2 (G+ ) . . . . . . . . . . . . . 5.6 Biorthogonal Systems, Bases, and Interpolation in A∼ ω 5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95 95 99 106 113 122 128 132
6
Orthogonal Decomposition of Functions Subharmonic in the Halfplane . . . p 6.1 The spaces hp (G+ ) and hω (G+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Orthogonal Projection L2ω (G+ ) → h2ω (G+ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133 133 135 140 141
7
Nevanlinna–Djrbashian Classes in the Halfplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Preliminary Definitions and Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Nevanlinna–Djrbashian Classes in the Halfplane . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143 143 154 160
p
Part II Delta-Subharmonic Extension of M.M. Djrbashian Factorization Theory 8
Extension of the Factorization Theory of M.M. Djrbashian . . . . . . . . . . . . . . . . . 8.1 Green-Type Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 R. Nevanlinna and M.M. Djrbashian Characteristics . . . . . . . . . . . . . . . . . . . . . . 8.3 Classes Nω of Delta-Subharmonic Functions with ω ∈ o(D) . . . . . . . . . . . . ∼(D) . . . . . . . . . . . . 8.4 Classes Nω of Delta-Subharmonic Functions with ω ∈ o 8.5 Embedding of the Classes Nω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Boundary Properties of Meromorphic Classes N {ω} ⊂ N . . . . . . . . . . . . . . . . 8.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163 163 172 177 180 185 187 194
9
Banach Spaces of Functions Delta-Subharmonic in the Unit Disc . . . . . . . . . . 9.1 Banach Spaces of Green-Type Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Banach Spaces of Potentials Formed by ∼ bω . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Smaller Banach Spaces of Potentials Formed by ∼ bω . . . . . . . . . . . . . . 9.1.3 Banach Spaces of Potentials Formed by bω . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Smaller Banach Spaces of Potentials Formed by bω . . . . . . . . . . . . . . 9.2 Banach Spaces of Delta-subharmonic Functions in the Disc . . . . . . . . . . . . . 9.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197 198 199 207 209 210 211 214
Contents
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10
Functions of Omega-Bounded Type in the Halfplane . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Blaschke-Type Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Green-Type Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 One More Property of Green-Type Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Representations of Classes of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 10.5 Riesz-Type Representations with a Minimality Property . . . . . . . . . . . . . . . . . 10.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215 215 224 230 234 237 243
11
Subclasses of Harmonic Functions with Nonnegative Harmonic Majorants in the Halfplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Boundary Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
245 245 250 253 260
12
Subclasses of Delta-subharmonic Functions of Bounded Type in the Halfplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Blaschke-Type Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Green-Type Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Weighted Classes of Delta-Subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 12.4 Boundary Property of Subclasses of Functions of Bounded Type in the Halfplane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261 261 269 273 278 288
13
Banach Spaces of Functions Delta-subharmonic in the Halfplane . . . . . . . . . . 13.1 Additional Statements on Green-Type Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . ∼ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Banach Spaces of Potentials P ∼ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Smaller Banach Spaces of Potentials P 13.4 Banach Spaces of Potentials Pω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Banach Spaces of Delta-subharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
289 290 294 298 303 306 308
A
Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 On Representation of Some Classes of Functions Holomorphic in the Unit Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 On a Generalization of the Jensen–Nevanlinna Formula. Canonical Representations of Meromorphic Functions of Unbounded Type . . . . . . . . C.1 On Representability of Some Classes of Entire Functions . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
309 310 328 341 351
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Part I Omega-Weighted Classes of Area Integrable Regular Functions
1
Preliminary Results
1.1
M.M. Djrbashian Operators Lω and His Omega-Kernels
We start by reducing M.M. Djrbashian’s general integrodifferential operator .Lω used in his factorization theory of functions meromorphic in the unit disc [12, 13, 16, 20–22] to some simple forms, which are used in this book. In the mentioned theory, a function .ω is said to be of the class .o, if .ω > 0 in .[0, 1), 1 .ω(0) = 1 and .ω ∈ L [0, 1]. Further, for any .ω ∈ o it is set f p(0) = 1 and
.
p(t) ≡ t t
1
ω(x) dx, x2
0 < t < 1,
(1.1)
and for a measurable in .|z| < R ≤ +∞ function u it is introduced the operator } ⎧ f 1 d iϕ u(tre )dp(t) . .Lω u(re ) ≡ − r dr 0 iϕ
(1.2)
This formula for .Lω was used by M.M. Djrbashian, since it writes his fractional integration and differentiation operator in a united form for the cases .ω(1−0) = 0 and .ω(1−0) = +∞ and even for .ω ≡ 1 when .Lω becomes the identical operator. ∼(D) of parameter-functions .ω, for which Now, we introduce some classes .o(D) and .o the operator .Lω of (1.2) can be written in some simplified forms.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_1
3
4
1 Preliminary Results
Definition 1.1 1◦ . .o(D) is the class of all positive, strictly decreasing, continuously differentiable in .[0, 1) functions .ω, such that .ω(0) = 1 and for some .α > 0
.
ω(t) ≤ O(1 − t)α
as
.
t → 1 − 0.
(1.3)
∼(D) is the set of all functions .ω(t) in .[0, 1), such that: 2◦ . .o
.
(i) .ω > 0 and is continuous and non-decreasing in .[0, 1), (ii) .ω(0) = 1 and .(1 − t)ω(t) → 0 as .t → 1 − 0, (iii) .ω satisfies the Lipschitz condition with .λt ∈ (0, 1] at all points .t ∈ [0, 1). Lemma 1.1 1◦ . Let .ω ∈ o(D), and let u be a subharmonic function in .|z| < R ≤ +∞. Then, the function .Lω u of the form (1.2) coincides with
.
f
1
Lω u(z) ≡ −
(1.4)
u(tz)dω(t)
.
0
◦ .2 .
almost everywhere in .|z| < R. ∼(D), and let .u(z) be a harmonic function in .|z| < R ≤ +∞. Then, the Let .ω ∈ o function .Lω u of the form (1.2) in .|z| < R coincides with Lω u(z) ≡ u(0) + Lω1 U (z),
(1.5)
.
where .Lω1 is of the form (1.4) and is applied to the harmonic function U (z) = |z|
.
f
∂ u(z), ∂|z|
and
1
ω1 (t) = t
ω(x) dx. x
(1.6)
Proof 1◦ . By (1.1), .ω(t) = p(t) − tp' (t). Thus, for any .z = reiϕ with .|z| = r < R
.
f
r
.
0
1 ⎛f r
f Lω u(xe )dx = − iϕ
0
0
⎞
⎡ ⎤ u(txe )dx d p(t) − tp ' (t) , iϕ
(1.7)
1.1 The operators Lω and omega-kernels
5
where the integrals are absolutely convergent. Denoting J (teiϕ ) = tp' (t)
f
r
.
0 < t ≤ 1,
u(txeiϕ )dx,
0
where the last integral obviously is a continuous function for .0 ≤ t ≤ 1, with at most a logarithmic singularity at .t = 0, observe that tp' (t) = t
f
1
ω(x) dx − ω(t) = t x2
.
t
f
1
t
ω' (x) dx. x
Therefore by (1.1) and (1.3) we easily get .J (teiϕ ) → 0 as .t → 1 − 0 and .t → +0. Consequently, by (1.7) f
r
Lω u(xeiϕ )dx = −
.
0
f 1⎡ f f
0
=−
0
0
0
⎡f r ⎤ f 1 1 p' (t)d t u(txeiϕ )dx = −r u(treiϕ )dp(t),
0
◦ .2 .
⎤ ⎡f r ⎤ f 1 u(txeiϕ )dx dp(t) − tp' (t)d u(txeiϕ )dx
r
0
0
and our statement holds by differentiation. Integrating by parts, for any .z = reiϑ with .|z| = r < R we get f .
−r
1
f u(treiϑ )dp(t) = ru(0) + r
0
0
r
⎡
⎤ ⎛ ⎞ ∂ t dt. u(teiϑ ) p ∂t r
Hence ⎧ f r⎡ ⎤ ⎛ ⎞ } ∂ ∂ t u(teiϑ ) p dt r ∂t r ∂r 0 } ⎤⎧ ⎛ ⎞ f r⎡ ∂ t '⎛ t ⎞ t iϑ − p dt, u(te ) p = u(0) + r r r ∂t 0
Lω u(reiϑ ) = u(0) +
.
and the equality .p(x) − xp' (x) = ω(x) leads to formulas (1.5) and (1.6).
u n
Remark 1.1 The general integral operator .Lω (1.4) in particular cases becomes the classical integral operators of Riemann–Liouville, Hadamard [32], Erdélyi [24]–Kober [60] and many other operators.
6
1 Preliminary Results
∼(D) the application of the It is easy to verify that in both cases .ω ∈ o(D) and .ω ∈ o operator .Lω to a holomorphic in a disc .|z| < R ≤ +∞ function f 1 means multiplication of its Taylor series coefficients by the moments .A0 = 1, .Ak = k 0 x k−1 ω(x)dx, .k ≥ 1, i.e. if f (z) =
∞ Σ
.
ak zk ,
Lω f (z) =
then
k=0
∞ Σ
ak Ak zk ,
k=0
where the function .Lω f is holomorphic in the same disc, since .limk→∞
.
√ k
Ak ≤
√ k
/ k
f
k
1
ω(x)dx → 1
as
√ k
Ak = 1. Indeed,
k → ∞.
(1.8)
0
On the other hand, for any .δ ∈ (0, 1)
.
√ k
/ Ak ≥
k
f
1
k
/ f
x k−1 ω(x)dx ≥ δ
1
1− k1 k
δ
ω(x)dx → δ
as
k → ∞,
(1.9)
δ
√ and the passage .δ → 1 − 0 gives .lim infk→∞ k Ak ≥ 1. This means that .Lω transforms the holomorphic in a neighborhood of the origin functions to functions of the same kind, and this mapping is one-to-one, since the converse transform means just a division of the Taylor coefficients by .Ak , which again does not change the convergence radius. Further, for .ω(x) = (1 − x)α / r(1 + α) (−1 < α < +∞) the operator .Lω becomes the classical Riemann–Liouville fractional integrodifferentiation with integration over the complex interval .[0, z]. Namely, for any .z = reiϑ formulas (1.4) and (1.5) respectively take the forms L (1−x)α u(reiϑ ) =
.
r(1+α)
r −α r(α)
f
L (1−x)α u(reiϑ ) = u(0) + r(1+α)
r
(r − t)α−1 u(teiϑ )dt,
0
r −α r(1 + α)
f
r 0
(r − t)α
0 < α < +∞,
∂ u(teiϑ )dt, ∂t
−1 < α < 0.
The next theorem proves that the operator .Lω is a one-to-one mapping in some wider sets of functions in starshaped, regular domains.1 Namely, we shall apply .Lω to deltasubharmonic functions, i.e., differences of two subharmonic functions. Note that the equality .u = v of two delta-subharmonic functions .u = u1 − u2 and .v = v1 − v2 means 1 A domain .G ⊂ C is said to be starshaped, if it contains the closed straight line interval .[0, z] along with any point .z ∈ G. For the definition of regular domains, see [33], Section 2.6.2, also formula (0.5). For a simpler case, see [31], Chapter 1, formula (1.6), also Theorem 1.1.
1.1 The operators Lω and omega-kernels
7
the equality .u1 + v2 = v1 + u2 . Besides, the subharmonic functions are a generalization of .log |f | of holomorphic functions f , while the delta-subharmonic functions are a generalization of .log |f | of meromorphic functions f . Theorem 1.1 Let .D ⊆ C be a star-shaped, regular domain. Then the following statements are true: 1◦ . Let .ω ∈ o(D) and let .Lω be defined by (1.4), then:
.
(i) If u is a harmonic function in D, then also the function .Lω u is harmonic in G. Besides, .Lω u ≡ 0 in D if and only if .u ≡ 0 in D. (ii) If u is a subharmonic function in D, with an associated Riesz measure .ν such that .minζ ∈supp ν |ζ | = d0 > 0, then the function .Lω u is continuous and subharmonic in D. (iii) If u and v are delta-subharmonic in D and the supports of their Riesz measures are distanced from the origin by some .d0 > 0, then .Lω u and .Lω v are deltasubharmonic in D and .Lω u ≡ Lω v in D if and only if .u ≡ v in D. ∼(D) and let .Lω be that defined by (1.5) and (1.6). 2◦ . Let .ω ∈ o
.
(i' ) If u is a harmonic function in D, then also the function .Lω u is harmonic in D. Besides, .Lω u ≡ 0 in D if and only if .u ≡ 0 in D.
.
(ii' ) If u is a subharmonic function in D, with an associated Riesz measure .ν such that .minζ ∈supp ν |ζ | = d0 > 0, then the function .Lω u is continuous and superharmonic in .D \ supp ν.
.
Proof 1◦ .
.
(i) If u is a real, harmonic function in D, then the function .f = u+iv, where v is the harmonic conjugate of u, is holomorphic in D, and it is easy to verify the validity of the Cauchy–Riemann polar equations for .Lω f at any point .z ∈ D. Thus, .Lω f is holomorphic and .Lω u is harmonic in D. Further, evidently f is holomorphic in a neighborhood .|z| < ρ of the origin, where also .Lω f is holomorphic, and the identity .Lω u ≡ Re Lω f ≡ 0 in .|z| < ε implies .f ≡ iC, and .u ≡ 0 in D.
8
1 Preliminary Results
(ii) Assuming that u is subharmonic in D, .ν is its associated Borel measure and .min{|ζ | : ζ ∈ suppν} = d0 > 0, observe that for any .δ ∈ (0, 1) and .R ∈ (0, +∞) the Riesz representation is true in the domain .δDR = {δz : z ∈ D, |z| < R}: ff u(z) = −
G(z, ζ )dν(ζ ) +
.
δDR
1 2π
f u(s) ∂δDR
∂G(s, z) ds, ∂n
z ∈ δDR ,
where G is the Green function of the domain .δDR , .∂/∂n is differentiation along the outer normal and ds is the curve length element (see, e.g. [31], Ch. I, Sec. 2). Since .ν is bounded in .δDR , we can write the latter formula also in the form ff u(z) =
.
⎪ z ⎪⎪ ⎪ log ⎪1− ⎪dν(ζ ) +U (z) ≡ P ∗ (z) +U (z), ζ δDR
z ∈ δDR ,
(1.10)
where .P ∗ is subharmonic and U is harmonic in .δDR , and hence also .Lω U is harmonic in .δDR . Besides, .P ∗ is harmonic in .δDR \ supp ν, since its integral is absolutely and uniformly convergent inside ⎪ any compact .K ⊂ δDR \ supp ν ⎪ due to the obvious inequality .⎪ log |1 − z/ζ |⎪ ≤ M1 < +∞ in .K, where .M1 is a constant depending on the distance from .K to .δDR \ supp ν. Assuming now that .K ⊂ δDR is any compact, we shall prove that the function ∗ .Lω P is continuous in .K and Lω P ∗ (z) =
.
ff
⎪ z ⎪⎪ ⎪ Lω log ⎪1 − ⎪dν(ζ ), ζ ζ ∈δDR
z ∈ δDR .
(1.11)
To this end, observe that for any .ζ ∈ δDR , .|ζ | ≥ d0 > 0, the function f 1 ⎪ ⎪ tz ⎪⎪ z ⎪⎪ ⎪ ⎪ log ⎪1 − ⎪dω(t) .J (z) ≡ Lω log ⎪1 − ⎪=− ζ ζ 0
(1.12)
is continuous in .δDR . Indeed, if the compact .K does not intersect ⎪ with the infinite interval .lζ ≡ {z : arg z = arg ζ, |ζ | ≤ |z| < +∞}, then .log ⎪1 − tz/ζ | ≤ M2 < +∞ (0 ≤ t ≤ 1), where .M2 is a constant depending solely on the distance from .K to the mentioned interval. Hence, J is harmonic in .K and, consequently, in .δDR \ lζ . For proving the continuous extension of J to .lζ , by integration by parts we get f 1 f 1 ⎪ z ⎪⎪ ω(t) ω(t) ⎪ dt = −Re dt, Lω log ⎪1 − ⎪ = −Re ζ /z − t λ −t ζ 0 0
.
λ=
ζ , z
(1.13)
where the last Cauchy -type integral is understood in the sense of its principal value for .λ ∈ [0, 1]. When a complex .λ tends to a point of .(0, 1], then .z = ζ /λ tends to a point of .lζ , and the continuity of the Cauchy- type integral when .λ
1.1 The operators Lω and omega-kernels
9
crosses .lζ holds by the properties of .ω ∈ o(D) and the well-known properties of the Cauchy-type integrals (see, eg., [29], Sec. 4.2, 4.4, 8.1). So, the function .Lω log |ζ −z| is continuous in .δDR and harmonic in .δDR \lζ . Besides, .Lω log |ζ − z| is subharmonic in .δDR , which is easy to verify by applying .Lω to both sides of the inequality for .log |ζ − z|, its integral mean and changing the integration order. To prove formula (1.11), observe that this formula holds at least in .|z| < d0 by applying .Lω to .P ∗ , due to the absolute convergence of its integral inside .|z| < d0 . On the other hand, .P ∗ is harmonic outside of the support of .ν, and hence, U ∗ is harmonic in the star-shaped domain .D \ .Lω P R ζ ∈supp ν lζ , and the righthand side integral is harmonic in the same domain and continuous in .δDR , due to its absolute and uniform convergence of its integral inside the mentioned domain and the already proved properties of .Lω log |ζ − z|. Hence, formula (1.11) is true U in .DR \ ζ ∈supp ν lζ by the uniqueness of harmonic function, it is true also in the whole .δDR , where .P ∗ has a continuous extension. Thus, the proof of the statement (ii) is complete by the arbitrariness of .δ and R. (iii) If u and v are subharmonic in D and .Lω u ≡ Lω v in D, then for u and v the Poisson-Jensen formula is true in the finite, regular, starshaped domain .δD. Besides, in .δD, there are decompositions .u = U + P1 and .v = V + P2 , where U and V are harmonic functions and .P1,2 are Green potentials. Consequently, ( ) ( ) ( ) ≡ Lω P2 − P1 in .δD, where .Lω U − V is a harmonic function, .Lω U − V ( ) while .Lω P2 − P1 is not harmonic in .δD for .δ close enough to 1 and the ( ) associated measures of u and v are different. So, .P1 ≡ P2 . Hence, .Lω U − V ≡ ( ) Lω P2 − P1 ≡ 0, which implies .U ≡ V in .δD by the already proved statement (ii), and consequently .u ≡ v in .δD. Exhausting D by the domains .δD, where we let .δ → 1 − 0, we get .u ≡ v in the whole D. Further, for two delta-subharmonic functions .u = u1 − u2 and .v = v1 − v2 , the identity .u ≡ v is understood in the sense that .u1 + v2 ≡ v1 + u2 . Hence, the identity .Lω u ≡ Lω v means ( ) ( .Lω u1 + v2 ≡ Lω v1 + u2 ), where .u1 + v2 and .v1 + u2 are subharmonic in D. Consequently .u = u1 − u2 ≡ v1 − v2 = v, i.e. .u ≡ v. 2◦ . (i.' ) The proof is the same as that of the statement .1◦ (i). (ii.' ) The proof is almost the same as that of .1◦ (ii). Therefore, we show only the differences. Applying the operator of the form (1.5)–(1.6) to .log |1 − z/ζ |, ∼(D) we get the same formula (1.13) with the only differences that .ω(t) ∈ o and the last integral is with the sign “+.” This excludes continuity at the point .z = ζ (λ = 1), since .ω(t) does not vanish as .t → 1 − 0. The rest of the proof differs from that of .1◦ (ii) only by a change of the domain .δDR by U .δDR,ε = δDR \ ζ ∈supp ν {ζ : |z − ζ | < ε}, where the superharmonicity of ∗ .Lω P in D is true due to the mentioned integral sign difference, and then letting .δ → 1 − 0 and .ε → +0. u n
.
The next lemma gives a decomposition of the operator .Lω .
10
1 Preliminary Results
Lemma 1.2 Let .ω1,2 ∈ o(D) and additionally, let .ω2 (x) ≡ 1 for .0 ≤ x ≤ ε < 1 with a fixed .ε. Then f 1 ⎛ ⎞ f 1 ⎛ ⎞ x x dω2 (t) ∈ o(D) dω1 (t) = − (1.14) .ω3 (x) ≡ − ω2 ω1 t t x x and ' .ω3 (x)
f
1
=− x
ω2'
f 1 ⎛ ⎞ ⎛x ⎞ x ' dt dt ' ω2 (t) . ω1 (t) = − ω1' t t t t x
(1.15)
Besides, if a function u is subharmonic in a starshaped domain D, and the associated Borel measure of u is supported in a ring .0 < d0 ≤ |ζ | < 1, then Lω3 u(z) = Lω1 Lω2 u(z) = Lω2 Lω1 u(z),
.
z ∈ D.
(1.16)
Proof The second equality in (1.14) holds by an integration by parts and a simple change of variable. Further, the integrals in (1.14) are uniformly convergent with respect to .x ∈ [0, 1], and hence .ω3 (0) = 1. If x sufficiently close to 1, then by (1.3) f
1
ω3 (x) =
ω2 (t)dω1
.
x
⎛x ⎞ t
≤ ω1 (x) ≤ O(1 − x)α1 ,
where .α1 > 0 is that of (1.3) for .ω1 . Further, changing the variable as .λ = x/t, by a well-known differentiation formula we get equalities (1.15): ⎪1 f 1 ⎛ ⎞ f 1 ⎛ ⎞ ⎛x ⎞ ⎪ x ' dλ dλ ' ' x ' ⎪ ω1 (λ)⎪ ω1 (λ) =− ω1 (λ) . = −ω2 − ω2 ω2' λ λ λ λ λ x x λ=x
' .ω3 (x)
Hence, .ω3' < 0 in .[0, 1) and is continuous in .[0, 1], and the equalities (1.14) hold. At last, u n the equalities (1.16) are easy to verify by (1.15). As we have seen, to get a holomorphic in .D function, which becomes f after the application of .Lω , it is just necessary to divide the Taylor coefficients of f by .Ak . This is the way in which the M.M. Djrbashian Cauchy and Schwarz type .ω-kernels are introduced in .D: Cω (z) =
.
∞ Σ zk , Ak k=0
and
Sω (z) = 2Cω (z) − 1,
z ∈ D,
(1.17)
1.1 The operators Lω and omega-kernels
11
where f A0 ≡ 1.
.
1
Ak ≡ k
x k−1 ω(x)dx,
k ≥ 1,
(1.18)
0
or f
1
Ak = −
.
x k dω(x) (k ≥ 0),
if ω(0) = 1 and ω(1) = 0.
(1.19)
0
Note that the functions .Cω and .Sω are holomorphic in .D because of the relations (1.8) and (1.9), besides for any .α ∈ (−1, +∞) C (1−x)α (z) ≡ Cα (z) =
.
r(1+α)
1 , (1 − z)1+α
S (1−x)α (z) ≡ Sα (z) = 1 − r(1+α)
2 (1 − z)1+α
(1.20)
and Lω Cω (z) = C0 (z) =
.
1 , 1−z
Lω Sω (z) = S0 (z) =
1+z . 1−z
In this book, also an analog of the operator .Lω with infinite integration contour is used for constructing in a sense similar theory in the upper halfplane .G+ = {z = x + iy : 0 < y < +∞}, where the Taylor series apparatus is replaced by that of the Laplace transform. It is natural to use the notation .Lω also for the new operator and again call it M.M. Djrbashian operator. ft For any acceptable functions u in .G+ , .ω in .(0, +∞) and .ω1 (t) = 0 ω(λ)dλ we set f
+∞
Lω u(z) ≡
.
0
u(z + it)dω(t) or
⎞ ⎛ ∂ u(z) . Lω u(z) ≡ Lω1 − ∂y
(1.21)
The way in which this operator acts to Laplace transforms is very similar to that in which the considered before .Lω was acting to Taylor series. Namely, if for some acceptable functions .ω and .μ in .(0, +∞) f
+∞
f (z) =
.
0
f eizt dμ(t), then Lω f (z) = 0
+∞
⎛ f eizt t
+∞
⎞ e−tλ ω(λ)dλ dμ(t)
0
for any .z ∈ G+ . The properties of this operator are more complicated than those of .Lω for the unit disc theory. Their study is given in those chapters of the book where they are used. We just notice that the introduced operator is a generalization of the Liouville fractional
12
1 Preliminary Results
integrodifferentiation, which holds as a particular case. Namely, for an acceptable function u(z) ≡ u(x + iy) given in .G+
.
L
tα r(1+α)
.
L
tα r(1+α)
u(z) =
f
1 r(α)
u(z) = L
+∞
(t − y)α−1 u(x + it)dt,
0 < α < +∞,
y
⎞ ⎛∂ u(z) , ∂y
t 1+α r(2+α)
−1 < α < 0.
Also, we introduce a Cauchy-type kernel .Cω in .G+ , which again is natural to call M.M. Djrbashian kernel. To introduce this kernel, we give some definitions. Definition 1.2 .oα (G+ ) .(−1 ≤ α < +∞) is the class of functions .ω given in .[0, +∞) and such that: (i) .ω - (is non-decreasing) in .(0, +∞), .ω(0) = ω(+0) and there exists a sequence .δk ↓ 0 such that .ω(δk ) ↓ (is strictly decreasing); (ii) .ω(t) x t 1+α for .A0 ≤ t < +∞ and some .A0 ≥ 0. Note that .f x g means that .m1 f ≤ g ≤ m2 f for some constants .m1,2 > 0. Evidently, if ω ∈ oα (G+ ) (α ≥ −1), then (ii) is true for any .A ∈ (0, A0 ].
.
+ Definition 1.3 .oN α (G ) is the set consisting of .ω ≡ 1 and all decreasing, continuous functions .ω > 0 in .(0, +∞), such that
ω(x) x x α
.
− 1 < α < 0 and any
for some
x ≥ A0 > 0,
where .A0 is a fixed number. Besides, we set f
x
ω1 (x) =
.
ω(t)dt < +∞,
0 < x < +∞.
0 + Assuming that .ω ∈ oα (G+ ) (α ≥ −1) or .ω ∈ oN α (G ) .(−1 < α < 0), we define
f Cω (z) =
+∞
.
eitz
0
f
dt , Iω (t)
Iω (t) = t
+∞
e−tx ω(x)dx,
z ∈ G+ ,
(1.22)
0
where we may write f Iω (t) =
.
0
+∞
e−tx dω(x),
if ω(0) = 0.
(1.23)
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
13
Note that, being an obvious generalization of the ordinary Cauchy kernel in the onedimensional case of .D, the .ω-kernel .Cω was first used A.H. Karapetyan in [57], where it was constructed in the multidimensional case of tube domains. + Further, note that for .ω ∈ oα (G+ ) the function .Cω and for .oN α (G ) the functions + .Cω and .Cω1 are holomorphic in .G . Indeed, for any .k ≥ 1 the integral of .Cω uniformly converges in .G+ δk = {z : Im z > δk } because of the estimate ⎪f ⎪ ⎪ ⎪ ⎪. Iω (t)⎪ ≥ ⎪ ⎪
δk−1
e δk
−tx
⎪ ⎪ ⎪ ⎪ dω(x)⎪⎪ ≥ e−tδk ⎪ω(δk−1 ) − ω(δk )⎪ > 0,
k ≥ 1,
and the same estimate with .ω1 . Besides, one can see that for any .α > −1 C
.
tα 1+α
1.2
(z) ≡ C0 (z) =
1 (−iz)1+α
and
Lω Cω (z) = C0 (z) =
1 , −iz
z ∈ G+ .
(1.24)
Evaluation of M.M. Djrbashian Cω -Kernels
This section gives some useful asymptotic estimates of .Cω -kernels in the unit disc and in the halfplane. As M.M. Djrbashian often stated, one of the most significant problems related to his factorization theory is the evaluation of the .Cω -kernels. The main assumption was that under some additional conditions on the behavior of the parameter-functions .ω in .(0, 1) or in .(0, +∞), the following estimates have to be true: |Cω (z)| ≤
.
M (z ∈ D), |(1 − z)2 ω' (|z|)|
|Cω (z)| ≤
M (z ∈ G+ ) |z2 ω' (Im z)|
(1.25)
which are natural to expect because of the equalities (1.20) and (1.24). Here, in the first subsection, we use a united evaluation method to prove (1.25) for both kernels, which we use under some conditions in which the derivative of .ω in .(0, 1) (or .(0, +∞)) decreases as .x → 1 − 0 (or .x → +0) not more rapidly than the function α α .(1 − x) (or .x ). The found estimates are exact on the positive radius .z = r ∈ (0, 1) and on the imaginary half-axis .z = iy, .y ∈ (0, +∞). In the second subsection, the .Cω -kernels are evaluated for certain scales of .ω, which are exponentially decreasing as .x → 1 − 0 (or .x → +0). These estimates differ from (1.25), though they also are exact on the positive radius of .D and on the imaginary half-axis of + .G .
14
1 Preliminary Results
ω Decreases Not More Rapidly than a Power Function
1.2.1
We start by the “model” argument on evaluation of the .Cω -kernel for the halfplane. Beforehand some necessary technical apparatus is to be prepared. We shall use the below easily verifiable inequalities, with .- and .- meaning the non-decreasing and non-increasing of a function. Namely, for any monotone in .(0, +∞) function .ϕ > 0 f
+∞
.
e−ty ϕ(t)dt
/f
1/y
1/y 0
⎧ ⎨≥ M , if ϕ 1 e−ty ϕ(t)dt , ⎩≤ M2 , if ϕ -
y > 0,
(1.26)
where .M1,2 are some positive constants. Besides, for small enough values .v > 0 f
+∞
.
e−tv ϕ(t)d[t]
/f
1/v
+0
1/v
⎧ ⎨≥ M , 3 e−tv ϕ(t)d[t] ⎩≤ M4 ,
if ϕ(t) if ϕ(t) -
,
(1.27)
where .[t] means the integral part of t and .M3,4 > 0 are some constants. We shall often use also the inequality .
t 4 t , < 1 − e−t < 31+t 1+t
0 < t < +∞.
The main tool of this section is the following, perhaps known statement. Lemma 1.3 Let .ϕ > 0 be a function defined in .(0, +∞). 1◦ . If . ϕ(t) - but . t −α ϕ(t) - in . (0, +∞) for some . α > 0, then
.
f
+∞
.
e−tx ϕ(t) dt x
0
ϕ(1/x) , x
0 < x < +∞.
(1.28)
2◦ . If .ϕ(t) - but .(1 − e−t )−α ϕ(t) - for some .α > 0, then for small enough .v > 0
.
f
+∞
.
+0
e−tv ϕ(t) d[t] x
ϕ(1/v) . v
(1.29)
3◦ . If . ϕ(t) - in . (0, +∞) but .t δ ϕ(t) - or . (1 − e−t )δ ϕ(t) - for a . δ ∈ (0, 1) , then (1.28) and (1.29) are true, respectively.
.
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
15
Proof 1◦ . Evidently
.
f
+∞
.
e−tx ϕ(t)dt ≥ x α ϕ(1/x)
0
f
1/x 0
e−tx t α dt =
ϕ(1/x) x
f
1
e−λ λα dλ.
0
On the other hand, by (1.26) f
+∞
e
.
−tx
0
f f ⎛ ⎛ 1 ⎞ +∞ −tx 1 ⎞ ϕ(1/x) +∞ −λ α e λ dλ. ϕ(t)dt ≤ 1+ e ϕ(t)dt ≤ 1+ M1 x M1 1/x 1
2◦ . If .v > 0 is small enough, then
.
f
+∞
.
+0
e−tv ϕ(t)d[t] ≥
f
1/v
+0
> M '' ϕ(1/v)
f
e−tv ϕ(t)d[t] ≥ M ' ϕ(1/v) 1/v
f
1/v +0
e−tv
⎛
t 1+t
⎞α d[t]
e−tv d[t]
1/2
⎤ ⎡ = M '' e−v ϕ(1/v) 1 + ev + · · · + e−v([1/v]−1) = M '' e−v ϕ(1/v)
ϕ(1/v) 1 − e−v[1/v] > M ''' . −v 1−e v
On the other hand, if .v > 0 is small enough, then by (1.27) f
+∞
.
+0
e
−tv
f 1 ⎞ +∞ −tv ϕ(t)d[t] ≤ 1 + e ϕ(t)d[t] M3 1/v f +∞ ⎞−α ⎛ 1 ⎞⎛ 1 − e−1/v ϕ(1/v) e−tv (1 − e−t )α d[t] ≤ 1+ M3 1/v f +∞ ϕ(1/v) ≤ M I V ϕ(1/v) e−tv d[t] ≤ M V . v 1/v ⎛
3◦ . The proof is similar and even more simple.
.
We start the evaluations by the case of the halfplane kernel .Cω of (1.22).
u n
16
1 Preliminary Results
Theorem 1.2 Let .ω > 0 be a non-decreasing, continuously differentiable function in (0, +∞), such that
.
1◦ . .ω(+0) = 0 and . lim e−εx ω(x) = 0 for any . ε > 0,
.
x→+∞
◦ .2 .
(i)
' .ω (x)
- but . x −α ω' (x) - for some . α > 0 or, alternatively,
(ii) .ω' (x) - but . x δ ω' (x) - for some . δ ∈ (0, 1), then Cω (iy) x
.
1 y 2 ω' (y)
0 < y < +∞.
,
(1.30)
If along with .1◦ and .2◦ (i) we have 3◦ .
.
.
ω' (+∞) = +∞ and .x −1 ω' (x) - or .x −1 ω' (x) - but .x −δ ω' (x) - for some .δ ∈ (0, 1), then there is a constant .M (≡ Mω ) > 0 for which |Cω (z)| ≤
.
M |z|2 ω' (y)
,
z = x + iy ∈ G+ .
(1.31)
Proof If .2◦ (i) is true, then by (1.22)–(1.23) and (1.28) f Cω (iy) x
+∞
.
e−yt
0
t ω' (1/t)
dt,
0 < y < +∞.
(1.32)
But .t[ω' (1/t)]−1 - and .x −α ω' (x) -. Hence .
1 t t −α = = t −(1+α) ' -. (1/t)−α ω' (1/t) ω' (1/t) ω (1/t)
Thus, the function .t[ω' (1/t)]−1 satisfies the condition .1◦ of Lemma 1.3, and (1.30) follows from (1.32) by (1.28). Under the assumption .2◦ (i), (1.30) is proved similarly. For proving (1.31), observe that the function .ϕ(t) = t n ω' (t) satisfies the condition .1◦ of Lemma 1.3 for any .n ≥ 0. Hence, an integration by parts gives 1 .Cω (z) = (iz)2
⎧f
f +∞
+∞
e
izt
0
f −2 0
e−σ t σ 2 ω' (σ )dσ 0 ⎤2 dt ⎡f +∞ −σ t ' e ω (σ )dσ 0 +∞
⎤2 } e−σ t σ ω' (σ )dσ 1 {I1 − I2 } , dt ≡ ⎤3 ⎡f (iz)2 +∞ −σ t ' e ω (σ )dσ 0
⎡f +∞ eizt
0
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
17
since .ω' (+∞) = +∞. One can easily verify that by (1.28) |I1,2 | ≤ M
.
'
f
+∞ 0
e−yt dt , tω' (1/t)
0 < y < +∞.
⎡ ⎡ ⎤−1 ⎤−1 If .x −1 ω' (x) - (i.e., . tω' (1/t) -) but .t 1−δ tω' (1/t) -, then |I1,2 | ≤ M ''
.
1 M '' 1 = ' , −1 ' ω (y) y y ω (y)
0 < y < +∞,
by Lemma 1.3(.3◦ ). Further, if .x −1 ω' (x) -, then .t −1−β ω' (t) - for a .β > 0 by .2◦ (i), and ◦ .I1,2 has the same estimate by Lemma 1.3(.1 ). u n The estimate (1.31) of the previous theorem is proved under the assumption that .ω' -, but it can be used for proving the following theorem for the case .ω' -. Theorem 1.3 Let .ω > 0 be a continuously differentiable, non-decreasing function in (0, +∞), such that
.
1◦ . .x −1 ω(x) -, .x −δ ω(x) - for a .δ ∈ (0, 1) and . ω' (x) -,
.
2◦ . .ω(+∞) = +∞ , but . lim e−εx ω(x) = 0 for any . ε > 0.
.
x→+∞
Then, there is a constant .M (≡ Mω ) > 0 for which |Cω' (z)| ≤
.
M , |z|2 yω' (y)
z = x + iy ∈ G+ .
Proof First, we shall verify that the function f F (t) =
.
t
ω(λ)dλ 0 < t < +∞,
0
satisfies the requirements of Theorem 1.2. Indeed, .F - and is continuously differentiable. Further, .F ' (+0) = F (+0) = 0, and . lim e−εt F (t) = 0 for any .ε > 0 since .F (t) ≤ t→+∞
tω(t) and 0 = lim e−εt ω(t) ≥ lim e−εt t −1 F (t) ≥ 0 for any
.
t→+∞
t→+∞
ε > 0.
18
1 Preliminary Results
Thus, F satisfies the condition .1◦ of Theorem 1.2, and the requirements .2◦ (i) and .3◦ evidently are fulfilled. Now observe that by Theorem 1.2 |CF (z)| ≤
.
M , |z|2 yω' (y)
z = x + iy ∈ G+ ,
since f
y
ω(y) =
.
ω' (t)dt ≥ yω' (y) ≥ 0,
0 < y < +∞.
0
It remains to observe that f CF (z) =
.
0
+∞
f +∞ 0
f
eizt dt e−σ t F ' (σ )dσ
+∞
= 0
f +∞ 0
eizt dt e−σ t ω(σ )dσ
= −iCω' (z). u n
The following two theorems are true for the disc kernels (1.17). Theorem 1.4 Let .ω > 0 be a non-increasing, continuously differentiable function in (0, 1), such that
.
1◦ . .ω(1 − 0) = 0 and . ω(+0) = 1, ◦ ' −α |ω' (x)| - for an . α > 0 or, alternatively, .2 . (i) .|ω (x)| - but . (1 − x) ' (ii) .|ω (x)| - but . (1 − x)δ |ω' (x)| - for a .δ ∈ (0, 1). .
Then, Cω (r) x
.
1 , (1 − r)2 |ω' (r)|
0 < r < 1.
(1.33)
If along with .1◦ and .2◦ (i) 3◦ . . ω' (1−0) = 0 and .(1−x)−1 |ω' (x)| -, or .(1−x)−1 |ω' (x)| - but .(1−x)−δ |ω' (x)| for a .δ ∈ (0, 1),
.
then, there is a constant .M (≡ Mω ) > 0 such that |Cω (z)| ≤
.
M , |1 − z|2 |ω' (|z|)|
z ∈ D.
(1.34)
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
19
Proof For proving (1.33), we shall use the representation ∞ Σ zk ≡ 1 + S1 (z), Ak
Cω (z) = 1 +
.
f z ∈ D,
1
Ak = −
x k ω' (x)dx,
k ≥ 1.
0
k=1
(1.35) As .Cω is holomorphic in .D, it suffices to prove (1.33) for r close enough to 1. To this end, we introduce the function .ω∗ (t) ≡ |ω' (e−t )| (0 ≤ t < +∞) and extend .Ak ≡ A(k) to the whole half-axis .[0, +∞) as follows: f
1
A(t) ≡
.
f
'
+∞
x |ω (x)|dx = t
0
e−x(t+1) ω∗ (x)dx,
0 ≤ t < +∞.
0
Evidently, .A(t) is a continuous, decreasing function on .[0, +∞), and f S1 (r) =
+∞
.
+0
e−vt
d[t] , A(t)
r = e−v ,
0 < v < +∞.
Now, observe that the function .ω∗ has the following properties, which are similar to those of .ω' in Theorem 1.2: (i) .ω∗ (t) - but . (1 − e−t)−α ω∗ (t) - for an . α > 0 or, alternatively, (ii) .ω∗ (t) - but . (1 − e−t )δ ω∗ (t) - for a . δ ∈ (0, 1). These properties make it possible to apply for small enough .v > 0 an argument similar to that in the beginning of the proof of Theorem 1.2. This leads to (1.33) for r close enough to 1, and consequently for all .r ∈ (0, 1). Proceeding to the proof of (1.34), one can verify that 1 .Cω (z) = 1 + (1 − z)2 +
⎧
z z2 z2 + −2 A1 A2 A1
}
∞ Σ Ak−1 Ak−2 − 2Ak Ak−2 + Ak Ak−1 k 1 z , 2 Ak Ak−1 Ak−2 (1 − z)
z ∈ D.
k=3
Hence Cω (z) =
.
O(1) + S2 (z) , (1 − z)2
z ∈ D,
(1.36)
20
1 Preliminary Results
where S2 (z) =
∞ Σ
.
b(k)zk ,
b(k) =
k=3
Ak−1 Ak−2 − 2Ak Ak−2 + Ak Ak−1 . Ak Ak−1 Ak−2
If .z = eiw (w = u + iv ∈ G+ ) then .|z| = e−v , and |S2 (z)| ≤
∞ Σ
.
f |b(k)||z|k =
+∞
e−xv |b(x)|d[x],
z ∈ D.
(1.37)
3−0
k=3
One can be convinced that f +∞
0 < b(x) =
.
f t/2 e−tx et/2 dt 0 o(σ, t)ω∗ (t/2 − σ )ω∗ (t/2 + σ )dσ 0 , f +∞ f +∞ f +∞ e−t (x+1) ω∗ (t)dt 0 e−tx ω∗ (t)dt 0 e−t (x−1) ω∗ (t)dt 0
⎨ } where .o(σ, t) = (eσ + e−σ )(1 + e−t ) − 2e−t/2 (e2σ + e−2σ ) . But .e2σ + e−2σ > eσ + e−σ for .σ > 0. Consequently, f t/2 ∗ ) t 2 ∗ 1+t dt 0 ω (t/2 − σ )ω (t/2 + σ )dσ M1 f +∞ f +∞ f +∞ e−t (x+1) ω∗ (t)dt 0 e−tx ω∗ (t)dt 0 e−t (x−1) ω∗ (t)dt 0 f +∞
0 < b(x) ≤
.
0
M1 ≤ 2 f +∞ 0
e−tx et
(
f +∞
e−t (x−1) t 3 [ω∗ (t)]2 dt M1 α1 (x) ⎡f ⎤2 ≡ 2 α2 (x) +∞ e−t (x+1) ω∗ (t)dt 0 e−tx ω∗ (t)dt 0
(1.38)
for .x ≥ 3. A suitablefestimate of .α2 is obtained by a simple argument: as .ω∗ -, the +∞ withdrawal of the part . 1/x of the integrals gives α2 (x) ≥
.
M2 ∗ [ω (1/x)]3 , x3
x ≥ 3.
(1.39)
For estimating .α1 , we represent it in the form f
1
α1 (x) =
e
.
−t (x−1) 3
∗
t [ω (t)] dt + e
0
≡ A1 (x) + A2 (x)
2
−(x−1)
f
+∞ 0
e−t (x−1) (t + 1)3 [ω∗ (t + 1)]2 dt
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
21
and observe that f A1 (x) x
.
1
e−tx (1 − e−t )3 [ω∗ (t)]2 dt ≤
f
0
+∞
e−tx (1 − e−t )3 [ω∗ (t)]2 dt.
(1.40)
0
As .ω∗ - and .(1 − e−x )−α ω∗ (x) - for an .α > 0, A2 (x) ≤ e
.
−x+1
(1 − e
−1 −2α
)
∗
f
[ω (1)]
+∞
2
e−t (x−1) (t + 1)3 dt
0 such that
.
|Cω' (z)| ≤
.
M , |1 − z|2 (1 − |z|)|ω' (|z|)
z ∈ D.
Proof It is not difficult to verify that the function f F (x) =
/f
1
ω(t)dt
.
1
0 < x < 1,
ω(t)dt, 0
x
satisfies all requirements of the previous theorem on .ω. Therefore |CF (z)| ≤
.
M , |(1 − z)2 F ' (|z|)|
z ∈ D.
(1.41)
Further, by (1.17) f1
x k ω(x)dx Ak+1 (ω) Ak = Ak (F ) = − 0f 1 , =− f1 (k + 1) 0 ω(x)dx 0 ω(x)dx
.
k ≥ 1.
Hence Cω' (z) =
.
∞ Σ 1 CF (z) 1 zk 1 1 + f1 = −f 1 + f1 + , A1 (ω) Ak (F ) A1 (ω) 0 ω(x)dx k=1 0 ω(x)dx 0 ω(x)dx
and by (1.41) |Cω' (z)| ≤
.
M1 + M2 , |(1 − z)2 F ' (|z|)|
z ∈ D.
(1.42)
f1 But .ω(x) = − x ω' (t)dt ≥ (1 − x)|ω' (x)| (0 < x < 1) since .|ω' | = −ω' -. Therefore ' ' .|F (x)| = M3 ω(x) ≥ (1 − x)|ω (x)| (0 < x < 1). This and (1.42) complete the proof. u n Remark 1.2 The used evaluation method is applicable also, if the condition of not more than power decrease is posed not on the derivative but immediately on the function .ω. This leads to some estimates similar to (1.25), where the power function in the denominator is linear and .ω' is replaced by .ω. However, we prefer the form (1.25) with .ω' as it turns to be better for later applications.
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
23
Remark 1.3 The .ω-function-parameters for which the .Cω -kernels are evaluated in Theorems 1.2–1.5 are strictly contained in the class of functions of regular behavior. Although, among them there are .ω-functions in .[0, 1) and .(0, +∞), such that |ω' (x)| x (1 − x)α logβ
.
1 1 as x → 1 − 0 and |ω' (x)| x x α logβ as x → +0 1−x x
for any given .α ∈ (−1, +∞) and .β > 0.
ω Decreases Exponentially
1.2.2
We start by two purely technical lemmas. Lemma 1.4 The following asymptotic relation is true for .0 < x < +∞: ⎧ −2√x e . . . (1 + x)(1+2α)/4 , . 1+α . x √ . f +∞ ⎨ e−2 x −xt−1/t α .oα (x) ≡ e t dt x , (1+α)/2+1/4 . (1 + x) 0 . √ . . . e−2 x . ⎩ log (e + 1/x), (1 + x)1/4 √
Proof Observe that .oα (x) = 2−(1+α) x −(1+α)/2 e−2
x
f +∞ 0
e−
√
if α > −1, if α < −1,
(1.43)
if α = −1. xt dμ(t),
where
⎡⎛ ⎞1+α ⎛ ⎞1+α ⎤ √ √ 1 − t + 2 − t 2 + 4t t + 2 + t 2 + 4t 1+α ⎧ ⎨√ t as t → +0, x ⎩t |1+α| as t → +∞.
μ(t) =
.
This obviously implies (1.43), except the case .α = −1, which is proved similarly.
u n
Lemma 1.5 If .γ > −1, .λ ∈ (−∞, +∞) are any fixed numbers and .δ = 2(γ + λ) + 1, then for .0 < y < +∞ f Iγ ,λ (y) ≡
+∞
.
√
e−yt+2 t t γ (1 + t)λ dt x e1/y y −δ−1/2 (1 + y)δ−1/2−γ , .
(1.44)
(1 + t)1/4 1+y dt x e1/y y −2 . log(e + 1/t) log(e + y)
(1.45)
0
f
+∞ 0
√
e−yt+2
t
24
1 Preliminary Results
Proof It is obvious that .I (y) x y −(1+γ ) as .y → +∞. Let .0 < y < 1/2. Then f Iγ ,λ (y) =
√
1
e−yt+2 t t γ (1 + t)λ dt +
.
f
+∞
e−y(1+t)+2
√ 1+t
(1 + t)γ (2 + t)λ dt
0
0
= O(1) + J (y), where f J (y) x e1/y
+∞
.
e−(
√ √ 2 y(1+t)−1/ y )
(1 + t)γ +λ dt ≡ K(y).
0
Now a change of variable gives ⎧f
⎡(
√ )δ ( √ )δ ⎤ dt 1 − (1 − y) t + 1 + (1 − y) t √ t 0 } f +∞ ⎛ ⎡ ⎞ ⎤ 2 1/y √ δ dt e e1/y √ −t (1−y) y 1 + (1 − y) t √ x δ+1 y + O(ye−1/y ) x δ+1/2 . e + y y t 1
K(y) =
.
1−y e1/y δ+1 y
1
e
−t (1−y) y
2
Hence, a suitable estimate of .Iγ ,λ follows for .0 < y < 1/2. Along with the above obtained estimate for the case .y → +∞, this leads to the desired statement. The relation (1.45) is proved similarly. u n Now we proceed to evaluation of .ω-kernels in the halfplane. Theorem 1.6 Let f ω(t) ≡ ωρ,α (t) =
.
t
e−ρ/σ σ α dσ,
0 < t < +∞,
(1.46)
0
where .ρ > 0 and .α are any fixed real numbers. Then the following estimate is true for the kernel (1.17): ⎧ . . ⎨1 + y,
1 × (1 + y)2+α , .Cω (iy) x y 3 ω' (y) . . ⎩ (1 + y) log−1 (e + y),
if α > −1, if α < −1,
0 < y < +∞.
(1.47)
if α = −1,
If .α > 0, then there is a constant .M ≡ Mρ,α > 0 such that |Cω (z)| ≤ M
.
1+y , |z|2 yω' (y)
z = x + iy ∈ G+ .
(1.48)
1.2 Evaluation of M.M. Djrbashian Cω -Kernels
25
Proof One can see that for any .α ∈ (−∞, +∞) −α
Cω (iy) = ρ
.
f
+∞ 0
eizt/ρ o−1 α (t) dt,
0 < y < +∞.
(1.49)
Hence, using (1.43), (1.44) and (1.45) we come to (1.47). Further, assuming that .α > 0 and twice integrating by parts from (1.49) we derive ρ 2−α .Cω (z) = z2
f
⎡
+∞
e
o''α (t)
izt/ρ
[oα (t)]2
0
−2
⎡ ' ⎤2 ⎤ oα (t) [oα (t)]3
z ∈ G+ .
dt,
Hence, for any .z = x + iy ∈ G+ ρ 2−α .|Cω (z)| ≤ |z|2 ≡
⎧f
+∞
e
−yt/ρ
0
f
o''α (t)
dt + 2 [oα (t)]2
⎡
+∞
e
−yt/ρ
0
o'α (t)
⎤2
}
dt [oα (t)]3
ρ 2−α {A1 (y) + A2 (y)} . |z|2
(1.50)
Using (1.43) we obtain that for .0 < y < +∞ f A1,2 (y) x
+∞
.
e−yt/ρ+2
√
t
t −1+α (1 + t)(3−2α)/4 dt.
0
Thus .A1,2 (y) x
1+y (0 < y < +∞) by (1.44). Hence (1.48) follows by (1.50). yω' (y)
u n
Some similar estimates are true for the kernels .Cω in the unit disc, for the following scale of parameter-functions: f ω(x) ≡ ωρ,α (x) = −K
.
1
ρ
e− 1−t (1 − t)α dt,
0 ≤ x ≤ 1,
(1.51)
x
where .K =
⎛f 1 0
ρ
e− 1−t (1 − t)α dt
⎞−1
and .ρ, α > 0 are any numbers.
Theorem 1.7 For any real .α and .ρ > 0 Cω (r) x
.
1 , (1 − r)3 |ω' (r)|
0 < r < 1.
(1.52)
26
1 Preliminary Results
If .α > 0, then there is a constant .M ≡ Mρ,α > 0 such that |Cω (z)| ≤
.
M , |1 − z|2 (1 − |z|)|ω' (|z|)|
z ∈ D.
(1.53)
Proof We shall use (1.43) and the relation f
+∞
.
e−yt+2
√
t d[t] x eρ/y y −2γ −3/2 ,
y → +0,
ρt γ
(1.54)
a
which is true for any fixed .a, ρ > 0 and .γ ∈ (−∞, +∞). Note that (1.54) is obtained by a separate evaluation of its left-hand side integral from above and from below, using integration by parts, estimation (by getting rid of .[t]), converse integration by parts and by the use of (1.44) (if .γ > −1) or, alternatively, by the repetition of the argument by which (1.44) is proved (if .γ ≤ −1). For proving (1.52), we use formula (1.35) for .Cω , where one can see that ρ
|ω' (x)| = Ke− 1−x (1 − x)α x e
.
− logρ1/x
⎛
log e/x log 1/x
⎞α ,
0 < x < 1.
Hence, by (1.43) f
+∞
Ak x
.
e−tρ(k+1)−1/t t α (1 + t)−α dt x e−2
√
ρ(k+1)
(k + 1)−3/4−α/2 ,
k ≥ 1,
0
and denoting .r = e−v by (1.54) we obtain that for .0 < r < 1 f S(r) x
+∞
e−vt+2
.
√ ρ(t+1)
⎡ ⎤−1 (t + 1)3/4+α/2 d[t] x eρ/v v −α−3 x (1 − r)3 |ω' (r)| .
1−0
So, (1.52) is proved. For proving (1.53) we use the representation (1.36) of .Cω and also the estimates (1.37) and (1.38), which evidently remain true for the considered exponential scale (with .ω∗ (t) ≡ |ω' (e−t )|). By (1.38), in (1.37) 0 < b(x) ≤ M1 α1∗ (x)/α2 (x),
.
2 < x < +∞,
where .α2 is the same as in (1.37) and α1∗ (x) =
f
.
0
+∞
e−tx et t 2 dt
f
t/2 0
ω∗ (t/2 − σ )ω∗ (t/2 + σ )dσ.
1.3 Volterra Equation and Hausdorff Moment Problem
27
One can verify that f
t/2
.
ω∗ (t/2 − σ )ω∗ (t/2 + σ )dσ x e−4ρ/t t 3/2+2α
as
t → +0.
0 √
Therefore, by (1.54) .α1∗ (x) < M2 e−4 ρ(x−1) x −5/2−α (3 < x < +∞), where .M2 > 0 is a √ √ constant. Along with this, .α2 (x) x√e−2 ρ(x+1)−4 ρx x −9/−3α/2 (3 < x < +∞) by (1.54). Consequently, .0 < b(x) < M3 e2 ρx x −1/4+α/2 (3 < x < +∞) for a constant .M3 > 0, and using (1.54), (1.36) and (1.37) we come to (1.53). u n
1.3
Volterra Equation and Hausdorff Moment Problem
In connection with his factorization theory, M.M. Djrbashian has posed the following Hausdorff moment problem: prove the existence of a function .β(x) of bounded variation in .[0, 1], for which f .
−
1
x n dβ(x) = λn ,
n = 0, 1, 2, . . . ,
(1.55)
0
where ⎛f λ0 = 1,
.
λn =
1
⎞ ⎛f ω1 (x)x n−1 dx
0
1
⎞−1 ω2 (x)x n−1 dx
,
n ≥ 1,
(1.56)
0
and the functions .ω1,2 are from the class .o∗ (D) which is defined as follows. Definition 1.4 .o∗ (D) is the set of all functions .ω > 0 in .[0, 1), such that: ω ∈ C([0, 1)), ω(0) = 1, . f 1 ω(x)dx < +∞, .
.
(1.57) (1.58)
0
f
1
|1 − ω(x)|
0
dx < +∞. x
(1.59)
Note that M.M. Djrbashian [12] solved this problem for .ω1,2 ∈ o∗ (D) in the particular case .ω1 ≡ 1. Later, V.I. Ostrovskii [67] solved the problem in a more general case by an witty idea based on the solution of the Volterra integral equation f ω1 (x) = −
1
ω2
.
x
⎛x ⎞ t
dβ(t),
a.e. x ∈ (0, 1),
(1.60)
28
1 Preliminary Results
by which he found a more delicate and restrictive condition, under which he solved the equation, and refuted the hypothesis of M.M. Djrbashian on the solvability of the Eqs. (1.55)–(1.56) under the single condition that the quotient .ω1 /ω2 is non-increasing in .[0, 1]. Following V.I. Ostrovskii, we shall prove a result on the solvability of the first kind Volterra equation of the form f
∗ .ω1 (x)
= 0
x
ω2∗ (x − t)dα(t),
a.e.
x ∈ (0, ∞),
(1.61)
which is connected with (1.60) by the relations β(x) ≡ α(− log x),
.
∗ ω1,2 (x) ≡ ω1,2 (− log x).
(1.62)
The main results are the next two theorems. Theorem 1.8 1◦ . Let .ω1,2 be any continuous, positive and monotone functions in .(0, 1), and let .ω2 be non-decreasing in .(0, 1). Further, let the following conditions be satisfied:
.
(i) the quotient .ω1 /ω2 is non-increasing in .(0, 1); (ii) the logarithm of at least one of functions .ω1,2 is convex in .(0, 1) with respect to .log x, i.e. at least for one of values .j = 1, 2 ωj (xA)ωj
.
(iii) . lim x ε x→+0
⎛x⎞ A
≥ [ωj (x)]2 ,
0 < x < A < 1;
ω1 (x) = 0 for any .ε > 0; ω2 (x)
(iv) .x ε ω1,2 (x) ∈ L1 (0, 1) for any .ε > 0. Then, the Eq. (1.60) has a non-increasing solution .β such that β(1) = 0,
.
β(x) ≤
ω1 (x) , ω2 (x)
0 < x < 1.
2◦ . If (iii) and (iv) are replaced by the more restrictive conditions .ω1,2 (0) = 1 and (1.59), then additionally .β(+0) = 1.
.
1.3 Volterra Equation and Hausdorff Moment Problem
29
Theorem 1.9 ∗ be any continuous, positive, and monotone functions on .(0, +∞), and let .ω∗ 1◦ . Let .ω1,2 2 be non-increasing in .(0, +∞). Further, let the following conditions be satisfied:
.
(i' ) the quotient .ω1∗ /ω2∗ is non-decreasing in .(0, +∞);
.
∗ is convex in .(0, +∞), i.e. at least (ii' ) the logarithm of at least one of functions .ω1,2 for one of values .j = 1, 2
.
ωj∗ (x + δ)ωj∗ (x − δ) ≥ [ωj∗ (x)]2 ,
.
0 < δ < x < +∞;
(iii' ) for any .s > 0
.
.
lim e−sx
x→+∞
∗ (x) ∈ L1 (0, +∞) (iv' ) .e−εx ω1,2
.
ω1∗ (x) = 0; ω2∗ (x)
(1.63)
for any .ε > 0.
Then, the Eq. (1.61) has a non-decreasing solution .α such that α(0) = 0,
.
α(x) ≤
ω1∗ (x) , ω2∗ (x)
0 < x < +∞.
(1.64)
2◦ . If .(iii' ) and .(iv' ) are replaced by the more restrictive conditions
.
∗ ω1,2 (+∞) = 1 and
f
⎪ ⎪ω∗ (x) − 1⎪ dx < +∞, 1,2
+∞ ⎪
.
0
(1.65)
then additionally .α(+∞) = 1. Corollary 1.1 It is easy to verify that, if .ω is a convolution of some continuously differentiable, non-decreasing functions .fk > 0 (k = 1, 2) vanishing on .(−∞, 0] and such that .f1 ≡ ω1∗ and .f2 ≡ ω2∗ (x) satisfy the requirements of Theorem 1.9, then the Laplace transform f F (z) =
.
+∞
e−zt ω(t)dt
0
is a holomorphic, non-vanishing function in the right halfplane .Rez > 0.
30
1 Preliminary Results
After some observations and a lemma, we shall prove only Theorem 1.9, since it easily implies Theorem 1.8 by the change of variable (1.62). Remark 1.4 The monotonicity of quotients and the logarithmic convexity of functions in Theorems 1.8 and 1.9 can be replaced by some more general conditions. Namely: (a) The statements of Theorem 1.8 remain valid, if (i) and (ii) are replaced by the weaker requirement that for sufficiently small .ε > 0 .
ω1 (t (1 − ε)) ω2 (τ (1 − ε)) , ≤ inf 0 0 such that .0 < x −A < x < x + A < +∞; (b)
min
.
x∈[a,a/2]
ω2 (x) ≥ ρa > 0 for sufficiently small .a > 0 and some constant .ρa depending
solely on a. Then, the equation f
+∞
.
0
ω1 (x) dx s+x
/f
+∞
ω2 (x) dx = s+x
0
f
+∞
e−st dα(t),
0 < s < +∞,
(1.75)
e−xt ω2 (t)dt, 0 < x < +∞.
(1.76)
0
has a non-decreasing solution .α satisfying the conditions f α(0) = 0, α(x) ≤
+∞
e
.
−xt
0
/f
+∞
ω1 (t)dt 0
The substitution .e−t = x, .s = 1, 2, . . . in (1.75) leads to a solution .β(x) ≡ α(− log x) of the moment problem f .
−
1
x n dβ(x) = λn ,
0
where .λn are defined by (1.74).
n = 0, 1, 2, . . . ,
(1.77)
1.3 Volterra Equation and Hausdorff Moment Problem
37
Proof Observe that the integrals ∗ .ω1,2 (x)
f
+∞
=
e−xt ω1,2 (t)dt,
0 < x < +∞,
(1.78)
0
satisfy the conditions of Theorem 1.9. Indeed, these obviously are continuous, decreasing functions in .(0, +∞). Further, because of (a) a simple transformation gives ⎤ ⎧ f t } f t ω1∗ (t) ' = sign − ω (x)ω (t − x)xdx + ω (x)ω (t − x)(t − x)dx 1 2 1 2 ω2∗ (t) 0 0 ⎞ ⎛ ⎞ ⎧f t/2 ⎡ ⎛ t t ω1 − σ ω2 +σ = sign 2 2 0 ⎛ ⎞⎤ } ⎞ ⎛ t t −σ σ dσ = 1. + σ ω2 −ω1 2 2
⎡ sign
.
Next, it is obvious that the logarithm of a Laplace transform of a nonnegative function is convex, and hence, ∗ ∗ ∗ ω1,2 (x + δ)ω1,2 (x − δ) ≥ [ω1,2 (x)]2 ,
0 < δ < x < +∞.
.
∗ (x) ∈ L1 (0, +∞) for any .s > 0, since by (1.78) Further, .e−sx ω1,2
f
+∞
.
0
∗ e−sx ω1,2 (x)dx =
f
+∞ 0
ω(t) dt < +∞. t +s
Finally, for using formula (1.72) we see that, if .0 < a < s and .x ≥ log 22/a , then esx ω2∗ (x) ≥ ρa esx
f
a
.
e−xt dt ≥
a/2
ρa (s−a)x e → +∞ 2
as
x → +∞. u n
Theorems 1.8 and 1.9 imply some useful formulas for the .Cω -kernels in .D and .G+ . Theorem 1.11 Under the conditions of Theorems 1.8 and 1.9 and the assumption that ∗ .ω1 ≡ 1 in .[0, 1] and .ω ≡ 1 in .[0, +∞), the following formulas are true for the Cauchy 1 type kernels in .D and in .G+ correspondingly f Cω2 (z) = −
1
.
0
dβ(t) (z ∈ D) 1 − zt
f and
Cω2∗ (z) =
+∞ 0
dα(t) (z ∈ G+ ), t − iz
where .β is non-increasing in .[0, 1) and .α is non-decreasing in .(0, +∞).
(1.79)
38
1 Preliminary Results
Proof By formulas (1.55), (1.56) and (1.17) with .ω1 ≡ 1, we get f .
⎛ f x k dβ(x) = k
1
− 0
1
ω2 (x)x k−1 dx
⎞−1
=
0
1 , An (ω2 )
k ≥ 1.
Consequently, for any .z ∈ D Cω2 (z) =
∞ Σ
.
k=0
k
f1 0
zk ω2 (t)t k−1 dt
=−
∞ Σ
f
1
zk
f
0
k=0
1
t k dβ(t) = − 0
dβ(t) . 1 − zt
fx Further, by (1.61) and (1.22) with .ω1∗ ≡ 1 we have . 0 ω2∗ (x − σ )dα(σ ) ≡ 1 (0 < x < +∞). Hence, for any .t > 0 .
1 = t
f
+∞
e−tx dx =
0
f
+∞
0
⎛f
+∞
=
e
−tx
0
ω2∗ (x)dx
e−tx
⎛f
⎞ ω2∗ (x − σ )dα(σ ) dx
x
0
⎞⎛ f
+∞
e
−tx
⎞ dα(x) ,
0
and consequently for any .z ∈ G+ f
+∞
C (z) =
.
e
ω2∗
0
f
+∞
= 0
izt
t
f +∞ 0
f
dt e−tx ω2∗ (x)dx
=
f
+∞
izt
+∞
e dt 0
e−tx dα(x)
0
dα(x) . x − iz u n
Remark 1.8 Passing to the real parts in the first equality of (1.79), we conclude that for ∼(D) any .ω ∈ o Re Cω (z) ≥ 0 and
.
Re Sω (z) ≥ 0,
z ∈ D.
Passing to the real parts in the second equality of (1.79), we conclude that for any .ω ∈ + oN α (G ) Re Cω (z) ≥ 0,
.
z ∈ G+ .
Indeed, .Re Cω ≥ 0 in .D, since .β - in .[0, 1], and .Re Sω ≥ 0 in .D, since .β(0) = 1 and Sω = 2Cω − 1. The inequality in .G+ is obvious, since .α - in .(0, +∞).
.
1.4 Notes
1.4
39
Notes
The operator .Lω was introduced by M.M. Djrbashian [12] in the form (1.2). Also, in [12] some of the properties of .Lω were investigated. In particular, the property that the Σ k application of .Lω to the Taylor series . ∞ k=0 ak z multiplies its coefficiens .ak by .Ak led to the definition of his Cauchy type .Cω -kernels (1.17) by the dividing the expansion coefficients of .C0 (z) ≡ (1 − z)−1 by .Ak and hence getting .Lω Cω = C0 . The statements of Lemma 1.1 were proved in [39]. For the first time, the kernel (1.22) has been used in [57] (see also [11]), where it was constructed in the multidimensional case of tube domains. In the case of the halfplane + the definition of the .C -kernel is obvious. Similar to the case of the disc, the .C .G ω ω kernels for .G+ are found by division of the generating function of the laplace transform representation of .C0 (z) ≡ −1/(iz) by .Iω . Section 1.2 presents the results of [43]. Section 1.3 actually unites the papers [40] and [41]. It somehow improves the result of I.V. Ostrovskii [67] and complemented it by Theorem 1.10 on the Hausdorff moment problem. Remark 1.4 is a consequence of the result of I.V. Ostrovskii [67].
p
2
Spaces Aω (D) in the Unit Disc .
This chapter contains some exhaustive generalizations of the main part of results of M.M. Djrbashian of 1945–1948 given in Addendum of this book. These results laid ground p for the theory of .Aα , or initially .H p (α), spaces and then for his factorization theory of meromorphic classes .N{ω} in the unit disc. The last letter .ω of the Greek alphabet is used in the same meaning as in M.M.Djrbashian’s later work on meromorphic functions [16, 22], this is to stress that p the spaces .Aω exhaust all functions holomorphic in .z ∈ D. p Also, the chapter improves some later results on .Aα spaces. The chapter gives the p analytic apparatus, which lies in the base of the developing theory of .Aω (D) spaces and their applications, where instead .(1 − r 2 )α dr some general weights of the form .dω(r 2 ) are used. p
2.1
The Spaces Aω (D), Representations
2.1.1
Definition and Elementary Properties of Aω (D) Spaces
p
p
We define .Aω (D) as the set of all those functions f holomorphic in .D, for which { ||f ||p,ω =
.
1 2π
}1/p
ff |ζ | −1). Further, p p .Lω (D) is a Banach space with the norm (2.1). Thus, it suffices to show that .Aω (D) is a p p p p closed subset of .Lω (D), i.e. if .{fn }∞ 1 ⊂ Aω (D) and .fn → f in .Lω (D), then .f ∈ Aω (D). p To this end, observe that, if .fn → f in .Lω (D), then by Fatou’s lemma f .
0
1
f g(t)|dω(t)| = 0
for g(ρ ) = lim inf 2
n→∞
0
2π
| | |fn (ρeiϑ ) − f (ρeiϑ )|p dϑ.
p
2.1 The Spaces Aω (D), Representations
43
Hence, by (i) there exists at least a sequence .ρk ↑ 1 such that .g(ρk2 ) = 0 and iϑ .||f (ρk e )||Lp [0,2π ] < +∞ (k = 1, 2, . . .). By a diagonal operation, one can choose a subsequence of .{fn }∞ 1 (we keep the same notation .{fn } for this and other subsequences) such that f .
2π
lim
n→∞ 0
| | |fn (ρk eiϑ ) − f (ρk eiϑ )|p dϑ = 0
for all
k = 1, 2, . . .
Hence, for any .k ≥ 1 the .H p norms of .fn (z) (n = 1, 2, . . .) are uniformly bounded in .|z| < ρk . Consequently, there exists a holomorphic function .F (z) to which a subsequence of .{fn (z)}∞ 1 tends uniformly inside .|z| < 1. Besides, ff .
|ζ | 3δk ) Cωδ (z) =
.
1 iz
f
+∞
0
|+∞ f deitz 1 +∞ itz Iδ' (t) 1 eitz || + e dt ≡ A + B, = Iδ (t) iz Iδ (t) |t=0 iz 0 [Iδ (t)]2
where |A| ≤
.
1 |z|[ω(δk ) − ω(δk+1 )]
and
|B| ≤
Aω(A) . |z|δk [ω(δk ) − ω(δk+1 )]2 u n
Theorem 5.1 Let .f ∈ Aω,γ (G+ ) for some .1 ≤ p < +∞, .−∞ < γ < 1 and some .ω satisfying the condition (i) of Definition 5.1 and such that .ω(t) = ω(A) < +∞ .(t > A) for a .A > 0. Then, p
ff 1 .f (z) = f (w)Cω (z − w)dμω (w). 2π G+ ff 1 = {Re f (w)}Cω (z − w)dμω (w), π G+
(5.15) z ∈ G+ ,
(5.16)
where both integrals are absolutely and uniformly convergent inside .G+ . Proof First, note that Apω,γ (G+ ) ⊂ A1ω,γ ' (G+ ),
.
1−
1−γ < γ ' < 1, p
(5.17)
5.2 Representation by an Integral Over a Strip
103
for any .p > 1 and .γ < 1. Indeed, if .f ∈ Aω,γ (G+ ) for some .p > 1, then by Hölder’s inequality for any .y > 0, we get p
f
+∞
.
−∞
}1/p { f +∞ dx dx p |f (x + iy)| |f (x + iy)| ' ≤ (1 + |x + iy|)γ (1 + |x + iy|)γ −∞ }1/q { f +∞ dx × , q(γ ' −γ /p) −∞ (1 + |x + iy|)
where .q(γ ' − γ /p) > 1. Further, under our assumption .dω(2y) = 0, .A/2 < y < +∞, and the uniform convergence of the integrals (5.15) and (5.16) in any compact inside .G+ is obvious by (5.14). Besides, by the above inclusion it suffices to prove the representations (5.15) and (5.16) only for .p = 1. So, let .f ∈ A1ω,γ ' (G+ ) where .ω(t) and 1 ' .γ are as required. Then, .f (z + iρ) ∈ H ' for any .ρ > 0. Hence, by Lemma 5.1 γ 1 .f (z + iρ) = 2π i
f
+∞
−∞
f (ξ + iρ) dξ ξ −z
for any fixed point .z = x + iy ∈ G+ . Consequently, for any .δ ∈ (0, y0 ) f
1 .f (z + iρ) = lim a→−∞ 2π b→+∞
f
+∞
e
iτ z
b
dτ
0
e−iτ ξ f (ξ + iρ)dξ
a
(f A−δ ) f +∞ 1 iτ z −τ x e e dω(x + δ) = lim a→−∞ 2π 0 0 b→+∞ (f b ) dτ −iτ ξ × e f (ξ + iρ)dξ I δ (τ ) a f +∞ f b f A/2 1 iτ (z+2iv−iδ) dτ = lim e−iτ ξ f (ξ + iρ)dξ dω(2v) e a→−∞ 2π I (τ ) δ a δ/2 0 b→+∞ f
A/2
≡ lim
a→−∞ b→+∞
Ja,b (v)dω(2v), δ/2
where Ja,b (v) =
.
≡
1 2π 1 2π
f
+∞ (
e−τ v
0
f
f
b
e−iτ ξ f (ξ + iρ)dξ
a +∞
−∞
Aa,b (τ )B(τ )dτ,
)(
) e−iτ (z−v+iδ) dτ Iδ (τ )
5 Spaces .Aω,γ (G+ ) in the Halfplane p
104
and it is assumed that .Aa,b (τ ) ≡ B(τ ) ≡ 0 for .τ ≤ 0. One can see that .Aa,b and B are bounded functions of .L1 (−∞, +∞), which are continuous in .(−∞, +∞)\{0}. Therefore, the following well-known formula on Fourier transforms of such functions is valid (see, e.g., [11], Ch. I, §3, p. 39, Theorem 1.12(2.◦ )) Ja,b (v) =
.
1 2π
f
+∞
−∞
F [Aa,b ](u) F [B](u) du,
where .F [B] is the Fourier transform of .B ∈ L1 (−∞, +∞). Using this formula we conclude that ) f b f +∞ 1 f (ξ + iρ)dξ eiτ (−u+iv−ξ ) dτ 2π a −∞ 0 (f +∞ ) dt × eit (u+z+iv−iδ) du Iδ (t) 0 ) f +∞ ( f b f (ξ + iρ) 1 1 dξ Cωδ (z − (u − iv) − iδ)du. = 2π −∞ 2π i a ξ − (u + iv)
Ja,b (v) =
.
1 2π
f
+∞ (
The last integral is absolutely convergent even for .a = −∞ and .b = +∞, provided .δ is small enough. Indeed, choosing k great enough, so that .y > 4δk , and assuming that .δ ∈ (0, δk+1 ), for .v ∈ (δ/2, A/2) we have .y + v − δ ≥ y − δk /2 > 3δk , and hence by (5.14) f I1 ≡
+∞ (f +∞
.
−∞
f
≤ 2M
−∞ +∞
−∞
) |f (ξ + iρ)| dξ |Cωδ (z − (u − iv) − iδ)|du |ξ − (u + iv)|
|f (ξ + iρ)|o(ξ )dξ,
where f o(ξ ) =
+∞
.
−∞
dσ (|σ | + 3δk )(|ξ − x − σ | + δ/2)
is a continuous function in .(−∞, +∞). Besides, one can show that o(ξ ) ≤
.
C C' , log(1 + |ξ |) < 1 + |ξ | (1 + |ξ |)γ
−∞ < ξ < +∞,
5.2 Representation by an Integral Over a Strip
105
where C and .C ' are some constants depending on x, .δ and .γ . Hence, .I1 < +∞, and consequently .
lim Ja,b (v) =
1 2π
=
1 2π
a→−∞ b→+∞
+∞ (
f
−∞
f
+∞ −∞
1 2π i
f
+∞
−∞
) f (ξ + iρ)dξ Cωδ (z−(u − iv) − iδ)du ξ − (u + iv)
f (u + iv + iρ)Cωδ (z − (u − iv) − iδ)du
+∞ (
) f (t + iv)dt Cωδ (z − (u − iv) − iδ)du −∞ t − (u + iρ) −∞ ) ( f +∞ f +∞ 1 Cωδ (t + z + iv − iδ) 1 = dt du f (u + iv) 2π −∞ 2π i −∞ t − (−u + iρ) f +∞ 1 = f (u + iv)Cωδ (z − (u − iv) + iρ − iδ)du, 2π −∞ 1 = 2π
f
1 2π i
f
+∞
f
+∞
since f I2 ≡
+∞
.
−∞
|f (u + iv)|du
−∞
|Cωδ (z − (t − iv) − iδ)| dt < +∞ |u − t − iρ|
and .Cωδ (z + i(y + v − δ)) ∈ Hγ1' in .G+ . The last inclusion simply follows from (5.14), and .I2 < +∞ holds in the same way as the inequality .I1 < +∞ above. Thus, for any fixed .z ∈ G+ and a small enough .δ > 0 1 .f (z) = 2π
ff δ/2 1.
Further, I2 (t) = e
.
−tA
f [ω(A) − ω(0)] + tA 0
A
e−txA [ω(xA) − ω(0)]dx.
5 Spaces .Aω,γ (G+ ) in the Halfplane p
110
Hence, for .tA > 1 f .I2 (t) > tA
1
e
−txA
1/4
[ ( ) ] ( ) tA 3tA A [ω(xA) − ω(0)]dx ≥ ω − ω(0) e− 4 1−e− 4 4
tA
> Mω''' e− 4 A1+α
tA 3tA 3 > Mω''' e− 4 A1+α 4 + 3tA 7
and .I2 (t) > e−tA [ω(A) − ω(0)] > MωI V A1+α for .0 < tA < 1. By these inequalities, ⎧ tA I1 (t) ⎪ ⎪ < MωV e− 2 A−(1+α) , tA > 1 ⎨ 2 I1 (t) [I2 (t)] . < −(1+α) tA A ⎪ 1 I2 (t)[I1 (t) + I2 (t)] ⎪ < MωV I e− 2 A−(1+α) , 0 < tA < 1. < ⎩ I V I2 (t) Mω u n
Hence, by (5.28) we come to (5.27). Now, we are ready to prove the main representation theorem of this chapter.
Theorem 5.2 Let .f ∈ Aω,γ (G+ ) for some .1 ≤ p < +∞, .ω ∈ oα (G+ ) with .−1 ≤ α < +∞ and .−∞ < γ < 1. Then, for any .z ∈ G+ p
ff 1 f (w)Cω (z − w)dμω (w). 2π G+ ff 1 = {Re f (w)}Cω (z − w)dμω (w), π G+
f (z) =
.
(5.29) (5.30)
where both integrals are absolutely and uniformly convergent inside .G+ . Proof The absolute and uniform convergence of the integrals in (5.29) and (5.30) inside p G+ is obvious by (5.19). Further, .Aω,γ (G+ ) ⊂ A1ω,γ ' (G+ ) for .1−(1−γ )/p < γ ' < 1, as in (5.17). Thus, it suffices to prove the representation (5.29) only for .p = 1 by the passage .A → +∞ in (5.15), and omit the proof of (5.30) since it simply holds by the same passage .A → +∞ in (5.16). We start by proving that for any .A > A0 , .ρ > 0 and any .δ ∈ (0, 1] .
|CA (z)| ≤
.
Mρ , |z|1−δ
z ∈ G+ ρ,
(5.31)
where the constant .Mρ depends only on .ρ and .δ, and .CA is that of (5.26). Indeed, .ωA (t) ∈ o−1 . Hence, if .0 < δ < 1, then by (5.18) CA (z) =
.
1 (−iz)1−δ
f
+∞ 0
eitz ϕA,δ (t)dt,
z ∈ G+ ,
(5.32)
5.3 General Representation
111
d D −δ [IA (t)]−1 ≥ 0 and the right-hand side integral of (5.32) is where .ϕA,δ (t) = dt convergent. On the other hand, for any .y > 0
1
CA (iy) =
.
f
y 1−δ
+∞
e−ty ϕA,δ (t)dt =
f
0
+∞
e−ty
0
dt , IA (t)
f +∞ where the last integral decreases by .A for any fixed y and .δ. Hence . 0 e−ty ϕA,δ (t)dt has the same property. Consequently, for .y > ρ and .A > A0 f
+∞
e
.
−ty
f ϕA,δ (t)dt ≤
0
+∞
0
e−tρ ϕA0 ,δ (t)dt ≡ Mρ < +∞,
and (5.31) holds by (5.32). For .δ = 1 (5.31) is obvious. Now fix any .z ∈ G+ and choose a number .A1 large enough to provide that 1 .I1 ≡ 2π
ff
ε , 4
|f (w)||CA (z − w)|dμω (w)
0 .
1 √ 2π
f
f
+∞
+∞
dω(2v) 0
0
f
e−t (y+v) |fv (t)|dt Iω (t)
+∞ {
≤ 0
}1/2 || || ||f-v || 2 C∼ dω(2v) ω (2i(y + v)) L (0,+∞) {f
≤ M||f ||L2
ω,0
(G+ )
+∞ 0
dω(2v) (y + v)3+2α−ε
}1/2 < +∞,
while the norm inequality in (5.43) is proved as follows: 1 2π
||o||2L2 (0,+∞) ≤
.
1 ≤ 2π
f
+∞ { f +∞
0
f
0
e−tv |f-v (t)|dω(2v) f
+∞
+∞
dω(2v) 0
0
}2
dt Iω (t)
|f-v (t)|2 dt ≤ ||f ||2L2
+ ω,0 (G )
.
If .Xδ (t) is the characteristic function of the interval .δ < t < +∞ .(δ > 0), then by (5.42) |2 | f +∞ | | 1 itz o(t) | dt || dx dω(2y) lim | √ e √ Iω (t) 2π δ 0 −∞ δ→+0 ( −ty ) |2 } f +∞ { f +∞ | f +∞ | 1 | e 1 |√ lim inf ≤ o(t)Xδ (t) dt || dx dω(2y) eitx √ | δ→+0 2π −∞ Iω (t) 2π −∞ 0 ) f +∞ −2ty f +∞ ( e 1 2 lim inf |o(t)| Xδ (t)dt dω(2y) = δ→+0 −∞ Iω (t) 2π 0 f +∞ −2ty f +∞ e 1 |o(t)|2 dt = ||o||2L2 (0,+∞) . = dω(2y) Iω (t) 2π 0 0
2 .||Pω f || 2 Lω,0 (G+ )
1 ≤ 2π
f
f
+∞
Thus, by (5.43) .||Pω f ||2 2
Lω,0 (G+ )
+∞
≤ ||f ||L2
+ ω,0 (G )
, i.e. .Pω maps .L2ω,0 (G+ ) to .A2ω,0 (G+ ),
and .||Pω || ≤ 1. Further, let .f ∈ A2ω,0 (G+ ), then .f (z + iρ) ∈ H 2 (G+ ) for any fixed .ρ > 0. Hence, by the Paley–Wiener theorem (see. e.g., [61], pp. 130–131) 1 f (z + iρ) = √ 2π
f
.
0
+∞
eitz f-ρ (t)dt,
z ∈ G+ ,
5 Spaces .Aω,γ (G+ ) in the Halfplane p
116
where .f-ρ is the Fourier transform of .fρ , and .||f (t + iρ)||L2 (−∞,+∞) = ||fρ ||H 2 (G+ ) = ||f-ρ ||L2 (0,+∞) . Besides, 1 . 2π
f
+∞
−∞
1 f (u + iv)Cω (z − w)du = √ 2π
f
+∞
eit (z+iv) f-v (t)
0
dt , Iω (t)
z ∈ G+ ,
for any .v > 0, and hence 1 Pω f (z) = √ 2π
f
+∞
.
1 =√ 2π
f
0 +∞
eitz
dt Iω (t)
f
+∞
{ } e−2tv etv f-v (t) dω(2v)
0
eit (z−iv) f-v (t)dt = f (z),
0
z ∈ G+ v.
Thus, this formula is true in the whole .G+ , the operator .Pω is identical on .A2ω,0 (G+ ) and, by the way, the representation (5.29) is proved. 2 (G+ ) is an arbitrary function. ◦ .2 . To prove the representation (5.30), suppose .f ∈ A ω,0 2 + Then .f (z + iρ) ∈ H (G ) for any .ρ > 0. Hence, as it is well-known (see e.g. [61], Ch. VI(E)), 1 f∼ρ (t) = lim √ R→+0 2π
f
R
.
−R
e−itu f (u + iρ)du = 0 for a.e.
t < 0,
and consequently -ρ (t) = f-ρ (t) + f-ρ (−t) = lim √2 .f R→+0 2π
f
R
−R
{ } e−itu Re f (u + iρ) du.
Thus, for any .v > 0 and .z ∈ G+ v .
1 π
ff
{ G+
} Re f (w) Cω (z − w)dμω (w) 1 =√ 2π 1 =√ 2π
f f
+∞ 0 +∞
eitz
dt Iω (t)
f
+∞
{ } e−2tv etv [f-v (t) + f-v (−t)] dω(2v)
0
eit (z−iv) f-v (t)dt = f (z),
0
and the representation (5.30) is true in the whole .G+ . u n
5.4 Orthogonal Projection and Isometry
117
Remark 5.4 In the case of power functions .ω(t) = t 1+α .(α > −1), formulas (5.29) and (5.30) become the representations of [19] (see also in [71]). For absolutely continuous measures .dω and spaces defined in a somehow different way, over multidimensional tube domains, the first line of (5.29) was obtained in [57, 58]. In the above proof of Theorem 5.4, we have almost established the following analog of the Paley–Wiener theorem. Theorem 5.5 For any .ω ∈ oα (G+ ) .(−1 ≤ α < 0) such that .ω(0) = 0), the class 2 (G+ ) coincides with the set of functions representable in the form .A ω,0 1 f (z) = √ 2π
f
+∞
.
0
o(t) dt, eitz √ Iω (t)
z ∈ G+ , o ∈ L2 (0, +∞).
If this representation is true, then .||f ||A2
+ ω,0 (G )
1 o(t) = √ Iω (t)
f
+∞
.
(5.44)
= ||o||L2 (0,+∞) and
e−tv f-v (t)dω(2v),
0 < t < +∞,
(5.45)
0
where .f-v is the Fourier transform of .f (u + iv) .(−∞ < u < +∞) on the level .v > 0. Proof If the representation (5.44) is true, then it can be easily verified that f is √ holomorphic in .G+ . Besides, one can see that .e−tρ o(t)/ Iω (t) ∈ L2 (0, +∞) for any 2 + .ρ > 0 Hence, .f (z + iρ) ∈ H (G ), the Nevanlinna condition (5.1) is satisfied, and by the Paley–Wiener theorem ||f ||2A2
.
+ ω,0 (G )
f +∞ f +∞ 1 dω(2y) |f (x + iy)|2 dx 2π 0 −∞ ||2 f +∞ || −yt || || e || || = dω(2y) || √I (t) o(t)|| 2 0 ω L (0,+∞) f +∞ −2yt f +∞ e 1 |o(t)|2 dt = ||o||2L2 (0,+∞) . = dω(2y) 2π 0 Iω (t) 0
=
Thus, .f ∈ A2ω,0 (G+ ) and the equality .||f ||A2 A2ω,0 (G+ )
+ ω,0 (G )
= ||o||L2 (0,+∞) and (5.45) are proved.
be any function, then the validity of formulas (5.44)–(5.45) is Now, let .f ∈ n u established in the proof of the previous Theorem 5.4. Remark 5.5 For absolutely continuous .dω, the above Paley–Wiener-type theorem holds from some general results [57] in mixed-norm weighted spaces over arbitrary tube domains of .Cn .
5 Spaces .Aω,γ (G+ ) in the Halfplane p
118
Remark 5.6 Let .S1 be the set of those .ω, which are continuous and strictly increasing in [0, A] for some .A ∈ (0, +∞), .ω(0) = 0, and .ω(x) = ω(A) for .x > A. Besides, let .S2 be the wider set of those .ω, which belong to .oα (G+ ) for some .α ∈ [−1, 0) and .ω(0) = 0. Then,
.
U .
A2ω,0 (G+ ) =
ω∈S1
U
A2ω,0 (G+ ),
ω∈S2
and these unions coincide with the set of all those functions, which are representable in the form f f (z) =
.
+∞
z ∈ G+ ,
eitz ψ(t)dt,
(5.46)
0
where .e−εt ψ(t) ∈ L2 (0, +∞) for any .ε > 0. √ Indeed, taking .ψ = o/ 2π Iω in the representation (5.44) of any .f ∈ A2ω,0 (G+ ) and recalling that .f (z + iε) ∈ H 2 (G+ ) by Proposition 5.1, we conclude that .e−εt ψ(t) ∈ L2 (0, +∞). Conversely, if f is representable in the form (5.46)), then .f (z + iε) ∈ H 2 (G+ ) for any .ε > 0. Hence, (see Proposition 5.1 and its proof), for any .A > 0, there is some function .ω with .ω(0) = 0, which is continuous in .[0, +∞), such that .ω(x) = ω(A) √ for .A < x < +∞, and .f ∈ A2ω,0 (G+ ). Now, it remains to take .o = 2π Iω ψ in the representation (5.44) of f . The next theorem, in particular, gives more information about the functions .o and .ψ in (5.44) and Remark 5.5. First, we prove the following lemma. Lemma 5.5 For arbitrary .ω ∈ oα (G+ ) with .−1 ≤ α < +∞ and any .ϕ ∈ L2 (−∞, +∞) 1 . 2π
f
+∞ −∞
1 ϕ(t)Cω (z − t)dt = √ 2π
f
+∞
eitz
0
ϕ (t) dt, Iω (t)
z ∈ G+ ,
(5.47)
where the integrals uniformly converge in .G+ and represent a holomorphic function f in + .G , such that for the integral operator .Lω of (1.21) 1 .Lω f (z) = 2π i
f
+∞
−∞
ϕ(t) 1 dt = √ t −z 2π
f
+∞ 0
eitz ϕ (t)dt,
z ∈ G+ .
(5.48)
5.4 Orthogonal Projection and Isometry
119
Proof The first integral in (5.47) is uniformly convergent in .G+ by the estimate (5.18). Hence, the function f represented by that integral is holomorphic in .G+ . Besides f (z) =
.
1 2π
f
f
+∞
+∞
ϕ(t)dt
−∞
1 R→+∞ 2π
eiσ (z−t)
0
f
+∞
= lim
0
eiσ z
dσ Iω (σ )
f
dσ Iω (σ )
R
1 e−iσ t ϕ(t)dt = √ 2π −R
f
+∞
eiσ z
0
ϕ (σ ) dσ, Iω (σ )
for any fixed .z = x + iy ∈ G+ , since f
| | f R | 1 | dσ e−iσ t ϕ(t)dt − ϕ (σ )|| e−σy || √ Iω (σ ) 2π −R || || f R || { }1/2 || 1 −iσ t || || ≤ C∼ (2y) e ϕ(t)dt − ϕ (σ ) → 0 as √ ω || 2π || 2 −R L (−∞,+∞)
+∞
.
0
R → +∞.
The uniform convergence of the second integral in (5.47) is obvious. Further, Lω f (z) =
.
1 2π
f
+∞ −∞
ϕ(t)Lω Cω (z − t)dt =
1 2π i
f
+∞
−∞
ϕ(t) dt, t −z
z ∈ G+ ,
since .Lω Cω (z) = (−iz)−1 .(z ∈ G+ ) and the change of integration order is valid, since by the estimate (5.18) for arbitrarily small .ε > 0 f
+∞
.
−∞
f
+∞
|ϕ(t)|dt 0
f
|Cω (z − t + iσ )|dω(σ ) +∞
f
+∞
dω(σ ) (1 + σ + |t|)2+α−ε −∞ 0 } { f +∞ f +∞ σ 1+α dσ ω(A0 ) dt ≤ M1 |ϕ(t)| + M 2 (σ + 1 + |t|)3+α−ε (1 + |t|)2+α−ε −∞ A0 } { f +∞ M3 ω(A0 ) ≤ M1 dt + |ϕ(t)| (1 + |t|)1−ε (1 + |t|)2+α−ε −∞ f +∞ |ϕ(t)| ≤ M4 dt 1−ε −∞ (1 + |t|) { f +∞ }1/2 √ dt ≤ M4 2π ||ϕ||L2 (−∞,+∞) < +∞. 2(1−ε) −∞ (1 + |t|) ≤ M1
|ϕ(t)|dt
5 Spaces .Aω,γ (G+ ) in the Halfplane p
120
The second equality in (5.48) is proved in the same way as (5.47): for any .z ∈ G+ 1 2π
Lω f (z) =
.
f
f
+∞
+∞
ϕ(t)dt
−∞
1 R→+∞ 2π
eiσ (z−t) dσ
0
f
+∞
= lim
f
eiσ z dσ
0
R
1 e−iσ t ϕ(t)dt = √ 2π −R
f
+∞
eiσ z ϕ (σ )dσ.
0
u n 2 (G+ ) along with The next theorem gives another canonical representation of .A∼ ω,0 an explicit isometry operator and its inverse between the Hardy space .H 2 (G+ ) and 2 (G+ ). First, we prove the following lemma. .A ∼ ω,0
Lemma 5.6 Let .ω ∈ oα (G+ ) .(−1 ≤ α < +∞) and let .ω(0) = 0. Then, the Volterra square of .ω, i.e. the function f
x
∼ ω(x) =
.
ω(x − t)dω(t),
0 < x < +∞,
∼ ω(0) = 0,
(5.49)
0
belongs to .o1+2α . Besides, for any .x ∈ (0, +∞) (f Iω2 (x) ≡
+∞
.
)2 f e−xt dω(t) =
0
+∞ 0
e−xt d∼ ω(t) ≡ I∼ ω (x).
(5.50)
Proof If .0 < x1 < x2 < +∞, then f ∼ ω(x2 ) − ∼ ω(x1 ) ≥
.
x1
[ ] ω(x2 − t) − ω(x1 − t) dω(t) ≥ 0,
0
since .ω is non-decreasing. Besides, f ∼ ω(δk ) − ∼ ω(δk+1 ) ≥
δk
.
[ ] ω(δk − t)dω(t) ≥ ω(δk − δk+1 ) ω(δk ) − ω(δk+1 ) > 0
δk+1
ω - in .[0, +∞) and .∼ ω(δk ) ↓ 0. If for the sequence .δk ↓ 0, over which .ω(δk ) ↓ 0. Thus, .∼ x > 2A0 , then obviously
.
f ∼ ω(x) ≥
x/2
.
A0
≥ M1
f ω(x − t)dω(t) ≥ M1
x/2
A0
(x − t)1+α dω(t)
] ( x )1+α [ ( x ) − ω(A0 ) ≥ M2 x 2(1+α) ω 2 2
5.4 Orthogonal Projection and Isometry
121
for some constants .M1 and .M2 , and there is a constant .M3 > 0 such that (f ∼ ω(x) =
A0
.
f
x−A0
+
0
A0
f
≤ ω2 (A0 ) + M1
f
)
x
+
ω(x − t)dω(t)
x−A0 x−A0
[ ] (x − t)1+α dω(t) + ω(A0 ) ω(x) − ω(x − A0 )
A0
≤ ω2 (A0 ) + M1 ω(A0 )x 1+α + M1 (x − A0 )1+α ω(x − A0 ) f x−A0 + M1 (1 + α) (x − t)α ω(t)dt A0
≤ ω2 (A0 ) + M1 ω(A0 )x 1+α + (M1 + M12 )(x − A0 )1+α ω(x − A0 ) ≤ M3 x 2(1+α) . For proving (5.50), observe that f Iω2 (σ ) = σ 2
+∞ −σ t
e
.
f
+∞ −λt
ω(t)dt
e
0
(f x +∞ −σ x
f ω(λ)dλ = σ 2
0
e
0
) ω(x − t)ω(t)dt dx,
0
where f
x
f ω(x . − t)ω(t)dt =
0
f
x
dω(σ ) 0
)
t
ω(x − t)d
ω(λ)dλ
0
f
x
=
(f
x
0
f
dy 0
σ
f
x
ω(y − σ )dy =
y
ω(y − σ )dω(σ ).
0
Therefore, twice integration by parts gives f 2 .Iω (σ )
=σ
+∞
2
e
f =σ
0 +∞
e 0
−σ x
−σ x
(f
f
x
dy
(f 0
0 x
y
) ω(y − σ )dω(σ ) dx
0
) ω(y − σ )dω(σ ) dx = I∼ ω (σ ). u n
Theorem 5.6 Let .ω ∈ oα (G+ ) with .−1 ≤ α < +∞, let .ω(0) = 0, and let .∼ ω 2 (G+ ) coincides with the set of functions be the Volterra square (5.49) of .ω. Then, .A∼ ω,0 representable in the form f (z) =
.
1 2π
f
+∞
−∞
ϕ(t)Cω (z − t)dt,
z ∈ G+ , ϕ ∈ L2 (−∞, +∞).
(5.51)
5 Spaces .Aω,γ (G+ ) in the Halfplane p
122
2 (G+ ), .L f ≡ ϕ is the unique function of the Hardy .H 2 (G+ ), such For any .f ∈ Aω 0 ω,0 that (5.51) is true with .ϕ = ϕ0 . Besides, .||ϕ0 ||H 2 (G+ ) = ||f ||A2 (G+ ) and .ϕ −ϕ0 ⊥H 2 (G+ ) ∼ ω,0
for any .ϕ ∈ L2 (−∞, +∞) with which (5.51) is true. The operator f Lω f (z) ≡
.
+∞
f (z + iσ )dω(σ ),
z ∈ G+ ,
(5.52)
0 2 (G+ ) −→ H 2 (G+ ), and the integral in (5.51) defines .(L )−1 in is an isometry .A∼ ω ω,0 2 + .H (G ). 2 (G+ ) coincides with the set of funcProof By Theorem 5.5 and Lemmas 5.6 and 5.5, .A∼ ω,0 2 (G+ ), then .L f ∈ H 2 (G+ ), tions representable in the form (5.51). Further, if .f ∈ A∼ ω ω,0 and obviously this is the unique function in .H 2 (G+ ), for which (5.48) and (5.51) are true. Besides, .||Lω f ||H 2 (G+ ) = ||f ||A2 (G+ ) by Theorem 5.4, and if .ϕ ∈ L2 (−∞, +∞) ∼ ω,0 is some other function for which (5.51) is true, then the Fourier transform of .ϕ − Lω f vanishes out of .(0, +∞), i.e., .ϕ − Lω f ⊥H 2 (G+ ). 2 (G+ ) −→ H 2 (G+ ) and the It remains to see that .Lω is a one-to one mapping .A∼ ω,0 integral (5.51) defines .(Lω )−1 , since these operators act as multiplication or division of the integrands in (5.47) and (5.48) by .Iω . u n
Remark 5.7 Under the conditions of√Theorem 5.6, .o = (2π Iω )−1 Lω f in the represen2 + −1 tation (5.44) of .A∼ ω,0 (G ) and .ψ = ( 2π Iω ) Lω f in (5.46), where .Lω f is the Fourier 2 transform of .Lω f ∈ L (−∞, +∞). Remark 5.8 The explicit isometry (5.51)–(5.52) between the Hardy space .H 2 (G+ ) and 2 + 2 + .A (G ) permits to convert any additive result known in .H (G ) to its similarity in ∼ ω 2 + .A (G ). In particular, this is true for the well-known approximations by rational functions ∼ ω 2 (G+ ). in .H 2 (G+ ), the images of which are the kernels .Cω in .A∼ ω,0
5.5
( p )∗ p p The Projection Lω,0 (G+ ) to Aω,0 (G+ ), the Space Aω,0 (G+ )
The sharp estimates of the .Cω -kernel, which are given in Theorems 1.2 and 1.3 permit to p p establish the following statements on the projection .Lω,0 (G+ ) → Aω,0 (G+ ) and reveal p the conjugate space of .Aω,0 (G+ ) .(p /= 2) for some classes of parameter-functions .ω on .(0, +∞), which decrease near the origin not more rapidly than power functions.
( p )∗ p p 5.5 The Projection Lω,0 (G+ ) to Aω,0 (G+ ), the Space Aω,0 (G+ )
123
Theorem 5.7 Let .1 < q < +∞, .q /= 2, and let the functions .ω1 ∈ ox1 (G+ ) and + .ω2 ∈ ox2 (G ) .(x1,2 > −1) be continuously differentiable in .[0, +∞) and satisfy one of the below conditions (A) or (B): (A) .t −1 ω2' (t) - or, alternatively, .t −1 ω2' (t) - but .t −δ ω2' (t) - for some .δ ∈ (0, 1). Besides, in .(0, +∞) t −α1,2 ω2' (t) -
.
t −β1,2 ω2' (t) -
and
(5.53)
for some .α1,2 > 0 and .β1,2 > 0 such that α1 < 1 + β1 + β2 ,
.
1 + α1 < q(1 + β2 )
and
q(α2 − β2 ) < 2 + β1 .
(5.54)
(B) .t −1 ω2 (t) - and .t −δ ω2 (t) - for some .δ ∈ (0, 1). Besides, .ω2' (t) - and .(5.53) is true for some .α1,2 ∈ (−1, 0) and .β1,2 ∈ (−1, 0) satisfying .(5.54). Then, Pω2 f (z) =
.
1 2π
ff G+
f (w)Cω2 (z − w)dμω2 (w),
z ∈ G+ ,
(5.55)
is a bounded operator acting from .Lω1 ,0 (G+ ) to .Aω1 ,0 (G+ ). q
q
Proof We intend to show that Schur’s test is satisfied for .g(t) = t −λ and K(z, ζ ) =
.
| ω2' (2η) || Cω2 (z − ζ )| , ' ω1 (2η)
z ∈ G+ , ζ = ξ + iη ∈ G+ ,
i.e. for some .λ the following estimates are true for any .z = x + iy ∈ G+ and .ζ = ξ + iη ∈ G+ respectively: 1 .I1 (z) ≡ 2π I2 (ζ ) ≡
1 2π
ff ζ =ξ +iη∈G+
K(z, ζ )[g(η)]p dμω1 (ζ ) ≤ a[g(y)]p , .
(5.56)
K(z, ζ )[g(y)]q dμω1 (z) ≤ b[g(η)]q ,
(5.57)
ff
z=x+iy∈G+
where .1/p = 1 − 1/q and .a, b > 0 are some constants. This will prove that the operator 1 .SF (z) = 2π q
ff G+
K(z, ζ )F (ζ )dμω1 (ζ ),
is bounded in .Lω1 ,0 and .||S|| ≤ a 1/q b1/p (see, e.g. [19], p. 35).
z ∈ G+ ,
(5.58)
5 Spaces .Aω,γ (G+ ) in the Halfplane p
124
Under (A), the kernel .Cω2 admits the estimate (1.31). Hence, M .I1 (z) ≤ 2π M 2
=
f f
+∞ 0
0
+∞
ω' (2η) η−λp 2' ω1 (2η)
dω1 (2η) ω2' (y + η)
f
+∞
−∞
η−λp ω2' (2η) dω1 (2η) =M y + η ω1' (2η) ω2' (y + η)
dξ |z − ζ |2 f +∞
ω2' (2η) η−λp dη. ω2' (y + η) y + η
0
One can see that f
y
.
0
ω2' (2η) η−λp dη = ω2' (y + η) y + η
f
y 0
ω2' (y
f
ω2' (2η)η−λp dη + η)(y + η)−β2 (y + η)1+β2
ηβ2 −λp dη 1+β2 0 (y + η) f 1 β2 −λp t dt = 2β2 y −λp ≡ C1 [g(y)]p , 1+β2 (1 + y) 0 β2
y
0. Thus, (5.56) is true for .λ ∈ (0, (1 + β2 )/p). For checking (5.56), observe that by the mentioned estimate of the kernel .Cω2 M ω2' (2η) .I2 (ζ ) ≤ π ω1' (2η) ≤M
ω2' (2η) ω1' (2η)
f
+∞
y f
0 +∞ 0
−λq
ω1' (2y) dy ω2' (y + η)
f
+∞
−∞
dx |z − ζ |2
ω1' (2y)y −λq dy . ω2' (y + η)(y + η)
Further, arguing as above, we get ω2' (2η) . ω1' (2η)
f 0
η
f ω2' (2η) η 2β1 ω1' (2y)(2y)−β1 y β1 −λq dy ω1' (2y)y −λq dy = ' ω2' (y + η)(y + η) ω1 (2η) 0 ω2' (y + η)(y + η)−α1 (y + η)1+α1 f η β1 −λq y dy 1 + β1 α1 α1 −β1 = C3 [g(η)]q , λ< ≤2 η , 1+α 1 q 0 (y + η)
( p )∗ p p 5.5 The Projection Lω,0 (G+ ) to Aω,0 (G+ ), the Space Aω,0 (G+ )
ω2' (2η) ω1' (2η)
f
+∞
η
125
f ω2' (2η) +∞ ω1' (2y)(2y)−α1 (2y)α1 −λq dy ω1' (2y)y −λq dy = ω2' (y + η)(y + η) ω1' (2η) η ω2' (y + η)(y + η)−β2 (y + η)1+β2 f +∞ α1 −λq y dy α1 − β2 ≤ 2β2 ηβ2 −α1 = C4 [g(η)]q , λ> . 1+β2 q (y + η) η
Thus, (5.55) is true for .λ ∈ ((α1 − β2 )/q, (1 + β1 )/q), and this interval is not empty, since α1 < 1 + β1 + β2 . Besides, both (5.54) and (5.55) are true for
.
( ) ( ) ( ) 1 + β2 n α1 − β2 1 + β1 α1 − β2 1 + β2 λ ∈ 0, , = , , p q q q p
.
and this interval is not empty, since .(α1 − β2 )/q < (1 + β2 )/p by .1 + α1 < q(1 + β2 ). q Hence, the operator (5.58) is bounded in .Lω1 ,0 (G+ ), and .Pω2 is of the same kind since .
| | |Pω F (z)| ≤ 1 2 2π ≤
1 2π
ff G+
| | |F (ζ )| |Cω2 (z − ζ )| dμω1 (ζ )
ff
G+
K(z, ζ )|F (ζ )|dμω1 (ζ ) = S|F (z)|,
|| || q for any .F ∈ Lω1 ,0 (G+ ). Hence, .||Pω2 F ||q,ω
z ∈ G+ ,
≤ ||S|F |||q,ω1 ,0 ≤ ||S|| ||F ||q,ω1 ,0 < +∞.
1 ,0 q Lω1 ,0 (G+ ).
q Lω1 ,0 (G+ )
for any .F ∈ Thus, .Pω2 F ∈ q For proving that .Pω2 F ∈ Aω1 (G+ ), we use the estimate of .Cω2 , Hölder’s inequality and (5.54), (5.55). Then for .Im z = y ≥ y0 > 0 and some constants .M1,2,3 > 0 ff .
f G+
|F (ζ )||Cω2 (z − ζ )|dμω2 (ζ ) ≤ M f
.
≤ M1
+∞ { f +∞
−∞
−∞
×
+∞
dξ 0
+∞
ω2' (2η)|F (ζ )|dω1 (2η)
ω1' (2η)ω2' (y + η)|z − ζ |2
}1/q |F (ζ )| dω1 (2η) q
0
{f
f
+∞
[ω2' (2η)]p dη
}1/p dξ
[ω1' (2η)]p−1 [ω2' (y + η)]p |z − ζ |2p {( f 1 f +∞ ) }1/p [ω2' (2η)]p dη + ≤ M2 ||F ||q,ω1 ,0 [ω1' (2η)]p−1 [ω2' (y + η)]p (1 + η)2p−1 0 1 0
5 Spaces .Aω,γ (G+ ) in the Halfplane p
126
[
ω2' (2) ω2' (y0 )
]q
M2 ||F ||q,ω1 ,0 [ω1' (2)]p−1 { f 1 pβ2 −(p−1)α1 } f +∞ η dη ηpα2 −(p−1)β1 dη β2 × + y 0 (1 + η)2p−1 (1 + η)2p−1 (y0 + η)pβ2 0 1 } {f 1 f +∞ ≤ M3 ||F ||q,ω1 ,0 ηpβ2 −(p−1)α1 dη + ηp(α2 −β2 )−(p−1)β1 −2p+1 dη < +∞.
≤
0
1
Using this, one can prove the holomorphity of .Pω2 F in .G+ . For proving that .Pω2 F satisfies (5.1), observe that for .Reiϑ = x + iy (.y ≥ ρ > 0) and some constant .M4 ff | | ω2' (2η)|F (ζ )|dμω1 (ζ ) |Pω F (Reiϑ )| ≤ M 2 ' ' iϑ 2 2π G+ ω1 (2η)ω2 (y + η)|Re − ζ | }1/q {f +∞ f M +∞ ≤ |F (ζ )|q dμω1 (ζ ) 2π −∞ 0 {f +∞ [ ' ] }1/p ω2 (2η) p dη × dξ ω2' (y + η) [ω1' (2η)]p−1 [(x − ξ )2 + (ρ + η)2 ]p 0 }1/p {f 1 f +∞ pβ2 −(p−1)α1 p(α2 −β2 )−(p−1)β1 −2p+1 ≤ M4 ||F ||q,ω1 ,0 η dη + η dη < +∞.
.
0
1
For proving all above assertions under the requirement (B), observe that f Cω2 (z) = i
.
y
+∞
Cω' 2 (x + it)dt,
z = x + iy ∈ G+ ,
where the integral is absolutely convergent by the estimate of .Cω' in Theorem 1.3. Using this estimate and the condition .t −β2 ω2' (t) - .(−1 < β < 0) we get .
f My β2 +∞ dt dt ≤ ' (y) 2 ω' (t)t 2 + t 2 )t 1+β2 ω |x + it| (x y y 2 2 f +∞ β 2 dt y 1 =M ' . ω2 (y) |x|2+β2 y/|x| (1 + t 2 )t 1+β2
| | |Cω (z)| ≤ M 2
f
+∞
If .y/|x| ≥ 1, then f
+∞
.
y/|x|
dt ≤ (1 + t 2 )t 1+β2
f
+∞
y/|x|
dt t 3+β2
1 = 2 + β2
(
|x| y
)2+β2 .
( p )∗ p p 5.5 The Projection Lω,0 (G+ ) to Aω,0 (G+ ), the Space Aω,0 (G+ )
127
Besides, if .y/|x| < 1, then f
+∞
.
y/|x|
(f
1
+∞ )
f
dt (1 + t 2 )t 1+β2 y/|x| 1 ] [( ) f 1 |x| β2 dt 1 − 1 + M1 + M1 = ≤ 1+β2 |β2 | y y/|x| t
dt ≤ (1 + t 2 )t 1+β2
( ≤ M2
+
|x| y
)2+β2 .
[ ]−1 + Consequently, .|Cω2 (z)| ≤ M5 |z|2 ω2' (y) .(z = x + iy ∈ G ) for some constant .M5 , and it remains to use the same argument as in the case (A). u n Remark 5.9 The requirements of Theorem 5.7 allow the equality .ω1 = ω2 .(0 < t < +∞). If .ω ≡ ω1,2 and .x ≡ x1 = x2 , then by the (5.29) Theorem 5.7, where q is replaced p p by p, states that (5.29) defines a bounded projection .Lω,0 (G+ ) −→ Aω,0 (G+ ) provided .(1 + x)(p − 1) < 1. Theorem 5.8 Let .1 < p < +∞, .p /= 2, let .1/p + 1/q = 1, and let a function .ω ∈ ox (G+ ) .(−1 < x < +∞, .(1 + x)(p − 1) > 1) satisfy the conditions of Theorem 5.7 p for .ω1 ≡ ω2 . Then the set of bounded linear functionals over .Aω,0 (G+ ) is completely described by the formula o(f ) =
.
1 2π
ff G+
f (z)g(z)dμω (z),
f ∈ Aω,0 (G+ ), g ∈ Aω,0 (G+ ), p
q
(5.59)
( p )∗ q and . Aω,0 (G+ ) = Aω,0 (G+ ) in the sense of isomorphism. Proof It is obvious that for any .g ∈ Aω,0 (G+ ), and even for any .g ∈ Lω,0 (G+ ), p formula (5.59) defines a bounded linear functional over .Aω,0 (G+ ) and .||o|| ≤ ||g||q,ω,0 . p For proving the converse statement, we extend .o to the whole .Lω,0 (G+ ) without a norm change by the Hahn-Banach theorem. Then, using the same notation .o for the extended q functional we conclude that there exists some .ϕ ∈ Lω,0 (G+ ) such that q
1 .o(f ) = 2π
ff G+
f (z)ϕ(z)dμω (z),
q
f ∈ Aω,0 (G+ ), p
and .||o|| = ||ϕ||q,ω,0 . Inserting here the representation (5.29) of f , by Fubini’s theorem one can change the order of integration, since by Theorem 5.7 ||Pω |ϕ|||q,ω,0 ≤ Mp,ω ||ϕ||q,ω,0 < +∞,
.
5 Spaces .Aω,γ (G+ ) in the Halfplane p
128
where .Mp,ω is a constant depending only on p and .ω. Finally, we get (5.59), where ff
1 .g(z) = 2π
G+
ϕ(ζ )Cω (z − ζ )dμω (ζ ),
z ∈ G+ ,
and .||g||q,ω,0 ≤ Mp,ω ||ϕ||q,ω,0 = Mp,ω ||o||.
u n
+ Remark 5.10 Obviously .Aω,0 (G+ ) = A∼ ω,0 (G ) for any continuously differentiable ' ' functions .ω and .∼ ω such that .ω (x) x ∼ ω (x) .(0 ≤ x < +∞). Besides, one can verify that the requirements of Theorem 5.8 are satisfied for p
p
f ω(x) =
x
.
0
( a) t α logλ 1 + dt, t
0 ≤ x < +∞,
where .α > −1, .λ ≥ 0 and .a > 0 are any numbers. Hence, the statement of Theorem 5.8 is true for the functions .ω ∈ oα (G+ ) .(−1 < α < +∞) such that ( a) ω' (x) x x α logλ 1 + , x
.
0 ≤ x < +∞
for some .α > −1, .λ ≥ 0 and .a > 0.
5.6
Biorthogonal Systems, Bases, and Interpolation in A2ω∼ (G+ )
The isometry (5.51) of Theorem 5.6 between the Hardy space .H 2 (G+ ) and the spaces 2 + 2 + .A (G ) permits to convert any result of additive character in .H (G ) into a similar ∼ ω 2 + statement in .A∼ ω (G ). In particular, for .p = 2 the results of [17, 18] on biorthogonal 2 (G+ ). systems and interpolation in .H p .(1 < p < +∞) imply similar statements in .A∼ ω Almost all these statements are given in the below propositions, the proofs of which are obvious and therefore are omitted. For simplicity, we consider the case when the knots are of multiplicity 1, i.e., everywhere below we assume that .{zk }∞ 1 is a sequence of pairwise different points in ∞ ∞ + .G . It is said that .{zk } 1 ∈ A, if the sequence .{zk }1 is uniformly separated, i.e., .
||
inf
k≥1
j =1, j /=k
| | | zj − zk | | | | z − z | = δ > 0. j k
(5.60)
Note that this relation implies the validity of the Blaschke condition
.
∞ Σ Im zk < +∞ 1 + |zk |2 k=1
(5.61)
2 + 5.6 Biorthogonal Systems, Bases, and Interpolation in A∼ ω (G )
129
which is necessary and sufficient for the convergence of the Blaschke product B(z) =
.
∞ || z − zk |1 + zk2 | z − zk 1 + zk2 k=1
with zeros at .{zk }∞ 1 to a holomorphic function everywhere in the finite complex plane, except the closure of the set .{zk }∞ 1 . The inequality (3.21) of [18] is transferred to the following proposition. 2 + Proposition 5.3 If .{zk }∞ ∈ A, then for any function .f ∈ A∼ ω (G ) the following 1 inequality is true: ∞ Σ .
|2 | Imzk |Lω f (zk )| ≤ C||f ||2A2 (G+ ) , ω
k=1
where .C > 0 is a constant independent of f . 2 (G+ ), note For giving a series of propositions on approximation and interpolation in .A∼ ω that the functions
1 z − zk
rk (z) =
.
and
ok (z) =
B(z) , z − zk
k = 1, 2, . . . ,
are of .H 2 (G+ ). Hence, all functions L−1 ω rk (z) = rk,ω (z)
.
and
L−1 ω ok (z) = ok,ω (z),
k = 1, 2, . . . ,
are of .A2ω (G+ ), and one can verify that .rk,ω (z) = −iCω (z − zk ). Theorem D and some other results of [18] imply the following statement. Proposition 5.4 If the sequence .{zk }∞ 1 does not satisfy the Blaschke condition, i.e., the series (5.61) is divergent, then the systems { .
2 (G+ ). are complete in .A∼ ω
− iCω (z − zk )
}∞ 1
and
{
ok,ω (z)
}∞ 1
5 Spaces .Aω,γ (G+ ) in the Halfplane p
130
Further, a transformation in the conditions (1.16), (1.17) of [18] (or (2.2), (2.3) of 2 (G+ ){z } ⊂ A2 (G+ ) of functions f for [17]) leads to the introduction of a subset .A∼ k ω ∼ ω which there exist some .g ∈ H 2 (G+ ) such that for almost all .−∞ < x < +∞ the non-tangential boundary values of .g(−z)B(z) from inside the lower halfplane .G− = {z : Im z < 0} coincide with those of the function .Lω f ∈ H 2 (G+ ) form inside .G+ . Evidently, 2 + .A (G ){zk } can be considered only under the condition (5.61). By Theorem 2 of [17], we ∼ ω get the following proposition. ∞ Proposition 5.5 The systems .{−iCω (z − zk )}∞ 1 and .{ok,ω (z)}1 are biorthogonal in 2 + .A (G ): ∼ ω
( .
) − iCω (z − zk ), oν,ω (z) ω = =
ff
[
G+
] − iCω (z − zk ) oν,ω (z)dμω (z)
⎧ ⎨1
if
ν = k,
⎩0
if
ν /= k.
The next proposition is implied by Lemmas B and 1.1 of [18]. 2 (G+ ), then: Proposition 5.6 If .f ∈ A∼ ω
1◦ f belongs to .A2ω (G+ ){zk } if and only if
.
ψ(z) =
.
2◦
.
1 2π i
f
+∞ −∞
Lω f (t) dt ≡ 0, B(t) t − z
z ∈ G+ ,
where .Lω f (t) and .B(t) are the boundary values of .Lω f ∈ H 2 (G+ ) and .B ∈ H 2 (G+ ). The following orthogonal decomposition is true: f (z) = F (z) + R(z)
.
(z ∈ G+ ),
||f ||22,ω = ||F ||22,ω + ||R||22,ω ,
[ ] 2 (G+ ){z } and .R = L−1 Bψ ∈ A2 (G+ ). where .F ∈ A∼ k ω ω ω By Theorems 4.1 and 5.2 of [18], we get the next proposition. { { }∞ }∞ Proposition 5.7 Each of the systems . − iCω (z − zk ) 1 and . ok,ω (z) 1 is a basis in 2 ∞ ∈ A. + .A (G ){zk }, if and only if .{zk } ∼ ω 1
2 + 5.6 Biorthogonal Systems, Bases, and Interpolation in A∼ ω (G )
131
Using formulas (4.29), (4.31) and the expansion formula in the end of the proof of Theorem 5.2 in [18], we get the next proposition. 2 (G+ ){z } is representable in Proposition 5.8 If .{zk }∞ k ω 1 ∈ A, then any function .f ∈ A∼ + .G by both series
f (z) =
∞ Σ
.
ck (f ) Cω (z − zk ) =
∞ Σ
) ( Lω f (zk )ok,ω (z) with ck (f ) = f, ok,ω ω ,
k=1
k=1
2 (G+ ) and uniformly inside .G+ . which converge in the norm of .A∼ ω
Theorem 4.2 of [18] implies the next proposition. 2 (G+ ) is representable in the Proposition 5.9 If .{zk }∞ ω 1 ∈ A, then any function .f ∈ A∼ form
f (z) =
∞ Σ
.
ck (f ) Cω (z − zk ) + ψ(z),
k=1 2 (G+ ) and uniformly inside .G+ , and the following where the series is convergent in .A∼ ω inclusions are true:
[ ] 1 2 + ψ(z) = L−1 ω B(z)ψ(z) ∈ A∼ ω (G ) and ψ(z) = 2π i
f
+∞
.
−∞
Lω f (t) dt ∈ H 2 (G+ ). B(t) t − z
By Theorems 5.1 and 5.2 of [18], we get the next proposition. + Proposition 5.10 Let .{zk }∞ 1 be a sequence of pairwise different points in .G . Then the following statements are true. ∞ 1◦ . If .{zk }∞ 1 ∈ A and .{wk }1 is a sequence of complex numbers for which
.
A=
∞ Σ
.
Im zk |wk |2 < +∞,
k=1 2 (G+ ){z } such that then there is a unique function .f0 ∈ A∼ k ω
Lω f0 (zk ) = wk
.
(k = 1, 2, . . .)
and
||f0 ||A2 (G+ ) ≤ Cδ A, ∼ ω
5 Spaces .Aω,γ (G+ ) in the Halfplane p
132
where .Cδ > 0 is a constant depending solely on .δ of (5.60). This function is expanded in the series f0 (z) =
∞ Σ
.
wk ok,ω (z). z ∈ G+ ,
k=1
◦ .2 .
5.7
2 (G+ ) and uniformly inside .G+ . which converges in the norm of .A∼ ω { }∞ Conversely, if the set of the sequences . (Im zk )1/2 f ' (zk ) 1 with all possible functions 2 + 2 .f ∈ A (G ) coincides with the space .l of sequences of complex numbers, which ∼ ω possess finite sums of squares of modules, then .{zk }∞ 1 ∈ A.
Notes
The results of Sects. 5.1, 5.2, 5.3 are published in [45] while those of Sects. 5.4, 5.5, 5.6 p in [47]. In Definition 5.1 of the spaces .Aω,γ (G+ ), the Nevanlinna condition (5.1) can be replaced by the requirement that .f ∈ H p in any halfplane .Im z > ρ > 0 without affecting p the further results and proofs. Defining the spaces .Aω,0 (G+ ) solely by the condition (5.2) with .γ = 0 requires the continuous differentiability of .ω, as it was done in [2].
6
Orthogonal Decomposition of Functions Subharmonic in the Halfplane
In this chapter, we prove a orthogonal decomposition for some classes of functions subharmonic in .G+ . The result is analogous to that for the unit disc .D, which is proved in Section 4.3.
6.1
The spaces hp (G+ ) and hω (G+ ) p
We shall introduce the spaces .hp (G+ ) and .hω (G+ ) after the following useful remark. p
Remark 6.1 It is well-known (see, eg. [61, Ch. VI]) that the Hardy space .hp (G+ ) .(1 ≤ p < +∞) of real, harmonic in the upper halfplane .G+ functions, which is defined by the condition {f ||u||hp (G+ ) ≡ sup
+∞
.
y>0
−∞
}1/p |u(x + iy)|p dx
< +∞,
is a Banach space becoming a Hilbert space for .p = 2. Since .|u|p is subharmonic in .G+ for any function u harmonic in .G+ , the results of [44, Ch. 7] on the equivalent definition of the holomorphic Hardy spaces .H p (G+ ) have their obvious analogs for .hp (G+ ). In particular, the space .hp (G+ ) .(1 ≤ p < +∞) coincides with the set of all functions harmonic in .G+ and such that f
p
||u||hp (G+ ) = lim lim inf
.
R→+∞ y→+0
R −R
|u(x + iy)|p dx < +∞
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_6
133
6 Orthogonal Decomposition in .G+
134
and for a sufficiently small .ρ > 0
.
1 R→+∞ R
f
π−β
lim inf
β
)π/x−1 ( | ( iϑ )|p |u Re | sin π(ϑ − β) dϑ = 0, π − 2β
(6.1)
where .β = arcsin(ρ/R) = π/2−x. Note that due to Hölder’s inequality, if the last relation is true for a .p > 1, then it is true also for .p = 1. Now, we introduce some classes of parameter-functions .ω which are used allover this chapter. Definition 6.1 .o∗α (G+ ) .(−1 ≤ α < +∞) is the set of functions .ω such that: (i) .ω is non-decreasing in .[0, +∞), strictly increasing in .[0, δ) for some .δ > 0, and .ω(0) = 0; (ii) .ω ∈ Lipλ [0, +∞) for some .0 < λ ≤ 1; (iii) .ω(t) x t 1+α , .A0 < t < +∞ for some .A0 > 0. Definition 6.2 .hω (G+ ) .(0 < p < +∞, .ω ∈ o∗α (G+ ), .−1 ≤ α < +∞) is the set of those real, harmonic in the upper halfplane .G+ functions for which (6.1) is true along with p
{f f ||u||p,ω ≡
.
G+
}1/p |u(z)|p dμω (z) < +∞,
(6.2)
where .dμω (x + iy) = dxdω(2y). For the .hω (G+ ) spaces, first we prove the following statement. p
Proposition 6.1 .hω (G+ ) .(1 ≤ p < +∞, .ω ∈ o∗α (G+ ), .−1 ≤ α < +∞) is a Banach space with the norm (6.2), which for .p = 2 becomes a Hilbert space with the inner product p
(u, v)ω ≡
.
1 2π
ff G+
u(z)v(z)dμω (z),
u, v ∈ h2ω (G+ ).
Proof Let .Lω (G+ ) .(1 ≤ p < +∞) be the Banach space of real functions, defined solely p p by (6.2). Then, it suffices to prove that .hω (G+ ) is a closed subspace of .Lω (G+ ) for any ∞ ⊂ hp (G+ ) converges to some .u ∈ Lp (G+ ) in the .1 ≤ p < +∞, i.e. if a sequence .{un } ω ω 1 p p + + norm of .Lω (G ), then .u ∈ hω (G ). To this end, observe that p
f
f
1/2
dω(2y)
.
0
+∞ −∞
|un (x + iy) − u(x + iy)|p dx → 0 as
n → ∞.
6.2 Orthogonal Projection L2ω (G+ ) → h2ω (G+ )
135
f1 Hence, by Fatou’s lemma . 0 g(t)dω(t) = 0 for f g(2y) ≡ lim inf
.
n→∞
+∞
−∞
|un (x + iy) − u(x + iy)|p dx.
(6.3)
As .ω ∈ o∗α (G+ ), there exists a sequence .ηk ↓ 0 such that .ω(ηk+1 ) < ω(ηk ). Introducing V the measure .ν(E) = E ω (i.e. the variation of .ω on the set E), we conclude that .ν([ηk+1 , ηk ]) > 0 for any .k ≥ 1 and obviously .g(t) = 0 in .[ηk+1 , ηk ] almost everywhere with respect to the measure .ν. On the other hand, .u(x + it) ∈ Lp (−∞, +∞) for almost every .t > 0 with respect to the measure .ν. Thus, there is a sequence .yk ↓ 0 such that simultaneously .g(2yk ) = 0 and .u(x + iyk ) ∈ Lp (−∞, +∞). Now, we choose a subsequence of .{un }∞ 1 , for which the limit (6.3) is attained for .y = y1 . From this subsequence, we choose another one, for which (6.3) is attained for .y = y2 , etc. Then, by a diagonal operation we choose a subsequence for which we keep the same notation ∞ .{un } , and by which 1 f g(2yk ) = lim
.
+∞
n→∞ −∞
|un (x + iyk ) − u(x + iyk )|p dx = 0
(6.4)
for all .k ≥ 1. Then, in virtue of Remark 6.1, for any .n ≥ 1 and .ρ > 0 the function un (z+iρ) belongs to .hp (G+ ). Note that in particular this is true for .ρ = yk .(k = 1, 2, . . .). p + By (6.4), for any fixed .k ≥ 1 the sequence .{un (z+iyk )}∞ n=1 is fundamental in .h (G ), and p + p + consequently .un (z+iyk ) → U (z+iyk ) ∈ h (G ) as .n → ∞ in the norm of .h (G ) over + + p + + .G . Hence, .un uniformly tends to U inside .G , and .U ∈ h (G ) in any halfplane .Gρ . Thus, by the results of Ch. 7 in [44] we conclude that (6.1) is true for U and, in addition, for any number .A > 0 .
{ ff
ff .
|x| 0. Hence, for any fixed .η > 0 the function .u(z + iη) is the real part of some function .f (z + iη) from the holomorphic Hardy space .H 2 (G+ ). Consequently, by the Paley-Wiener theorem 1 .f (z + iη) = √ 2π
f
+∞
eitz f-η (t)dt,
z ∈ G+ ,
0
where the limit by norm 1 f-η (t) = l.i.m. √ R→+∞ 2π
f
R
.
−R
e−itξ f (ξ + iη)dξ
(6.9)
is the Fourier transform of f on the level .iη, and ||f (ξ + iη)||2L2 (−∞,+∞) = ||f (z + iη)||2H 2 (G+ ) = ||f-η ||2L2 (0,+∞) .
.
Note that one can prove the independence of the function .etη f-η (t) of .η > 0. Further, for any .η > 0 and .ζ = ξ + iη .
1 2π
f
+∞ −∞
1 u(ξ + iη)Cω (z − ζ )dξ = √ 2π
f
1 = √ 2 2π
+∞
ei(z+iη)t uη (t)
0
f
+∞ 0
dt tIω (t)
[ ] dt . ei(z+iη)t f-η (t) + f-η (t) tIω (t)
From (6.9) and the Paley-Wiener theorem, it follows that for any .t > 0 1 √ R→+∞ 2π
-η (−t) = lim .0 = f
f
R −R
e−itξ f (ξ + iη)dξ = f-η (t),
where the limit is in the norm of .L2 (−∞, +∞). Consequently, for any .z ∈ G+ η .
1 π
f
+∞ −∞
1 u(ξ + iη)Cω (z − ζ )dξ = √ 2π
f 0
+∞
dt , ei(z+iη)t f-η (t) tIω (t)
and hence } f +∞ f +∞ 1 izt dt −2tη tη e e {e fη (t)}dω(2η) .Tω u(z) = Re √ tIω (t) 0 2π 0 } { f +∞ 1 i(z−iη)t e fη (t)dt = Re {f (z)} = u(z), = Re √ 2π 0 {
i.e. the operator .Tω is identical in .h2ω (G+ ).
u n
6 Orthogonal Decomposition in .G+
140
6.3
Orthogonal Decomposition
We start by the following remark. Remark 6.2 In virtue of Remark 7.2 and Theorem 7.2, if .ω ∈ o∗α (G+ ) .(−1 ≤ α < +∞) and .ν is the associated Riesz measure of a subharmonic in .G+ function .u ∈ L1ω (G+ ), such that .
1 R→+∞ R
f
)π/x−1 ( | ( iϑ )|2 |u Re | sin π(ϑ − β) dϑ = 0 π − 2β
π−β
lim inf
β
(6.10)
for any .ρ > 0 and .β = arcsin(ρ/R) = π/2 − x, then the conditions (f
ff .
G+
2Im ζ
) ω(t)dt dν(ζ ) < +∞
ff and
0
G+ ρ
Im ζ dν(ζ ) < +∞ (∀ρ > 0) (6.11)
provide the convergence of the Green type potential ff Pω (z) =
.
G+
log |bω (z, ζ )|dν(ζ )
in .G+ . Besides, the function u is representable in .G+ in the form u(z) = Pω (z) +
.
1 π
ff G+
{ } u(w) Re Cω (z − w) dμω (w) ≡ Pω (z) + uω (z).
(6.12)
If .α = −1, then the first condition in (6.11) implies the second one, as it is not difficult to verify, the statements of Theorem 7.2 remain true, besides, all above statements of this remark are valid. The next theorem proves that (6.12) is an orthogonal decomposition for some .ω-weighted classes of functions subharmonic in .G+ . Theorem 6.2 If .ω ∈ o∗α (G+ ) with .−1 ≤ α < 0, then: 1◦ . Both summands .Pω and .uω in the right-hand side of the representation (6.12) of any subharmonic function .u ∈ L2ω (G+ ) ∩ L1ω (G+ ) satisfying (6.10) are of .L2ω (G+ ). ◦ 2 + .2 . The operator .Tω is identical on .hω (G ) and it maps all satisfying (6.10) Green type 1 + 2 + potentials .Pω ∈ Lω (G ) ∩ Lω (G ) to identical zero. ◦ 2 + 2 + .3 . Any harmonic function .u ∈ hω (G ) is orthogonal in .Lω (G ) to any Green type 1 + 2 + potential .Pω ∈ Lω (G ) ∩ Lω (G ) satisfying (6.10). .
6.4 Notes
141
Proof Let .u ∈ L1ω (G+ ) ∩ L2ω (G+ ) be a subharmonic in .G+ function satisfying (6.10). Then, by Remark 6.2, u is representable in the form (6.12), where .uω ∈ h2ω (G+ ) by Theorem 6.1. Hence, also .Pω ∈ L2ω (G+ ) and satisfies (6.10). Further, if .Pω ∈ L1ω (G+ ) and satisfies (6.10), then applying the operator .Tω to both sides of the equality (6.12) written for .Pω we get .Tω Pω (z) ≡ 0, .z ∈ G+ . Since .Tω is the orthogonal projection of 2 + 2 + .Lω (G ) to its harmonic subspace .hω (G ), we conclude that ( .
Tω u, Pω
) ω
) ) ) ( ( ( = Tω uω , Pω ω = Tω∗ uω , Pω ω = uω , Tω Pω ω = 0.
At last, if u is a function of .h2ω (G+ ) and a Green type potential .Pω ∈ L1 (G+ ) ∩ L2 (G+ ) satisfies (6.10), then by Theorem 6.1 ( ) ) ) ) ( ( ( u, Pω ω = Tω u, Pω ω = Tω∗ u, Pω ω = u, Tω Pω ω = 0.
.
u n
6.4
Notes
The results of this chapter are published in [54].
7
Nevanlinna–Djrbashian Classes in the Halfplane
7.1
Preliminary Definitions and Statements
We start by a preliminary statement on a representation formula in some weighted classes of harmonic functions. Theorem 7.1 Let U be a harmonic function in .G+ , such that for .ρ > 0 small enough
.
f | ( π(ϑ − β) )π/x−1 1 π −β || | dϑ = 0, . |U (Reiϑ )| sin R→+∞ R β π − 2β ff |U (z)| dμω (z) < +∞, lim inf
G+
(7.1) (7.2)
where .β = arcsin Rρ = π2 − x and .dμω (x + iy) = dxdω(2y), where .ω ∈ oα (G+ ) for some .α ≥ −1 (see Definition 1.2). Then 1 .U (z) = π
ff
z ∈ G+ ,
(7.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_7
143
G+
U (w)Re {Cω (z − w)}dμω (w),
where the integral is absolutely and uniformly convergent in .G+ .
7 Nevanlinna–Djrbashian Classes in .G+
144
Proof It suffices to repeat the proofs of Theorems 5.1 and 5.2 starting by a different representation formula. Namely, assuming that the function .ω satisfies the requirement (i) of Definition 5.1 and is such that .ω(t) = ω(A) for some .A > 0 and any .t ≥ A, we use the Poisson representation } 1 1 − U (ξ + iρ)dξ i(ξ − z) i(ξ − z) −∞ f f +∞ 1 +∞ U (ξ + iρ)dξ e−i(ξ −z)τ dτ, z ∈ G+ , = Re π −∞ 0
U (z + iρ) =
.
1 2π
+∞ {
f
which is true for any .ρ > 0. This leads to the analog of formula (5.16). Then we let A → +∞, as in Theorem 5.1 .(p = 1, γ = 0). n u
.
Proceeding to delta-subharmonic functions U in .G+ , we assume that .ν is the associated measure of U , i.e., we assume that .U = U1 − U2 , where .U1,2 are subharmonic in .G+ , possess associated Riesz measures .ν1,2 , and .ν = ν1 − ν2 . Besides, we assume that the measure .ν is minimally decomposed in the Jordan sense, i.e., .ν = ν+ − ν− , where .(supp ν+ ) ∩ (supp ν− ) = ∅ and .ν± are the positive and negative variations of .ν. Further, we consider Tsuji’s characteristic of the form 1 .L(y, U ) ≡ 2π
f
+∞
−∞
f
+
U (x + iy)dx +
+∞
n+ (t)dt,
y
0 < y < +∞,
(7.4)
where ff n+ (t) =
.
G+ t
dν− (ζ ),
G+ t = {ζ : Im ζ > t}.
Assuming that .L(y, −U ) is defined similarly, by means of .U − and .ν+ , one has to note that generally .L(y, U ), .L(y, −U ), or both these quantities can be infinite. Therefore, for the sake of generality, we shall not use the conditions under which .L(y, ±U ) .(0 < y < +∞) are finite and connected by B.Ya. Levin’s formula of the special form L(y, U ) = L(y, −U ),
.
0 < y < +∞
(see [44, Chapter 4]), which is a similarity of the well-known equilibrium for the Nevanlinna characteristics. We define the class of acceptable .ω-functional parameters and the considered weighted classes of delta-subharmonic functions as follows.
7.1 Preliminary Definitions and Statements
145
∼α (G+ ) .(−1 < α < +∞) is the set of those functions .ω with .ω(0) = 0, Definition 7.1 .o which are continuous and strictly increasing in .[0, +∞), continuously differentiable in .(0, +∞) and such that for some .A > 0 ω' (x) x x α ,
A < x < +∞.
.
(7.5)
∼α (G+ ), then .ω(x + ρ) ∈ oα (G+ ) for any .ρ ≥ 0 (see One can see that if .ω ∈ o Definition 1.2). ∼α (G+ ) .(−1 < α < +∞) is the set of those functions that Definition 7.2 .Nω with .ω ∈ o + are delta-subharmonic in .G and satisfy the condition f
+∞
.
[L(y, U ) + L(y, −U )] dω(2y) < +∞.
(7.6)
0
The below lemma gives some simple properties of the classes .Nω . ∼α (G+ ) .(−1 < α < +∞), then Lemma 7.1 If .U ∈ Nω with some .ω ∈ o ff .
(f
ff G+
|U (z)|dμω (z) < +∞,
G+
) ω(t)dt dν± (ζ ) < +∞,
2Im ζ 0
(7.7)
and for any .ρ > 0 ff .
G+ ρ
Im ζ dν± (ζ ) < +∞.
(7.8)
Proof The first inequality in (7.7) follows from (7.4) and (7.6) while the second one is true, since f .
+∞>
f
+∞
+∞
dω(2y) 0
y
f
ff =
G+
dν± (ζ )
f n± (t)dt =
Imζ 0
ff
+∞
dω(2y) 0
( ) 1 Im ζ − y dω(2y) = 2
G+ y
ff
G+
( ) Im ζ − y dν± (ζ ) (f
)
Im ζ 0
ω(t)dt dν± (ζ ).
Further, it is obvious that for almost all .ρ ∈ (0, +∞) ff .
+∞>
G+ ρ
( ) Im ζ − ρ dν± (ζ ) ≥
ff G+ 2ρ
( ) Im ζ − ρ dν± (ζ ),
where .Im ζ −ρ > Im ζ2 . This proves (7.8) for any .ρ > 0, since its integral is non-increasing by .ρ. u n
7 Nevanlinna–Djrbashian Classes in .G+
146
Note that in the next lemmas of this subsection, we use the Green-type potentials constructed by means of the elementary factor { f bλ (z, ζ ) = exp −
} τ λ dτ , [τ − i(z − ζ )]1+λ
2η
.
0
ζ = ξ + iη ∈ G+ ,
(7.9)
which for any .λ > −1 is a holomorphic in .G+ function with a unique, simple zero at .z = ζ . Henceforth, referring to some theorems of [44, Chapter 4], we shall mean their obvious delta-subharmonic extensions. ∼α (G+ ), .−1 < α < +∞, then for any .ρ > 0 and Lemma 7.2 If .U ∈ Nω with .ω ∈ o + + .λ > 1 + α .(α = max{α, 0}) the function ff oλ,ρ (z) ≡ U (z + iρ) −
.
G+ ρ
log |bλ (z, ζ − iρ)|dν(ζ )
(7.10)
is harmonic in .G+ ρ , and .oλ,ρ (z + iρ) satisfies the conditions of Theorem 7.1 for .ωρ (x) = ω(x + 4ρ) .(0 < x < +∞). Proof By (7.5) f
+∞
.
[L(y, U ) + L(y, −U )] y α dy < +∞
ρ
for any .ρ > 0, and one can see that .L(t + ρ, ±U (z)) = L(t, ±U (z + iρ)) .(0 < t < +∞). Hence f .
+∞
[L(y, U (z + iρ)) + L(y, −U (z + iρ))] (y + ρ)α dy < +∞.
(7.11)
0
If .α ≥ 0, then the above inequality means that .U (z + iρ) belongs to the class .Nm 1+α,0 of [44, Chapter 4], i.e, more precisely, to its delta-subharmonic extension defined by (7.6) with .ω(y) ≡ y 1+α . Therefore, the Green-type potential in (7.10) is convergent, and by Theorems 2.2 and 3.1 in [44, Chapter 4] for any .λ > 1 + α oλ,ρ (z) = Re
.
λ2λ π
ff G+
U (ζ + iρ)
(Im ζ )λ−1 [−i(z − ζ )]λ+1
dσ (ζ ),
z ∈ G,
(7.12)
+ where .σ is Lebesgue’s area measure and .oλ,ρ is a harmonic function of .Nm 1+α,0 in .G , i.e.,
ff M0 ≡
.
G+
| | |oλ,ρ (ζ )|(Im ζ )α dσ (ζ ) < +∞.
7.1 Preliminary Definitions and Statements
147
Besides, using (7.12), one can verify that for any .y > 0 f M(y) ≡
| |oλ,ρ (x + iy)|dx ≤ M1
+∞ |
.
≤
−∞
ff G+
ff
|U (ζ + iρ)|
(Im ζ )λ−1 dσ (ζ ) (y + Im ζ )λ
| | |U (ζ + iρ)|(Im ζ )α dσ (ζ ) ≡ M2 . y 1+α G+
M1 y 1+α
Consequently ff .
G+ 1
| | |oλ,ρ (z + iρ)|dμω (z) = 2 ρ f
+∞
≤ M3
f
+∞
M(y + ρ)ω' (2y + 4ρ)dy
1
f
M(y + ρ)(y + 2ρ)α dy = M3
1
+∞
M(y)(y + ρ)α dy ≤ M0 M3
1+ρ
and ff .
G+ \G+ 1
| | |oλ,ρ (z + iρ)|dμω (z) = ρ
f
1
M(y + ρ)dωρ (2y)
0
f
1
≤ M2 0
dωρ (2y) M2 ≤ 1+α ω(2 + 4ρ), 1+α (y + ρ) ρ
i.e. .oλ,ρ (z + iρ) ∈ Nωρ or, which is the same, .oλ,ρ (z + iρ) satisfies the condition (7.2) of Theorem 7.1 for .ωρ (x) and .γ = 0. If .−1 < α < 0, then (7.11) means that .U (z + iρ) ∈ Nm 1,−α . Hence, for any .λ > 1 the potential in (7.10) is convergent, the function .oλ,ρ is harmonic in .G+ and representable in the form (7.12). Besides, .oλ,ρ ∈ Nm 1,−α , i.e. M0' ≡
ff
.
G+
| | |oλ,ρ (ζ )|(1 + Im ζ )α dσ (ζ ) < +∞.
Similarly, for any .y > ρ M(y) ≤
.
≤
M1 y 1+α M4 y 1+α
ff G+
ff
G+
| | |U (ζ + iρ)|(y + Im ζ )α dσ (ζ ) | | |U (ζ + iρ)|(1 + Im ζ )α dσ (ζ ) ≡ M5 , y 1+α
7 Nevanlinna–Djrbashian Classes in .G+
148
and therefore ff .
G+ 1
f
+∞
≤ M3
G+ \G+ 1
+∞
M(y + ρ)ω' (2y + 4ρ)dy
1
f
M(y + ρ)(y + 2ρ) dy = M3 α
1
ff
f
| | |oλ,ρ (z + iρ)|dμω (z) = 2 ρ
| | |oλ,ρ (z + iρ)|dμω (z) = ρ
f
+∞ 1+ρ
1
M(y)(y + ρ)α dy ≤ M0' M4 ,
M(y + ρ)dω(2y + 4ρ)
0
f ≤ M5 0
1
dω(2y + 4ρ) M5 ≤ 1+α ω(2 + 4ρ). (y + ρ)1+α ρ
These inequalities prove that .oλ,ρ (z + iρ) ∈ Nωρ , i.e. .oλ,ρ (z + iρ) satisfies the condition (7.2) of Theorem 7.1 with .ωρ and .γ = 0. It remains to see that by (7.12) and Theorem 5.3 the function .oλ,ρ (z + iρ) .(λ > 1 + α + ) satisfies the condition (7.1) for any .ρ > 0 and any .α > −1. ∼α (G+ ), .−1 < α < +∞. Then by Lemma 7.2 and Let .U ∈ Nω with some .ω ∈ o Theorem 7.1 for any fixed .λ > 1 + α + and .ρ > 0 1 .oλ,ρ (z + iρ) = Re π
ff G+
oλ,ρ (w + iρ)Cωρ (z − w)dμωρ (w),
z ∈ G+ ,
where .ωρ (x) = ω(x + 4ρ) .(0 < x < +∞). Inserting here (7.10) and changing the integration order, for .z ∈ G+ one can formally write 1 U (z + . 2iρ) = Re π ff +
ff G+
{ G+ ρ
U (w + 2iρ)Cωρ (z − w)dμωρ (w)
} log |bλ (z + iρ, ζ − iρ)| − Re Jω,λ (z, ζ, ρ) dν(ζ ),
(7.13)
where Jω,λ (z, ζ, ρ) =
.
1 π
ff G+
log |bλ (w + iρ, ζ − iρ)|Cωρ (z − w)dμωρ (w).
To justify these formulas, note that the potential ff .
G+ ρ
| | log |bλ (z + iρ, ζ − iρ)|dν(ζ )
(λ > 1 + α + )
(7.14)
7.1 Preliminary Definitions and Statements
149
is convergent in .G+ (and even in .G+ −ρ ) by the results of [44, Chapter 4]. In addition, we shall prove some estimates for .|Jω,λ (z, ζ, ρ)|, which allow to apply Fubini’s theorem and hence, to justify (7.13) completely. For simplicity, we assume that .λ .(> 1+α + ) is a natural number. Then, the integration by parts in (7.9) gives w−ζ
bλ (w, ζ ) =
.
w−ζ
exp
{ λ Σ1[ k=1
2η
k −i(w − ζ )
]k } ,
(7.15)
where .ζ = ξ + iη ∈ G+ , and hence | | λ λ | | | | |. log |bλ (w, ζ )|| ≤ log | w − ζ | + λ2 η ≤ 2vη + λ2 η | w − ζ | |w − ζ | |w − ζ |2 |w − ζ |
(7.16)
for .w = u + iv ∈ G+ . If .z = x + iy, where .y ≥ d0 > 0, and .w = u + iv, then, by the estimate (5.19) for arbitrarily small .ε > 0 and any .ρ ∈ [0, 1) .
1 π
ff G+
| | |log |bλ (z + iρ, ζ − iρ)|||Cω (z − w)|dμω (w) ρ ρ ≤
1 π
(f
2η 0
f +
+∞ )
f dωρ (2v)
2η
+∞ −∞
| log |bλ (w + iρ, ζ − iρ)|| du ≡ A1 + A2 , |z − w|2+α−ε
where by (7.16), (7.9) and (7.5) | | | t + i(v + η) | |dt | log | t + i(v − η + 2ρ) | 0 −∞ } f +∞ λ2λ (η − ρ) dt + dωρ (2v) (v + η)(d0 + v)α−ε −∞ |t + i(y + v)|2 f 2η { f 2(η−ρ) 1 = sign(τ +v−η + 2ρ)dτ (d0 + v)2+α−ε 0 0 } λ2λ (η − ρ) dωρ (2v) + (v + η)(d0 + v)1+α−ε f +∞ f +∞ dω(2v + 4ρ) dω(2v + 4ρ) λ2λ ω(6) λ < 2(η − ρ) + λ2 (η − ρ) + 1+α−ε 2+α−ε v(d0 + v)1+α−ε (d0 + v) d0 1 0
1 .A1 ≤ π
f
2η
{
1 (d0 + v)2+α−ε
f
+∞
7 Nevanlinna–Djrbashian Classes in .G+
150
and +∞ { 2(v + η)(η − ρ)
} du λ2λ (η − ρ) + dωρ (2v) π(d0 + v)2+α−ε −∞ |w − ζ + 2iρ|2 (v + η)(d0 + v)1+α−ε 2η } f +∞ { λ2λ (η − ρ) 2(v + η)(η − ρ) + dωρ (2v) = (v − η + 2ρ)(d0 + v)2+α−ε (v + η)(d0 + v)1+α−ε 2η f +∞ f +∞ dω(2v + 4ρ) dω(2v + 4ρ) λ2λ ω(6) λ + 1+α−ε + λ2 (η − ρ) . < 6(η − ρ) 2+α−ε (d0 + v) v 2+α−ε 3d0 0 1 f
A2 ≤
.
f
+∞
By these inequalities, the integral (7.14) is convergent, and therefore, the function Jω,λ (z, ζ, ρ) is holomorphic in .G+ . Besides,
.
| | |Jω,λ (z, ζ, ρ)| ≤ M ' (η − ρ) + M '' , ω,λ ω,λ
.
y ≥ d0 ,
0 ≤ ρ < 1,
(7.17)
' '' are some positive constants independent of .ρ. In view of (7.8) and where .Mω,λ and .Mω,λ (7.17), the change of integration order, which leeds to (7.13), is valid. u n
Remark 7.1 One can see that the argument, which proved (7.17) remains true in the case when .λ = ρ = 0. Hence | | |Jω,0 (z, ζ, 0)| ≤ ηM ' , ω,d0
.
y ≥ d0 , η > 0,
(7.18)
' where .Mω,d is a constant depending only on .ω and .d0 . 0
For letting .ρ → +0 in formulas (7.13)–(7.14) and coming to the final, canonical representations we use also the following statement. ∼α (G+ ) with some .−1 < α < +∞. Then: Lemma 7.3 Let .ω ∈ o + (A) For any .ρ ∈ [0, 1), .ζ ∈ ξ +iη ∈ G+ ρ and any natural number .λ > 1+α the function
{ } bωρ (z + iρ, ζ − iρ) ≡ bλ (z + iρ, ζ − iρ) exp −Jω,λ (z, ζ, ρ) ,
.
(7.19)
where .ωρ (x) = ω(x + 4ρ), is holomorphic in .G+ , where it can vanish only at the point .z = ζ − 2iρ (if .ζ − 2iρ ∈ G+ ), which is a first order zero. (B) Formula (7.13) can be written in the following form: for any .ρ ∈ (0, 1) 1 .U (z + 2iρ) = Re π ff +
ff G+
G+ 2ρ
U (w + 2iρ)Cωρ (z − w)dμωρ (w)
log |bωρ (z + iρ, ζ − iρ)|dν(ζ ),
z ∈ G+ ,
(7.20)
7.1 Preliminary Definitions and Statements
151
+ and, for any .ρ ∈ [0, 1), .ζ = ξ + iη ∈ G+ 2ρ and .z ∈ Gη−2ρ ,
{ f } 2(η−2ρ) .bωρ (z + iρ, ζ − iρ) = exp − Cωρ (z − ζ + 2iρ + it)[ωρ (t) − ωρ (0)]dt . 0
(7.21) Proof (A) We have proved that for any .ρ ∈ [0, 1) the function .Jω,λ (z, ζ, ρ) is holomorphic in .G+ . This and the properties of .bλ (z, ζ ) imply our statements on holomorphity of + .bωρ (z, ζ ) defined by (7.19) and the zero of .bωρ (z, ζ ) in .G . (B) For proving formulas (7.20) and (7.21), we calculate the integral (7.14) under the assumption that .z = x + iy ∈ G+ , .y > η − 2ρ .(0 ≤ ρ < 1) and .λ ≥ 0 is an integer. By (7.14), (7.15) and (7.9) Jω,λ (z, ζ, ρ) =
.
ff λ Σ (2(η − ρ))k k=1
1 − π
πk ff G+
G+
[ f Re
{ Re
2(η−ρ) 0
1 [−i(w − ζ )]k
} Cωρ (z − w)dμωρ (w)
] dτ Cωρ (z − w)dμωρ (w) τ − i(w − ζ + 2iρ)
≡ K1 + K2 ,
(7.22)
where, in notation .w = u + iv, K1 =
.
f λ Σ (2(η − ρ))k k
k=1
g1 (v) ≡
.
1 2π
f
f
+∞
g1 (v)dωρ (2v),
0 +∞ {
K2 =
g2 (v)dωρ (2v), 0
1
+
} 1 Cωρ (z − w)du, [i(w − ζ )]k
[−i(w − ζ )]k f +∞ { f 2(η−ρ) 1 1 dτ g2 (v) ≡ w − ζ + iτ + 2iρ 2π i 0 −∞ } 1 − Cωρ (z − w)du. w − ζ − iτ − 2iρ −∞
+∞
7 Nevanlinna–Djrbashian Classes in .G+
152
Observe that } 1 Cωρ (z − w)du −i(w − ζ ) i(w − ζ ) −∞ } f +∞ { 1 1 (−1)k−1 ∂ k−1 1 Cωρ (w + z)du − = (k − 1)! ∂ηk−1 2π i −∞ w + ζ w+ζ f +∞ Cωρ (u + iv + z) (−1)k−1 ∂ k−1 1 du = (k − 1)! ∂ηk−1 2π i −∞ u − (iv − ζ )
g1 (v) =
.
=
(−1)k−1 ∂ k−1 2π(k − 1)! ∂ηk−1
f
+∞ {
1
+
(−1)k−1 ∂ k−1 Cω (z − ζ + 2iv), (k − 1)! ∂ηk−1 ρ
since the function .Cωρ (s +iv +z) .(v > 0, .z ∈ G+ ) belongs to .H 1 by the variable .s ∈ G+ . Hence K1 =
.
λ Σ (2(η − ρ))k
k!
k=1
=
λ Σ (2(η − ρ))k
k!
k=1
(−1)k−1
k−1
(−1)
∂ k−1 ∂ηk−1
f
+∞ 0
Cωρ (z − ζ + 2iv)dωρ (2v)
[ ( ) Σ ]k λ ∂ k−1 1 2η 1 . = k −i(z − ζ ) ∂ηk−1 −i(z − ζ ) k=1
(7.23)
Besides, f
2(η−ρ)
g2 (v) = −
dτ
.
0
1 2π i
f
+∞ {
1 u − (iv − ζ + iτ + 2iρ) −∞ } 1 − Cωρ (u + iv + z)du. u − (iv − ζ + iτ + 2iρ)
Hence we conclude that: (a) If .0 < η ≤ 2ρ or, alternatively, .2ρ < η < +∞ and .v > η − 2ρ, then f
2(η−ρ)
g2 (v) = −
.
0
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ.
(b) If .2ρ < η < +∞ and .0 < v < η − 2ρ, then f
η−2ρ−v
g2 (v) =
.
0
Cωρ (z − ζ + 2iρ − iτ )dτ −
f
2(η−ρ)
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ.
η−2ρ−v
7.1 Preliminary Definitions and Statements
153
Consequently, for .0 < η ≤ 2ρ f
f
+∞
K2 = −
2(η−ρ)
dωρ (2v)
.
f
0
0 2(η−ρ)
=− 0
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ
z−ζ dτ = log , τ − i(z − ζ + 2iρ) z − ζ + 2iρ
(7.24)
and for .2ρ < η < +∞ f
f
η−2ρ
K2 =
η−2ρ−v
dωρ (2v)
.
0
0
f 0
η−2ρ−v
f
+∞
−
2(η−ρ)
dωρ (2v) η−2ρ
f
2(η−ρ)
dωρ (2v) f
0
f
η−2ρ
=
f
η−2ρ
−
η−2ρ−v
dωρ (2v) 0
0
f 0
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ
Cωρ (z − ζ + 2iρ − iτ )dτ
η−2ρ−v 0
f
+∞
−
2(η−ρ)
dωρ (2v) 0
0
f
η−2ρ
η−2ρ−v
dωρ (2v) 0
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ
dωρ (2v) f
=
f
η−2ρ
+
f
Cωρ (z − ζ + 2iρ − iτ )dτ
0
f
f
η−2ρ
+ 0
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ
Cωρ (z − ζ + 2iρ − iτ )dτ
η−2ρ+v
dωρ (2v) 2v
Cωρ (z − ζ + 2iv + 2iρ + iτ )dτ
Cωρ (z − ζ + 2iρ + iτ )dτ + log
z−ζ z − ζ + 2iρ
.
Integrating by parts, we obtain that for .2ρ < η < +∞ K2 = log
.
z−ζ z − ζ + 2iρ
f + 0
2(η−2ρ)
Cωρ (z − ζ + 2iρ + it)[ωρ (t) − ωρ (0)]dt.
Consequently, by (7.22), (7.23), (7.24) and (7.15) { } [ ]−1 exp. −Jω,λ (z, ζ, ρ) = bλ (z + iρ, ζ − iρ) ⎧ ⎪ ⎨1, if 0 < η ≤ 2ρ, f 2(η−2ρ) × ⎪ Cωρ (z − ζ + 2iρ + it)[ωρ (t) − ωρ (0)]dt, if 2ρ < η < +∞. ⎩− 0
Inserting this representation into (7.13) and (7.19), we get (7.20) and (7.21).
u n
7 Nevanlinna–Djrbashian Classes in .G+
154
7.2
Nevanlinna–Djrbashian Classes in the Halfplane
The next theorem gives the canonical representation of delta-subharmonic functions from Nω , which contains Green type potentials formed by the Blaschke type factor
.
{ bω (z, ζ ) = exp
f
2Im ζ
−
.
} Cω (z − ζ + it)ω(t)dt ,
Im z > Im ζ,
(7.25)
0
i.e. the case .ρ = 0 of (7.21). ∼α (G+ ) with .−1 < α < +∞. Then the Theorem 7.2 Let .U ∈ Nω for some .ω ∈ o inequalities (7.7) and (7.8) are true, the Green type potentials ff Pω(±) (z) ≡
.
G+
log |bω (z, ζ )|dν± (ζ )
(7.26)
by the positive and negative variations of the associated measure of U are convergent, and the following representation is true in .G+ : ff U (z) =
.
G+
log |bω (z, ζ )|dν(ζ ) +
1 π
ff G+
{ } U (w)Re Cω (z − w) dμω (w).
(7.27)
Proof For proving the convergence of the potentials (7.26), we shall use the inequalities (7.7) and (7.8). We assume that .z = x + iy, .y > d0 , where .d0 > 0 is a fixed number. Besides, we note that under (7.8), the Green potentials | | |z − ζ | | |dν± (ζ ) ≡ log |b0 (z, ζ )|dν± (ζ ) ≡ log | z−ζ| G+ G+ ρ ρ f
(±) .Pρ (z)
f
are convergent in .G+ . Further, for any .0 < ρ < d0 /32 we set ff .
ff G+ ρ
log |bω (z, ζ )|dν± (ζ ) =
G+ ρ
| | | bω (z, ζ ) | | |dν± (ζ ) log | b0 (z, ζ ) | G+ ρ
ff log |b0 (z, ζ )|dν± (ζ ) +
≡ Pρ(±) (z) + Vρ(±) (z), (±)
where the functions .Vρ
are harmonic in .G+ d0 . This is because by (7.19) ff
(±) .Vρ (z)
= −Re
G+ ρ
Jω,0 (z, ζ, 0)dν± (ζ ),
where .Jω,0 (z, ζ, 0) is holomorphic in .G+ and admits the estimate (7.18) which along with (7.8) provides the uniform convergence of the above integrals in .G+ d0 . On the other hand,
7.2 Nevanlinna–Djrbashian Classes in the Halfplane
155
for .0 < Im ζ < ρ .(< d0 /32) the function .log |bω (z, ζ )| is harmonic in .G+ ρ . By (7.25) and (7.7) ( ) f f ( f 2 Im ζ ) | | | log |bω (z, ζ )||dν± (ζ ) ≤ Cω i d0 ω(t)dt dν± (ζ ) < +∞. 2 G+ G+ \G+ 0 ρ (7.28)
ff .
Consequently, the functions .Qρ(±) (z) ≡ + .G , d0
ff
G+ \G+ ρ
log |bω (z, ζ )|dν± (ζ ) are harmonic in
and it remains to see that the definition Pω(±) (z) ≡ Pρ(±) (z) + Vρ(±) (z) + Qρ(±) (z),
.
0 < ρ < d0 /32 (0 < d0 < y),
is correct since its right-hand side does not depend on .ρ. Proceeding to the passage .ρ → +0 in (7.20), observe that by our notation .ωρ (x) = ω(x + 4ρ) the representation (7.20) can be written in the form ff U (z) . =
log |bωρ (z − iρ, ζ − iρ)|dν(ζ )
G+ 2ρ
+
ff
1 π
G+ 2ρ
{ } U (w)Re Cωρ (z − w − 2iρ) dμω (w),
z ∈ G+ 2ρ ,
(7.29)
where .ρ ∈ (0, 1). A simple calculation gives f 4ρ f +∞ e−4tρ 0 e−tσ dω(σ ) 1 − e−4tρ dt, eitz .Cωρ (z) − Cω (z) = e dt − f +∞ Iω (t) Iω (t) 4ρ e−tσ dω(σ ) 0 0 f +∞ e4tρ − 1 dt. Cωρ (z) − Cωρ (z − 2iρ) = − eitz Iωρ (t) 0 f
+∞
itz
Besides, one can be convinced that for .0 < ρ < ρ0 /2 < d0 /32 f
+∞
e
.
−tσ
f dω(σ ) ≥
4ρ
+∞ 4ρ0
e−tσ dω(σ ) ≥ e−4ρ0 t Iωρ0 (t),
and f Iωρ (t) ≥ e
.
4ρt
+∞
4ρ0
e−tσ dω(σ ) ≥ e−4(ρ0 −ρ)t Iωρ0 (t).
7 Nevanlinna–Djrbashian Classes in .G+
156
Therefore, assuming that .z = x + iy (.y > 16ρ0 ) and using the well-known theorem on the asymptotic of the Laplace transforms at infinity, we get | | |Cω (z) − Cω (z)| ≤ ω(4ρ) ρ
f
+∞
.
0
e−ty/2 dt + 4ρ Iω (t)Iωρ0 (t)
f
+∞
e−ty
0
tdt Iω (t)
M1 ω(4ρ) + ρM2 ,. (1 + y)3+α f +∞ | | tdt ρM3 |Cω (z) − Cω (z − 4iρ)| ≤ 4ρ e−ty/2 , ≤ ρ ρ Iωρ0 (t) (1 + y)3+α 0 ≤
(7.30) (7.31)
where .M1,2,3 ≡ M1,2,3 (ρ0 ) are some constants independent of z and .ρ. Using (7.30), (7.31) and (7.7), we conclude that for .0 < ρ < ρ0 /2 < d0 /32 and .y > d0 | ff | .| |
ff G+ 2ρ
U (w)Cωρ (z − w − 4iρ)dμω (w) −
ff ≤
G+ 2ρ
G+
| | U (w)Cω (z − w)dμω (w)||
| | |U (w)| |Cωρ (z − w − 4iρ) − Cωρ (z − w)| dμω (w)
ff
+
G+ 2ρ
| | |U (w)| |Cωρ (z − w) − Cω (z − w)| dμω (w)
ff .
+
G+ \G+ 2ρ
|U (w)||Cω (z − w)|dμω (w) ff
≤ [M1 ω(4ρ) + (M2 + M3 )ρ] +
Cω (8iρ0 ) π
ff G+ \G+ 2ρ
G+
|U (w)|dμω (w)
|U (w)|dμω (w) −→ 0
as
ρ → +0.
It remains to prove that for any fixed .z = x +iy .(y > 16ρ0 ) the difference of the potentials of (7.27) and (7.29) vanishes as .ρ → +0. To this end, first observe that by (7.25) and (7.7), similar to (7.28), | ff | .| |
G+ \G+ 2ρ
| | log |bω (z, ζ )|dν± (ζ )||
≤ Cω (8iρ0 )
(f
ff G+ \G+ 2ρ
2Im ζ 0
) ω(t)dt dν± (ζ ) → 0 as ρ → +0.
7.2 Nevanlinna–Djrbashian Classes in the Halfplane
157
Further, the modulus of the integrand in ff .
+ G+ 2ρ \G2ρ
log |bωρ (z − iρ, ζ − iρ)|dν± (ζ )
(0 < ρ < ρ0 )
(7.32)
0
has an integrable majorant, which is independent of .ρ ∈ (0, ρ0 ). Indeed, denoting .ζ = ξ + iη by (7.21) we obtain | | |log |bω (z − iρ, ζ − iρ)|| ≤ ρ
f
2η
.
4ρ
|Cωρ (z − ζ + it − 4iρ)| ω(t)dt f
≤ Cωρ0 (14iρ0 )
2η
ω(t)dt. 0
Hence, as .ρ → +0 the potential (7.32), the support of which is disjoint from z, tends to the part of the potential (7.26) supported in the strip .G+ \ G+ 2ρ0 . At last, using (7.19) we get ff A± ≡
.
ff log |bω (z, ζ )|dν± (ζ ) −
G+ 2ρ
0
G+ 2ρ
log |bωρ (z − iρ, ζ − iρ)|dν± (ζ ) 0
| | ff | bλ (w − iρ, ζ − iρ) | 1 |Cω (z − 4iρ − w)dμω (w) | = Re log | | ρ + b (w, ζ ) π λ G G+ 2ρ0 2ρ ff [ ] 1 + log |bλ (w, ζ )| Cωρ (z − 4iρ − w) − Cω (z − w) dμω (w) π G+ 2ρ ff
{
1 π ff
ff
−
.
≡
G+ \G+ 2ρ
G+ 2ρ
} log |bλ (w, ζ )|Cω (z − w)dμω (w) dν± (ζ )
} { Re A1 + A2 + A3 dν± (ζ ).
0
For evaluation of the quantities .A1,2,3 , note that using (7.9) with .λ = 0 and (7.15) one can prove the following inequalities: f
−∞
f
| | log |bλ (u + iv, ζ )||du ≤ 2λπ η,
+∞ |
.
|
| − iρ, ζ − iρ)| || | du ≤ Mλ,d0 ρλπ η, log |bλ (u + iv, ζ )|
+∞ | log |b (u + iv λ |
−∞
|
7 Nevanlinna–Djrbashian Classes in .G+
158
where the constant .Mλ,d0 depends only on .λ and .d0 . Besides, obviously in .A1 | | |Cω (z − w − 4iρ)| ≤ Cω (v + d0 /2) ≤ ρ ρ0
.
Md0 , (1 + v)2+α
0 < ρ < ρ0 , 16ρ0 < y,
where .Md0 depends only on .d0 . Therefore, a calculation based on (7.9) gives f |A1 | ≤ 2ρMρ0
+∞
.
0
dω(2v) < +∞. (1 + v)2+α
Evaluating the difference of kernels in .A2 , by (7.30) and (7.31) we get | | |Cω (z − w − 4iρ) − Cω (z − w)| ≤ ρ
.
Md' 0 (1 + v)3+α
[ ] M1 ω(4ρ) + (M2 + M3 )ρ ,
and hence [ ] |A2 | ≤ 2ηMρ''0 M1 ω(4ρ) + (M2 + M3 )ρ
f
+∞
.
0
dω(2v) . (1 + v)3+α
At last, it is obvious that f |A3 | ≤ 2λπ η
.
2ρ
Cω (id0 )dω(2v) = 2λπ ηCω (id0 )ω(2ρ).
0
Consequently, by (7.8) ff |A± | ≤ o(1)
.
G+ 2ρ
(1 + η)dν± (ζ ) −→ 0
as
ρ → +0.
0
We conclude this section by a representation theorem for the defined below weighted classes of delta-subharmonic functions, the harmonic parts of which are written as integrals over strips. Definition 7.3 In assumption that .A > 0 is a fixed number and .ω with .ω(+0) = 0 is a strictly increasing function in .(0, A), such that .ω(x) = ω(A) for .x > A, the class A + + and satisfy .Nω (G ) is the set of those functions U , which are delta-subharmonic in .G (7.6) and (7.1). Note that for .NA ω the proves of all preceding statements of this section become even more A−4ρ simple. In particular, one can simply take .λ = 0 and state that .oρ (z + iρ) ∈ Nωρ (.ρ < A/4) in the similarity of Lemma 7.2. As a result, using Theorem 7.1 we come to the following theorem, which we give without a proof.
7.2 Nevanlinna–Djrbashian Classes in the Halfplane
159
Theorem 7.3 Let .U ∈ NA ω , where .A > 0 is an arbitrary fixed number and .ω .(ω(+0) = 0) is a strictly increasing function in .(0, A), such that .ω(x) = ω(A) for .0 < x < +∞. Then, all statements of Theorem 7.2 remain true. Remark 7.2 Let a subharmonic in .G+ function U be such that ff . |U (z)|dμω (z) < +∞, G+
where ∼α (G+ ) .(−1 < α < +∞) or, alternatively, (I) .ω is a function of .o (II) .ω is a strictly increasing function in .(0, A) .(A > 0), such that .ω(+0) = 0 and .ω(x) = ω(A) .(A < x < +∞) and, additionally, U satisfies (7.1). Then U belongs to .Nω under (I) and to .NA ω under (II). Hence, U has the representation (7.27). Remark 7.3 If a sequence of numbers .{zk } ⊂ G+ satisfies the density condition Σf .
k
2Im zk
ω(t)dt < +∞,
(7.33)
0
then one can prove that the Blaschke -type product Bω (z, {zk }) ≡
||
.
bω (z, zk )
k
is uniformly convergent in .G+ . Therefore, in particular, when .U = log |f |, where f is a meromorphic function in .G+ , and U belongs to .Nω or .NA ω , and .ω is as required in Theorem 7.2 or Theorem 7.3, the representation (7.27) becomes the following canonical factorization: } { ff Bω (z, {an }) 1 .f (z) = log |f (w)|Cω (z − w)dμω (w) + iC , z ∈ G+ , exp Bω (z, {bm }) π G+ where C is a real number and .{an }, .{bm } ⊂ G+ are the zeros and poles of f , which satisfy (7.33).
7 Nevanlinna–Djrbashian Classes in .G+
160
7.3
Notes
The results of the chapter are published in [47].
Part II Delta-Subharmonic Extension of M.M. Djrbashian Factorization Theory
8
Extension of the Factorization Theory of M.M. Djrbashian
This chapter gives the extension of the part of M.M. Djrbashian’s factorization theory of functions meromorphic in the unit disc, which relates with the classes including R. Nevanlinna’s class N, up to an exhaustive theory of Riesz-type representations for all delta-subharmonic functions in .D. The theory is constructed by a new, simplified method based on the one-to-one property of the M.M. Djrbashian operator .Lω .
8.1
Green-Type Potentials
In this section, we shall study some properties of the Green-type potentials constructed by means of M.M. Djrbashian’s Blaschke-type factor of the form .∼ bω (z, ζ ) ≡ exp{−oω (z, ζ )} ≡ exp
| | ∼ bω (z, ζ )|
ω≡1
{
f −
1
|ζ |
[
] } ( zζ ) (z ) ω(x) −1 dx , Cω x + Cω ζ x x (8.1)
ζ − z |ζ | ≡ exp{−o . b0 (z, ζ ) = , 0 (z, ζ )} ≡ ∼ 1 − ζz ζ
∼(D) (see Definitions 1.1, 2.1 and 4.1). It is where it is assumed that .ω ∈ o(D) or .ω ∈ o easy to see that this factor differs from the factor .bω considered in Part I of this book by some holomorphic, non-vanishing in .D multiplier, namely .
) } { f |ζ | ( ) f 1( ( ) ∼ ω(x) zζ z ω(x) bω (z, ζ ) Cω x dx − −1 dx = exp Cω ζ x x x bω (z, ζ ) |ζ | |ζ |2
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_8
(8.2)
163
164
8 Extension of M.M. Djrbashian Factorization Theory
∼(D) and a fixed .ζ ∈ D the for any .z, ζ ∈ D. Hence, for any .ω ∈ o(D) or .ω ∈ o ∼ function .bω is holomorphic in .D and has a unique, simple zero at the point .z = ζ (see bω | and .Lω log |∼ b0 |, where .Lω is the Theorem 4.1.(1◦ )). Besides, both functions .Lω log |∼ integral operator of the form (1.4), are continuous and subharmonic in .D. In addition, it is easy to see that ∼ b0 (z, ζ ) = exp
.
{
f −
1
[
|ζ |
] } 1 1 dx + , − 1 zζ x 1 − ζz x 1− x
z, ζ ∈ D.
Besides, using (8.1), we simply come to the representation .Lω log ∼ bω (z, ζ ) = −
f
1
[
|ζ |
] 1 1 ω(x) dx −1 z + zζ x 1− ζx 1− x
(8.3)
which is true for all .z ∈ D except, perhaps, the radial interval .[ζ, ζ /|ζ |). Note that for .ω ∈ ∼(D) the above formulas remain in force, since the application of .Lω means multiplication o of the k-th Taylor coefficient of a function holomorphic in the neighborhood of the origin by .Ak . It is not difficult to verify that for any .ζ ∈ D \ {0} Lω log b0 (z, ζ ) = − log
.
1 + |ζ |
1{
f 0
} z ζz + ω(x)dx xz − ζ 1 − ζ xz
(8.4)
∼(D), and consequently by (8.3) we get in both cases .ω ∈ o(D) and .ω ∈ o ∼ bω (z, ζ ) b0 (z, ζ ) f |ζ | f 1 f 1 ω(x) ω(x) ζ zω(x) 1 − dx − dx − dx. = log |ζ | x − ζ /z x − ζ z 1 − ζ zx 0 |ζ | 0
Fω (z, ζ ) ≡ Lω log
.
(8.5)
It is easy to see that the function .Fω is continuous in .D, except, perhaps the point .z = ζ /|ζ |. Calculating .Uω ≡ Re Fω for any .ϑ ∈ (arg ζ, arg ζ + 2π ) we get Uω (eiϑ , ζ ) ≡ Re Fω (eiϑ , ζ ) } f 1{ 1 1 ζ eiϑ − Re = log ω(x)dx, + |ζ | x − ζ e−iϑ 1 − ζ eiϑ x 0
.
and it is obvious that for .ω ≡ 1 we have .F0 ≡ U0 ≡ 0.
(8.6)
8.1 Green-Type Potentials
165
Lemma 8.1 If .ζ ∈ D \ {0} is a fixed point, then: ∼(D), 1◦ . For any .ω ∈ o(D) or .ω ∈ o
.
| | Lω log |∼ bω (eiϑ , ζ )| = 0,
0 ≤ ϑ ≤ 2π.
.
| | 2◦ . For any .ω ∈ o(D) the function .Lω log |∼ bω | is continuous and subharmonic in .D and harmonic in .D, except the radial interval .[ζ, ζ /|ζ |).
.
| | Lω log |∼ bω (z, ζ )| < 0,
z ∈ D.
.
| | ∼(D), the function .Lω log |∼ 3◦ . For any .ω ∈ o bω | is continuous and superharmonic in .D, except the point .z = ζ , and harmonic in .D except the radial interval .[ζ, ζ /|ζ |).
.
Proof 1◦ . Passing to the real parts in (8.3), we get
.
| | bω (reiϑ , ρeiϕ )| Lω log |∼
.
f
1
= −(1 − r ) 2
ρ
ρ) 2 x cos(ϑ − ϕ) + r | | | r i(ϑ−ϕ) |2 |1 − rρ ei(ϑ−ϕ) |2 x ρ xe
1−r
| |1 −
(x
ρ
+
ω(x) dx x
(8.7)
| | for .ζ = ρeiϕ .(0 < ρ < 1) and .z = reiϑ /∈ [ρeiϕ , eiϕ ]. Hence, .Lω log |∼ bω (eiϑ , ζ )| = 0 for any .ϑ ∈ [0, 2π ], except .ϑ = arg ζ . This exception will disappear when we prove the statements .2◦ , .3◦ . | | ◦ .2 . Note that for .ω ∈ o(D) the function .Lω log |∼ bω | is subharmonic and continuous in .|z| < 1/|ζ | by Theorem 1.1(ii), while its representation proves that it vanishes on the unit disc. Consequently, by the maximum principle for subharmonic functions, | | bω (z, ζ )| ≤ 0, Lω log |∼
z ∈ D.
.
3◦ . The statement holds by Theorem 1.1(ii.' ).
u n
.
| | bω |. The next lemma gives a useful estimate for the integral means of .Lω log |∼ ∼(D) and .ζ = ρeiϕ ∈ D with .0 < d0 ≤ ρ < 1, Lemma 8.2 For any .ω ∈ o(D) or .ω ∈ o .
1 2π
f
2π 0
| ( iϑ )|| | |Lω log |∼ bω re , ζ ||dϑ ≤ Md0 ,ω
f
1
|ζ |
ω(t)dt,
where .Mω,d0 is a constant depending only on .ω and .d0 .
0 < r < 1,
(8.8)
166
8 Extension of M.M. Djrbashian Factorization Theory
| ( iϑ )| ( ( ) ) Proof By formula (8.7), .Lω log |∼ bω re , ζ | = u1 reiϑ , ζ + u2 reiϑ , ζ , where
( ) u2 reiϑ , ζ = −(1 − r 2 )
f
1
ρ
f
ρ x ω(x) ρ + x dx, | | |2 | 2 rρ r i(ϑ−ϕ) | x ρ |1 − xei(ϑ−ϕ) | |1 − ρ x e ) ( 1 − r ρx + ρx + r 2 ω(x) dx. |2 | | |2 rρ r |1 − xei(ϑ−ϕ) | |1 − ei(ϑ−ϕ) | x ρ x
( ) ϑ −ϕ u1 reiϑ , ζ = −2r(1 − r 2 ) sin2 2
.
1
It is easy to verify the following inequalities: ⎧ |ρ | ⎨ ρ | sin(ϑ − ϕ)| ≥ | | iϕ iϕ .δ(ϑ, r, ρe ) ≡ min | e − x iϑ | ≥ r ⎩ρ , ρ≤x≤1 r
2ρ π r |ϑ
− ϕ|, |ϑ − ϕ| ≤ π 2
r
sin2
⎧ ⎨1,
π 2,
≤ |ϑ − ϕ| ≤ π,
if π2 ≤ |ϑ − ϕ| ≤ π, ϑ −ϕ ≤ 2 ⎩ (ϑ−ϕ) , if |ϑ − ϕ| ≤ π . 2 2 4
Hence f ( ){ ρ | ( iϑ )| ϑ − ϕ }2 1 (1 − r 2 )ω(x) dx |u1 re , ζ | ≤ 2 1 + 1 sin | |2 |2 | rρ r i(ϑ−ϕ) | x d0 r 2 ρ |1 − xei(ϑ−ϕ) | |1 − x e ρ f 1 ω(x) ≤ A(ϑ, r, ρeiϕ )(1 − r 2 ) | dx, | rρ i(ϑ−ϕ) |2 | ρ 1− e x
.
where, by the above inequalities 1 ){ ρ −1 ϑ − ϕ }2 2( δ (ϑ, rρeiϕ ) sin 1+ ≤ A(d0 ) < +∞. d0 r 2 d0
A(ϑ, r, ρeiϕ ) =
.
Consequently, | ( iϑ )| 2 .|u1 re , ζ | ≤ A(d0 )(1 − r )
f
1 ρ
| |1 −
ω(x)
| dx.
rρ i(ϑ−ϕ) |2 x e
Further, noting that
.
1 2π
f 0
2π
| |1 −
( r 2 ρ 2 )−1 = 1 − | rρ i(ϑ−ϕ) |2 x2 e dϑ
x
8.1 Green-Type Potentials
167
and integrating by parts, we get
.
1 2π
f
2π
| ( iϑ )| |u1 re , ζ |dϑ ≤ A(d0 )(1 − r 2 )
0
f
1 ρ
ω(x)dx 1−
ρ2r 2 x2
f
1
≤ A(d0 )
ω(x)dx. ρ
For proving a similar inequality for .u2 and completing the proof, note that 1−r
(x
.
ρ
+
) ρ) r (ρ + r2 = − x (x − rρ), x ρx r
and hence, | ( iϑ )| r .|u2 re , ζ | ≤ (1 − r 2 ) d02
f
1 ρ
| | |x − ρ ||x − rρ| r | | | ω(x)dx | |1 − r xei(ϑ−ϕ) |2 |1 − rρ ei(ϑ−ϕ) |2 ρ x
In the above integral, | |2 | rρ rρ ||2 | | | |1 − ei(ϑ−ϕ) | ≥ |1 − | ≥ 1 − r > 0, x x
.
and therefore | ( iϑ )| r .|u2 re , ζ | ≤ 2 d02
f
| | |x − ρ |
1
ρ
r
| | ω(x)dx. |1 − r xei(ϑ−ϕ) |2 ρ
Further, noting that
.
1 2π
f
2π 0
| |1 −
| x 2 r 2 ||−1 | − = | , |1 | xr i(ϑ−ϕ) |2 ρ2 e dϑ
ρ
and integrating by parts, we get 1 . 2π
f 0
2π
| ( iϑ )| |u2 re , ζ |dϑ ≤ 2 d02
f ρ
1
| | |x − ρ |
| |1 −
r | ω(x)dx r 2x2 | 2 ρ
≤
2 d02
f
1
ω(x)dx. ρ
| | The next two lemmas give some comparison of .Lω log |∼ bω | and .Lω log |b0 |. ∼(D), the function Lemma 8.3 For any .ζ ∈ D \ {0}, in both cases .ω ∈ o(D) and .ω ∈ o | | | | uω (z, ζ ) ≡ Lω log |∼ bω (z, ζ )| − Lω log |b0 (z, ζ )|
.
168
8 Extension of M.M. Djrbashian Factorization Theory
is harmonic in .D, continuous in .D, and the following inequalities are true in .D: uω (z, ζ ) > 0, if ω ∈ o(D)
.
∼(D). uω (z, ζ ) < 0, if ω ∈ o
and
(8.9)
∼(D), formula (8.6), where the integral is Proof In both cases, .ω ∈ o(D) and .ω ∈ o understood in the sense of its principal value when .ϑ = arg ζ , can be written in the form 1 .uω (e , ζ ) = log − |ζ |
f
1
iϑ
0
| | | x − ζ e−iϑ | |. | ω(x)d log | 1 − ζ eiϑ x |
Hence, integrating by parts we get f
1
uω (eiϑ , ζ ) =
.
0
⎧ | | ⎨ | x − ζ e−iϑ | |dω(x) ≥ 0 log || | ⎩≤ 0, 1 − ζ e−iϑ x
| | due to the properties of .ω in Definition 1.1 and the relation .log |
ω ∈ o(D), ∼(D), ω∈o |
x−ζ e−iϑ | | 1−ζ e−iϑ x
= O(1 − x) as
x → 1−0. So, the function .uω is harmonic in .D, continuous in .D. Consequently, the above inequalities on .∂D pass to the same kind inequalities inside .D by the extremum principle u n of harmonic functions. Thus, the relation (8.9) is proved.
.
∼(D) and .|z| ≤ r < |ζ | < 1, then the following Lemma 8.4 If .ω ∈ o(D) or .ω ∈ o asymptotic formula is true uniformly with respect to z and .ϑ ∈ [0, 2π ]: oω (z, ζ ) = Sω (ze−iϑ )
f
1
.
|ζ |
} { f 1 ω(x) ω(x) dx dx + O (1 − |ζ |) x |ζ | x
as
ζ → eiϑ . (8.10)
Proof One can see that ( Cω
.
zζ |ζ |
(
) + Cω
z|ζ | ζ
)
− 1 = Sω (ze−i arg ζ ),
since .2Cω − 1 = Sω . Therefore, an integration by parts in the exponent of (8.1) gives 1
ω(x) dx |ζ | x ( ) ( )}( f 1 ) f 1{ z zζ zx zζ ω(t) + Cω' − 2 Cω' + dt dx. x ζ ζ t x |ζ | x
oω (z, ζ ) =Sω
.
( |ζ | ) f ζ
8.1 Green-Type Potentials
169
Σ∞
Further, .Cω' (z) = Consequently,
k=1 kz
| | and hence .max|z|≤r |Cω' (z)| = Cω' (r) < +∞.
k−1 /A , k
| ( ) ( )| { ( r )} | zζ ' zζ r z ' zx || ' |. − C | x 2 ω x + ζ Cω ζ | ≤ |ζ | Cω (r) + Cω |ζ | , and the desired asymptotic relation holds, since f
1
(f
) f 1 ω(x) ω(t) dx < +∞. dt dx ≤ (1 − |ζ |) t |ζ | x
1
.
|ζ |
x
Now, we are ready to prove the below theorem on Green-type potentials. ∼(D), and let .ν be a nonnegative Borel measure Theorem 8.1 Let .ω ∈ o(D) or .ω ∈ o supported in a ring .0 < d0 ≤ |ζ | < 1 and such that the Blaschke -type condition is satisfied: ff ( f
1
.
D
|ζ |
) ω(x)dx dν(ζ ) < +∞.
(8.11)
1◦ . Then, the Green type potential
.
∼ω (z) = P
ff
.
D
log |∼ bω (z, ζ )|dν(ζ )
(8.12)
is convergent in .D, where it represents subharmonic function with the Riesz measure .ν. ∼ω is continuous and subharmonic in .D and 2◦ . (i) For .ω ∈ o(D), the function .Lω P U harmonic outside the closure of the union of radial intervals . ζ ∈supp ν [ζ, ζ /|ζ |); ∼(D), if additionally the closure of the union of radial intervals (ii) For .ω ∈ o U . ζ ∈supp ν, |ζ | 0. In a similar way, we come to the estimate | | | I '' (t) Iω' (t) || | ω 3 3tδ − 2[ . |[ ]3 | ≤ C5 (1 + t) e , | Iω (t)]2 Iω (t) |
0 ≤ t < +∞,
(10.35)
where .C5 > 0 is some constant. It follows from the representation (10.32) and the estimates (10.33), (10.34), and (10.35) that Re Cω (z) =
.
y 1 + Re ψ(z), ω(A) |z|2
z ∈ G+ ρ,
(10.36)
where the function 1 1 .ψ(z) = − 2 (iz) [ω(A)]2
f 0
A
1 xdω(x) + (iz)2
f
+∞ itz
e
0
{
Iω'' (t) Iω' (t) [ ]2 − 2 [ ]3 Iω (t) Iω (t)
} dt
10 Functions of Omega-Bounded Type in .G+
232
admits the estimate |ψ(z)| ≤
.
C6 C5 + 2 2 |z| |z|
f
+∞
e−t (ρ−3δ) (1 + t)3 dt ≡
0
C7 , |z|2
z ∈ G+ ρ, u n
with some constants .C6,7 > 0. Hence, the estimate (10.32) holds by (10.36). We shall use also the following statement on Green-type potentials.
Theorem 10.6 If .ω ∈ oA (G+ ) and a Borel measure .ν ≥ 0 satisfies the condition (10.21), then for any .ρ > 0, the following inequality is true for the corresponding Green-type potential: f .
sup
+∞ |
y>ρ −∞
| |P ∼ ω (x + iy)|dx < +∞.
(10.37)
Proof Assuming that .0 < ρ/2 < A, we represent the Green-type potential in the halfplane .G+ ρ/2 in the form (10.23), i.e., as the sum of integrals .P0 (z, ρ/2) and .Uω (z, ρ/2), and estimate these integrals separately. To this end, first we observe that the condition (10.21) implies that ff .
G+ ρ
Im ζ dν(ζ ) < +∞,
ζ = ξ + iη.
Therefore, by (10.7) f .
+∞
sup
y>ρ/2 −∞
ff |P0 (x + iy, ρ/2)| dx ≤ 2π
G+ ρ /2
η dν(ζ ) < +∞.
Further, we represent .Uω (z, ρ/2) in .G+ ρ/2 as the sum of integrals .Uω (z, ρ/2) and (1)
(2)
Uω (z, ρ/2), as in (10.25). Then, by (10.2) and by the estimate (10.32), we obtain that for .0 < η < ρ/2 and .y > ρ
.
f | | ||| | |∼ | log (z, ζ ) ≤ b | | ω
η
.
f
η
≤ M1 f
0 η
≤ M2 0
{
{
} |Re Cω (z − ζ + it)| + |Re Cω (z − ζ − it)| ω(t)dt
0
} 1+y+η−t 1+y−η+t + ω(t)dt (x − ξ )2 + (y − η + t)2 (x − ξ )2 + (y + η − t)2
y ω(t)dt, (x − ξ )2 + M3 y 2
10.3 One More Property of Green-Type Potentials
233
where .M1,2,3 > 0 are some constants. Consequently f .
+∞ |
sup
y>ρ −∞
| | (1) | |Uω (x + iy, ρ/2)|dx f
ff ≤ M2 sup y>ρ
π M2 ≤√ M3
ff
G+ \G+ ρ/2
dν(ζ )
(f
G+ \G+ ρ/2
η
+∞ −∞
ydx (x − ξ )2 + M3 y 2
f
η
ω(t)dt 0
)
ω(t)dt dν(ζ ) < +∞.
0
Further, by the representation (10.4) ff Uω(2) (z, ρ/2) = −
.
G+ ρ/2
Re Jω (z, ζ )dν(ζ ),
where .Jω is the integral (10.5). Further, again by the estimate (10.32) for .y > ρ, we get f
| | y .|Re Jω (z, ζ )| ≤ M4 π
+∞
| | |Lω log |b0 (u, ζ )|| |z − u|2
−∞
f
y = M4 π
+∞
−∞
du (x − u)2 + y 2
f
du | | log |b0 (u + it, ζ )||dω(t),
A| 0
where .M4 > 0 is some constant. Consequently, by (10.7) we obtain that for any .y > ρ and .η > ρ/2 f
| |Jω (x + iy, ζ )|dx
+∞ |
.
−∞
f
f
| | log |b0 (u + it, ζ )||du y π 0 −∞ [ f +∞ ] f η = 2π M5 η dω(t) + tdω(t) ≤ M6 η, ≤ M5
A
dω(t)
η
+∞ |
f
+∞ −∞
dx (x − u)2 + y 2
0
where .M5,6 > 0 are some constants. Consequently, f .
sup
ff | | (2) | |Uω (x + iy, ρ/2)| dx ≤ M7
+∞ |
y>ρ −∞
for some constant .M7 > 0.
G+ ρ/2
ηdν(ζ ) < +∞ u n
10 Functions of Omega-Bounded Type in .G+
234
10.4
Representations of Classes of Harmonic Functions
We start by the following theorem on representations of some weighted classes of harmonic functions in .G+ . Theorem 10.7 If .ω ∈ oA (G+ ), then: 1◦ . The class of harmonic in .G+ functions U , for which
.
f .
+∞
|U (x + iy)|dx < +∞
(10.38)
|Lω U (x + iy)|dx < +∞,
(10.39)
sup
y>ρ −∞
for any .ρ > 0 and f .
+∞
sup
y>0 −∞
coincides with the set of functions representable in the form U (z) =
.
1 π
f
+∞
−∞
Re Cω (z − t)dμ(t),
z ∈ G+ ,
(10.40)
where .μ is a function of bounded variation on .(−∞, +∞). 2◦ . If the representation (10.40) is true, then the following analog of the Stieltjes inversion formula holds:
.
f μ(x) = lim
.
y→+0 0
x
Lω U (t + iy)dt
a.e.
x ∈ (−∞, +∞).
(10.41)
Proof 1◦ . First, let us verify that, if U is harmonic in .G+ , then also .Lω U is harmonic there. Indeed, U is uniformly continuous in any compact inside .G+ . Hence, if .z = x + iy ∈ G+ , then for any number .ε > 0, there is some .δ ∈ (0, y) such that
.
| ( )| |U (z + iσ ) − U z + iσ + ρeiϑ |
0 is small enough. Besides, it is easy to see that
.
1 2π
f
2π
0
f A f 2π ) ) ( ( 1 Lω U z + ρeiϑ dϑ = dϑ U z + iσ + ρeiϑ dω(σ ) 2π 0 0 f A f 2π ) ( 1 = U z + iσ + ρeiϑ dϑ dω(σ ) 2π 0 0 f A U (z + iσ )dω(σ ) = Lω U (z). = 0
Now, suppose that a harmonic in .G+ function U is such that the relations (10.38) and (10.39) are true. Then it is well-known (see, e.g., [44], Lemma 1.3 on p. 48) that (10.38) implies y−ρ .U (z) = π
f
+∞ −∞
U (t + iρ)dt , (x − t)2 + (y − ρ)2
z = x + iy ∈ G+ ρ,
for any .ρ > 0. From this representation, it follows that for any .ρ > 0 the function U is the real part of a function .fρ holomorphic in .G+ ρ and can be written as a Laplace transform. Indeed, } f 1 +∞ U (t + iρ)dt π −∞ −i(z − iρ − t) { f +∞ } f +∞ 1 = Re U (t + iρ)dt eiτ (z−iρ−t) dτ π −∞ 0 [ f +∞ ] } {f +∞ 1 e−iτ t U (t + iρ)dt dτ ≡ Re {fρ (z)}, z ∈ G+ eiτ (z−iρ) = Re ρ, π −∞ 0 (10.42) {
U (z) = Re
.
where all integrals are absolutely and uniformly convergent inside .G+ ρ . Hence, { } Lω U (z) = Re Lω fρ (z) ,
z ∈ G+ ρ,
.
where .Lω fρ is a holomorphic in .G+ ρ function representable as a Laplace transform. Indeed, f
+∞
Lω fρ (z) =
.
f
fρ (z + iσ )dω(σ )
0
f
+∞
=
+∞
dω(σ ) f
0
0 +∞
= 0
{ eiτ (z−iρ+iσ )
{
eiτ (z−iρ)
Iω (τ ) π
f
+∞
−∞
1 π
f
+∞
−∞
} e−iτ t U (t + iρ)dt dτ
} e−iτ t U (t + iρ)dt dτ,
10 Functions of Omega-Bounded Type in .G+
236
where all integrals are absolutely and uniformly convergent inside .G+ ρ , including .Iω . Further, the condition (10.39), which is true for the function .Lω U harmonic in .G+ , implies the representation Lω U (z) =
.
f
y π
+∞
−∞
dμ(t) , (x − t)2 + y 2
z = x + iy ∈ G+ ,
(10.43)
where .μ is a function of bounded variation on .(−∞, +∞). Hence, the function .Lω U is the real part of some Laplace transform, namely, of the holomorphic in .G+ function f
{
+∞
F (z) =
eiτ z
.
0
1 π
f
} e−iτ t dμ(t) dτ,
+∞ −∞
z ∈ G+ ,
and .Lω U = Re F in .G+ . Further, by (10.42) .Re F = Re Lω fρ in .G+ ρ for any .ρ > 0, and hence F (z) = Lω fρ (z) + iCρ ,
.
z ∈ G+ ρ,
where .Cρ is a real constant depending on .ρ. But for any .ρ > 0 .
lim F (iy) = lim Lω fρ (iy) = 0,
y→+∞
y→+∞
since the generating functions of the Laplace transforms representing F and .fρ are bounded. Thus, .Cρ = 0 for any .ρ > 0. So, for any .ρ > 0, the function .Lω fρ has a holomorphic continuation to the whole halfplane .G+ , where z ∈ G+ .
F (z) ≡ Lω fρ (z),
.
Consequently, by the uniqueness of the generating functions of Laplace transforms (see, e.g., [86], Chapter 2, Section 6), 1 . π
f
+∞ −∞
e
−iτ t
Iω (τ ) dμ(t) = π
f
+∞ −∞
e−iτ t U (t + iρ)dt,
0 < τ < +∞,
where the right-hand side does not depend on .ρ > 0. Hence, for any .ρ > 0 .
1 π
f
+∞ −∞
e−iτ t U (t + iρ)dt =
1 π Iω (τ )
f
+∞ −∞
e−iτ t dμ(t),
0 < τ < +∞,
10.5 Riesz-Type Representations with a Minimality Property
237
and, coming to the Laplace transforms of these functions, by (10.42), we get f
{
+∞
fρ (z) =
.
e
iτ (z−iρ)
e
iτ (z−iρ)
0
f =
0
f
1 = π =
{
+∞
+∞ {f +∞
f
+∞
−∞
f
+∞
−∞
1 π Iω (τ ) e
−∞
1 π
1 π
e
−iτ t
f
+∞
} U (t + iρ)dt dτ e
−∞
iτ (z−iρ−t)
0
−iτ t
} dμ(t) dτ
} dτ dμ(t) Iω (τ )
Cω (z − iρ − t)dμ(t),
z ∈ G+ ρ.
Consequently, for any .ρ > 0 1 .U (z) = π
f
+∞ −∞
Re Cω (z − iρ − t)dμ(t),
z ∈ G+ ρ,
and letting .ρ → +0, we obtain the representation (10.40). Conversely, let the representation (10.40) be true. Then, by the estimate (10.32), U is harmonic in .G+ , and the relation (10.38) is true. As to the relation (10.39), it follows by application of the operator .Lω to both sides of formula (10.40), and this gives (10.43). ◦ .2 . The statement is obvious in virtue of formula (10.43). u n
10.5
Riesz-Type Representations with a Minimality Property
We assume that U is a .δ-subharmonic function in the upper halfplane .G+ , and its Rieszassociated measure .ν is minimally decomposed in the Jordan sense, i.e., .ν = ν+ − ν− , where .ν± are the positive and the negative variations of the measure .ν, which are some non-negative Borel measures with non-overlapping supports in .G+ . We deal with the Tsuji characteristics of the form L(y, ±U ) =
.
1 2π
f
+∞
−∞
(±U )+ (x + iy)dx +
f
+∞ y
n∓ (t)dt,
0 < y < +∞,
where .a + = max{a, 0}, .a = a + − a − and ff n± (t) =
.
G+ t
dν∓ (ζ ),
G+ t = {ζ : Im ζ > t}.
Now, we introduce the classes of .δ-subharmonic functions in .G+ , which we shall study.
10 Functions of Omega-Bounded Type in .G+
238
+ Definition 10.2 For any .ω ∈ oA (G+ ), the class .Nm ω is the set of .δ-subharmonic in .G functions U for which
.
[ ] sup L(y, U ) + L(y, −U ) < +∞
for any
y>ρ
ρ ∈ (0, A)
(10.44)
and .
[ ] sup L(y, Lω U ) + L(y, −Lω U ) < +∞.
(10.45)
y>0
Remark 10.2 In contrast to the theory in the unit disc of the complex plane [16, 22, 39], the delta-subharmonic extensions of which is given in Chap. 8 and which is based on the equilibrium between the growth and the decrease R. Nevanlinna characteristics and their generalizations, such an equilibrium, i.e., a version of B.Ya. Levin formula (see [64], Ch. 4, Sec. 2), is not true for all functions delta-subharmonic in the halfplane (see [44], Ch. 3). Therefore, it is natural to define the class .Nm ω by the restrictions (10.44) and (10.45) which are on both growth and decrease Tsuji characteristics, as it is done also in Chapter 4 of [44]. The following theorem gives the descriptive Riesz-type representations of the classes .Nm ω. Theorem 10.8 If .ω ∈ oA (G+ ), then: 1◦ . The class .Nm ω coincides with the set of functions of the form
.
U (z) =
.
1 π
f
+∞ −∞
ff
+
G+
{ } Re Cω (z − t) dμ(t) | | log |∼ bω (z, ζ )|dν(ζ ),
z ∈ G+ ,
(10.46)
where .μ is a function of bounded variation on .(−∞, +∞) and .ν = ν+ − ν− , where ν± (ζ ) are non-negative Borel measures in .G+ , such that
.
(f
ff .
G+
0
Im ζ
) ω(x)dx dν± (ζ ) < +∞.
(10.47)
2◦ . The measure .μ in (10.46) is revealed by the Stieltjes inversion formula:
.
f μ(x) = lim
.
y→+0 0
x
Lω Uω (t + iy)dt,
x ∈ (−∞, +∞).
(10.48)
10.5 Riesz-Type Representations with a Minimality Property
239
Proof 1◦ . Let .U ∈ Nm ω . Then by (10.44) for any .ρ ∈ (0, A)
.
(f
ff .
G+ ρ
η−ρ
0
) ω(x)dx dν± (ζ ) (f
ff ≤
G+ ρ
η 0
) ω(x)dx dν± (ζ ) < +∞
(ζ = ξ + iη).
(10.49)
Indeed, the first inequality is obvious. For proving the second one, observe that for any ρ ∈ (0, A)
.
(f
ff .
G+ ρ
η
0
) ( ff ω(x)dx dν± (ζ ) = ff ≤
G+ A
G+ A
(f ff
≤ ω(A)
A 0
)( f
ff +
+ G+ ρ \GA
)
ω(x)dx dν± (ζ ) + ω(A) ff
G+ ρ
η dν± (ζ ) ≤ ω(A)
η 0
) ω(x)dx dν± (ζ )
ff + G+ ρ \GA
ηdν± (ζ )
( ρ) dν± (ζ ) < +∞, η− 2 G+ ρ/2
since the condition (10.44) is true for .ρ/2 in particular. By (10.49) and Theorem 10.2, the Green-type potentials in .G+ ρ , with the measures .ν± , are convergent, and hence the function ff | | .U0 (z) = U (z) − log |∼ bω (z − iρ, ζ − iρ)|dν(ζ ) G+ ρ
is harmonic in .G+ ρ . Consequently, also the function ff Lω U0 (z) = Lω U (z) −
.
G+ ρ
| | Lω log |∼ bω (z − iρ, ζ − iρ)|dν(ζ )
(10.50)
is harmonic in .G+ ρ . Besides, by the continuity of .Lω U0 and of the Green-type potential, the function .Lω U in the above formula is continuous and delta-subharmonic in .G+ ρ, and hence in the whole .G+ . Further, it is obvious that f .
sup
| |Lω U0 (x + iy)|dx ≤ sup
+∞ |
y>0 −∞
f
| |Lω U (x + iy)|dx
+∞ |
y>0 −∞
| | | +∞ |f f | | | | |∼ | + sup Lω log bω (x + iy − iρ, ζ − iρ) dν(ζ )| dx | | G+ y>0 −∞ | ρ f
≡A + B,
10 Functions of Omega-Bounded Type in .G+
240
where [ ] A ≤ sup L(y, Lω U ) + L(y, −Lω U ) < +∞
.
y>ρ
by (10.45), and .B < +∞ by the estimate (10.29). Thus, Lω U0 (z + iρ) =
.
f
y π
+∞
−∞
dμ(t) , (x − t)2 + y 2
z = x + iy ∈ G+ ,
where .μ is of bounded variation on .(−∞, +∞). Moreover, the measure .dμ(t) of the above representation is absolutely continuous and is equal to .Lω U (t + iρ)dt. For proving this, observe that for any function f continuous in .(−∞, +∞) and such that .limx→±∞ f (x) = 0 f lim
.
+∞
y→+0 −∞
f f (x)Lω U0 (x + iy + iρ)dx =
+∞ −∞
f (x)dμ(x)
(see, e.g., [30], Ch. 1, Theorems 5.3 and 3.1(c)). Then, for any interval .[a, b] ⊂ (−∞, +∞), introduce the sequence of functions .{fn }∞ 1 assuming that .fn ≡ 1(a ≤ x ≤ b) and .fn (x) ≡ 0(x /∈ [a − 1/n, b + 1/n]) and continuing .fn to the remaining intervals of .(−∞, +∞) as a continuous, linear function. Then, by the continuity of .Lω U (t + iρ) and the relation (10.30) f
+∞
.
−∞
f fn (x)Lω U (x + iρ)dx = lim
+∞
y→+0 −∞
f
=
fn (x)Lω U (x + iy + iρ)dx
+∞
−∞
fn (x)dμ(x)
for any .n = 1, 2, . . .. On the other hand, f
b
.
f Lω U (x + iρ)dx = lim
+∞
n→∞ −∞
a
f
b
fn (x)Lω U (x + iρ)dx =
dμ(x). a
Thus, .dμ(t) ≡ Lω U (t + iρ)dt. Besides, .Lω U (t + iρ) ∈ L1 (−∞, +∞) and y−ρ .Lω U0 (z) = π
f
+∞
−∞
Lω U (t + iρ) dt, (x − t)2 + y 2
z = x + iy ∈ G+ ρ.
Consequently, by (10.50) ff Lω U (z) −
.
| | y−ρ Lω log |∼ bω (z − iρ, ζ − iρ)|dν(ζ ) = + π Gρ
f
+∞
−∞
Lω U (t + iρ) dt (x − t)2 + y 2
10.5 Riesz-Type Representations with a Minimality Property
241
for any .z = x + iy ∈ G+ ρ , and .
} { ff | | 1 lim y Lω U (iy) − Lω log |∼ bω (z − iρ,ζ − iρ)|dν(ζ ) 2 y→+∞ G+ ρ f +∞ 1 = Lω U (t + iρ)dt. 2π −∞
On the other hand, by the representation (10.10) .
y y→+∞ 2
ff
lim
G+ ρ
| | Lω log |∼ bω (z − iρ, ζ − iρ)|dν(ζ ) = −
(f
ff G+ ρ
η−ρ
) ω(t)dt dν(ζ ),
0
where .ζ = ξ + iη. Thus 1 . lim yLω U (iy) + 2 y→+∞
ff
(f
)
η−ρ
ω(t)dt dν(ζ )
G+ ρ
0
=
1 2π
f
+∞
−∞
Lω U (t + iρ)dt,
(10.51)
where all quantities are finite. Now, observe that by the relation (10.30) and Theorem 10.5 the integrals in (10.50) taken by the components .dν+ and .dν− of the measure .dν are ordinary Green potentials (+) (−) in .G+ ρ , i.e., there exist some non-negative Borel measures .νω and .νω with nonoverlapping supports in .G+ ρ and such that ff .
and ff .
G+ ρ
G+ ρ
(η − ρ)dνω(±) (ζ ) < +∞
| | Lω log |∼ bω (z − iρ, ζ − iρ)|dν± (ζ ) =
ff G+ ρ
log |b(z − iρ, ζ − iρ)| dνω(±) (ζ ),
where .b(z, ζ ) = (z−ζ )/(z−ζ ) is the ordinary Blaschke factor. Note that the measures (±) dνω are independent of .ρ, since they are the positive and the negative variations of the Riesz measure of the function .Lω U . Further, by a passage to the limit in the last formula, we obtain that for any .ρ ∈ (0, A)
.
(f
ff .
G+ ρ
η−ρ 0
) ω(t)dt dν± (ζ ) =
ff G+ ρ
(η − ρ)dνω(±) (ζ ) < +∞.
(10.52)
10 Functions of Omega-Bounded Type in .G+
242
Inserting this equality in (10.51), we come to the B.Ya. Levin formula for the function Lω U . Then, by some simple rearrangement of terms, we get the following equilibrium relation for the Tsuji characteristics:
.
.
1 lim yLω U (iy) + L(ρ, −Lω U ) = L(ρ, Lω U ), 2 y→+∞
0 < ρ < A.
Hence, by (10.45) ff .
G+
η dνω(±) (ζ ) < +∞,
which implies (10.47) in virtue of (10.52). So, the relation (10.47) is true. Consequently, the Green-type potential in (10.46) converges and the harmonic in .G+ function ff U (z) −
.
G+
| | log |∼ bω (z, ζ )|dν(ζ ),
z ∈ G+ ,
satisfies the conditions (10.38) and (10.39) of Theorem 10.7. Indeed, U satisfies (10.38) in virtue of (10.44), and the Green-type potential by the estimate (10.37). As to the condition (10.39), U satisfies this condition by (10.39), and the Green-type potential by the estimate (10.29). Thus, the considered function is of the form (10.40), and U is of the form (10.46). Conversely, let U be representable in the form (10.46). Then, obviously, U is a .δsubharmonic function in .G+ . Further, the Green-type potential in (10.46) satisfies the condition (10.38) by (10.37), and it satisfies the condition (10.39) by (10.29). Besides, the function 1 . π
f
+∞ −∞
Re{Cω (z − t)}dμ(t)
in (10.46), which is harmonic in .G+ , satisfies these conditions by Theorem 10.7. Thus, m .U ∈ Nω . ◦ .2 . The relation (10.48) follows from (10.41) and (10.30), since y .Lω U (z) = π
f
+∞
−∞
dμ(t) + (x − t)2 + y 2
ff G+
log |b(z, ζ )|dνω (ζ ),
z = x + iy ∈ G+ ,
where .νω is a Borel measure such that its positive and negative variations satisfy the condition ff . Im ζ dνω(±) (ζ ) < +∞. G+
10.6 Notes
243
Remark 10.3 In particular, the above theorem implies that the class of those functions f meromorphic in .G+ , for which .log |f | ∈ Nm ω , coincides with the set of functions admitting a factorization } { f +∞ ∼ {ak }) B(z, 1 .f (z) = Cω (z − t)dμ(t) + iC , exp ∼ {bn }) π −∞ B(z,
z ∈ G+ ,
(10.53)
where .μ is a function of bounded variation on .(−∞, +∞), C is a real number, and .{ak } ⊂ G+ , .{bn } ⊂ G+ are the zeros and the poles of f , which satisfy the density conditions Σf .
k
Im ak
ω(t)dt < +∞,
0
Σf n
Im bn
ω(t)dt < +∞.
0
If this factorization is true, then the following analog of the Stieltjes inversion formula is valid: f x .μ(x) = lim Lω log |f (t + iy)|dt, a.e. x ∈ (−∞, +∞). y→+0 0
Remark 10.4 Note that a change of the integration orders in the exponent of the factorization (10.53) and in the harmonic part of the representation (10.46) gives a Laplace transform and the real part of a Laplace transform.
10.6
Notes
The results of the chapter are published in [50, 52].
Subclasses of Harmonic Functions with Nonnegative Harmonic Majorants in the Halfplane
11
In this chapter, we consider some .ω-weighted classes of harmonic functions possessing nonnegative harmonic majorants in the upper halfplane .G+ . The descriptive representations for these classes are found. A description of the boundary values of the functions from the considered classes is given by means of the notion of .ω-capacity on the real axis, which becomes an analog of O. Frostman’s .α-capacity in a particular case. So, we consider some classes of harmonic in .G+ functions u, the .ω-partial derivatives m .Lω u of which belong to the E.D. Solomentsev [83] class .N described by Theorem 10.5, i.e., f .
sup
+∞ |
y>0 −∞
| |Lω u(x + iy)|dx < +∞,
where .Lω is the fractional differentiation operator (1.21). Besides, we use the Cauchy-type kernel of the form (1.22). Everywhere in this chapter, we assume that .ω is a function of the class .oαN (G+ ) introduced by Definition 1.3. Note that in particular .oαN (G+ ) contains all functions of the form .ω(x) = x α with .−1 < α ≤ 0.
11.1
Preliminary Lemmas
The statements of this section are obvious for .ω ≡ 1, and therefore we omit this case in the proofs. We start by some properties of the kernel .Cω .
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_11
245
11 Subclasses of Harmonic in .G+ Functions
246
Lemma 11.1 If .ω ∈ oαN (G+ ), then the function .Cω is holomorphic in .G+ , and for any .ρ > 0, there is a constant .Mρ,ω > 0 depending only on .ρ and .ω, such that |Cω (x + iy)| ≤
.
Mρ,ω , y 1+α
−∞ < x < +∞,
ρ < y < +∞.
(11.1)
Proof Let .0 < ρ < y < +∞ be any numbers. Then, for .0 < t < 1/A0 | | | ei(x+iy)t | e−ρt e−yt | | .| ≤ f +∞ | = f +∞ | Iω (t) | t e−xt ω(x)dx t 1/t e−xt ω(x)dx 0 ≤
e−ρt e−ρt t α 1 1 ≤ , f +∞ f +∞ C1 t 1/t e−xt x α dx C1 1 e−λ λα dλ
while for .1/A0 < t < +∞ | | | ei(x+iy)t | e−ρt e−yt | | .| ≤ fA | = f +∞ | Iω (t) | t e−xt ω(x)dx t 0 0 e−xt ω(x)dx 0 ≤
e−ρt e−ρt 1 1 . ≤ f A0 ω(A0 ) t ω(A0 ) 1 − e−tA0 e−xt dx 0
Thus, for any .ρ > 0, the integrand in (1.22) has a summable majorant. Hence, the integral (1.22) is uniformly convergent in the halfplane .y > ρ, and the function .Cω is holomorphic in .G+ . Besides, similarly we obtain that for .y > 0 great enough and any .x ∈ (−∞, +∞) f +∞ −yt e dt 1 + f +∞ −λ α e λ dλ ω(A0 ) 1/A0 1 − e−tA0 0 1 f 1/A0 f +∞ 1 1 −yt α ≤ e t dt + e−yt dt C3 0 C4 ω(A0 ) 1/A0 f y/A0 f +∞ 1 Kω 1 −u α e u du + e−u du ≤ 1+α , ≤ 1+α yC4 ω(A0 ) y/A0 y C3 0 y
1 .|Cω (z)| ≤ C2∗
f
1/A0
e−yt t α dt
where .Kω > 0 is a constant depending solely on .ω. Thus, (11.1) is true.
u n
Note that .Lω Cω (z) = 1/(−iz), for any .z ∈ G+ . Indeed, if .ω ∈ oαN (G+ ), then by Fubini’s theorem ( Lω Cω (z) = Lω1
.
) f +∞ ( ) ∂ ∂ Cω (z) = Cω (z + iσ ) dω1 (σ ) − − ∂y ∂y 0
11.1 Preliminary Lemmas
247
f
+∞
=−
∂ ∂y
0
f
(f 0
f
+∞
=
+∞
ω(σ )dσ 0
0
) ei(z+iσ )t dt ω(σ )dσ Iω (t)
+∞
tei(z+iσ )t dt 1 = . Iω (t) −iz
For applying .Lω to some other functions, several definitions are to be given. Definition 11.1 A domain .G ⊆ C is said to be .∞-starshaped, if it contains the infinite interval .z + ih, .0 < h < +∞ along with any point .z ∈ G. Definition 11.2 Let .ω ∈ oαN (G+ ). Then, a function u given in an .∞-starshaped domain G is said to be of the class .Mω , if there exists an angular domain .A(δ0 , R0 ) = {z : |π/2 − arg z| ≤ δ0 , |z| ≥ R0 } with some .0 < δ0 ≤ π/2 and .0 < R0 < +∞, such that f .
| | | | ∂ u(z + iσ )|ω(σ )dσ < +∞ | | ∂y
+∞ |
sup z∈K 0
(11.2)
for any compact .K ⊂ G ∩ A(δ0 , R0 ). + Lemma 11.2 Let .ω ∈ oN α (G ), and let G be an .∞-starshaped domain containing all + halfplanes .Gρ = {z = x + iy : y > ρ} with .ρ ≥ ρ0 , where .ρ0 > 0 is a fixed number. Then, the operator .Lω maps one-to-one the set of functions .u ∈ Mω harmonic in G, such that | f +∞ | | |∂ | . sup (11.3) u(x + iy)|| dx < +∞, | ∂y y>ρ0 −∞
to itself, up to a summand .a0 + a1 x with some real constants .a0 , a1 . Proof To see that .Lω u is continuous in G, we set .u1 ≡ −∂u/∂y and observe that |f | | | |. Lω u(z0 ) − Lω u(z)| =| | f ≤
+∞ 0 ρ
0
f u1 (z0 + iσ )ω(σ )dσ −
+∞
0
| | u1 (z + iσ )ω(σ )dσ ||
| | |u1 (z0 + iσ ) − u1 (z + iσ )|ω(σ )dσ
f
+ ρ
+∞ |
| |u1 (z0 + iσ )|ω(σ )dσ +
f
+∞ |
| |u1 (z + iσ )|ω(σ )dσ
ρ
for any .z, z0 ∈ G, .ε > 0 and .ρ > 0. In the right-hand side of this inequality, the last two integrals become less than .ε/3 for .ρ large enough. Indeed, the function .u1 is of the harmonic Hardy space .h1 (G+ ); hence .|u1 (x + iσ )| ≤ Ku1 /σ and .ω(σ ) x σ α (−1 < α
0 for which the compact .K0 = K + id0 is in the domain .G ∩ A(δ0 , R0 ). Hence, for any .z ∈ K ( f d0 f +∞ ) f +∞ | | | | |u1 (z + iσ )|ω(σ )dσ = |u1 (z + iσ )|ω(σ )dσ ≡ I1 + I2 , . + 0
0
d0
where f sup I1 ≤
.
z∈K
max
z∈K, 0≤σ ≤d0
|u1 (z + iσ )|
d0
ω(σ )dσ < +∞,
0
while by (11.2) f
+∞ |
| |u1 (z + iσ )|ω(σ )dσ =
.
d0
f ≤
f
+∞ |
| |u1 (z + id0 + iσ )|ω(σ + d0 )dσ
0 +∞ |
| |u1 ((z + id0 ) + iσ )|ω(σ )dσ ≤ sup
0
f
+∞ |
| |u1 (z + iσ )|ω(σ )dσ < +∞,
z∈K0 0
since .ω is a decreasing function. Hence, using Fubini’s theorem we conclude that for any z ∈ G and a small enough .r > 0
.
.
1 2π
f
2π
) ( ∂ Lω1 − u(z + reiθ ) dθ ∂y 0 ) f 2π f +∞ ( ∂ 1 u(z + reiθ + iλ)dθ dω1 (λ) =− 2π 0 ∂y 0 ) f +∞ ( ∂ = u(z + iλ) dω1 (λ) = Lω u(z). − ∂y 0
Lω u(z + reiθ )dθ =
0
1 2π
f
2π
Thus, .Lω u is harmonic in G. Further, because of (11.3), the harmonic in G function .u1 belongs to the class .Nm in + 1 + + .Gρ (also, to the harmonic .h (Gρ )) with any .ρ > ρ0 . By Theorem 10.5, for any .z ∈ Gρ f +∞ f f +∞ u1 (t + iρ) 1 1 +∞ u1 (t + iρ)dt ei(z−iρ−t)σ dσ dt = Re π i −∞ t + iρ − z π −∞ 0 ) ( f +∞ f +∞ 1 ei(z−iρ)σ e−itσ u1 (t + iρ)dt dσ = Re π 0 −∞ f +∞ ei(z−iρ)σ ϕ(σ )dσ, ≡ Re
u1 (z) = Re
.
0
11.1 Preliminary Lemmas
249
where the function .ϕ is bounded in .0 ≤ σ < +∞. Hence ( ) f +∞ ∂ .Lω u(z) = Lω1 u1 (z + iλ)dω1 (λ) − u(x + iy) = ∂y 0 f +∞ f +∞ = Re dω1 (λ) ei(z+iλ−iρ)σ ϕ(σ )dσ 0
f = Re
0
+∞
( f i(z−iρ)σ ϕ(σ ) e
0
+∞
e
−σ λ
) dω1 (λ) dσ,
0
where the Laplace transform ( f f +∞ .F (z) = ei(z−iρ)σ ϕ(σ ) 0
+∞
z = x + iy ∈ G+ ρ,
) e−σ λ dω1 (λ) dσ
0
+ is a holomorphic function in .G+ ρ , and the condition .Lω u ≡ 0 in a halfplane .Gρ (ρ > ρ0 ) implies .F ≡ iC with some real constant C. Note that .ϕ is a bounded function in .0 ≤ σ < +∞, while because of the properties of .ω the function
f
+∞
e
.
−σ λ
f
+∞
dω1 (λ) =
0
e−σ λ ω(λ)dλ
0
is bounded and continuous in .0 < δ < σ < +∞ for any .δ > 0. Besides, it is well-known (see, e.g., [86], p. 182, Corollary 1a) that the relation .ω(λ) x λα (λ → +∞, −1 < α < 0) implies f
+∞
.
e−σ λ dω1 (λ) =
f
0
+∞
e−σ λ ω(λ)dλ x
0
1 → 0 as σ 1+α
σ → +∞.
Thus, the generating function of the Laplace transform F , i.e., the function f
+∞
ϕ(σ )
.
e−σ λ dω1 (λ),
0 < σ < +∞,
0
is bounded, and hence .|F (iy)| ≤ My −1 as .y → +∞, where .M > 0 is some constant. Consequently .C = 0, and by the uniqueness of the generating function of the Laplace transform f
+∞
ϕ(σ )
.
e−σ λ dω1 (λ) ≡ 0,
0 < σ < +∞,
0
where f .
0
+∞
e−σ λ dω1 (λ) > 0,
0 < σ < +∞.
11 Subclasses of Harmonic in .G+ Functions
250
Thus, .ϕ ≡ 0, i.e., ∂ .u1 (x + iy) = − u(x + iy) = Re ∂y
f
+∞
ei(z−iρ)σ ϕ(σ )dσ ≡ 0,
y > ρ.
0
But the function .u1 is harmonic in the whole G, and hence .u1 ≡ 0 in the whole G by the uniqueness of harmonic function. Hence, if .Lω u ≡ 0 in G, then u(z) ≡ a0 + a1 x,
.
z = x + iy ∈ G, u n
where .a0 and .a1 are some real numbers depending solely on the function u.
11.2
Representations
The main result of this section is on the representations of the introduced below classes of functions. m + + Definition 11.3 If .ω ∈ oN α (G ), then .Nω is the class of real, harmonic in .G functions u which belong to .Mω and satisfy the condition (11.3) for some .ρ0 > 0, along with
f .
sup
+∞ |
y>0 −∞
| |Lω u(x + iy)|dx < +∞.
(11.4)
Theorem 11.1 1◦ . For any .ω ∈ oαN (G+ ), the class .Nm ω coincides with the set of functions representable in the form
.
1 .u(z) = a0 + a1 x + π
f
+∞
−∞
Re Cω (z − t)dμ(t),
z = x + iy ∈ G+ ,
(11.5)
where .a0 and .a1 are real constants and .μ is a function of bounded variation in (−∞, +∞).
.
2◦ . If the representation (11.5) is true, then for any .z = x + iy ∈ G+
.
u(z) =a0 + a1 x ] )−1 f +∞ f +∞ [ ( f +∞ −tσ −itλ itz 1 + Re e ω(σ )dσ e dμ(λ) dt. e t π 0 −∞ 0
.
(11.6)
11.2 Representations
251
Besides, .
lim u(x + iy) = a0 + a1 x,
y→+∞
−∞ < x < +∞,
(11.7)
while the measure .μ in formulas (11.5) and (11.6) can be revealed by the following analog of the Stieltjes inversion formula: f μ(t) = lim
.
y→+0 0
t
Lω u(x + iy)dx,
−∞ < t < +∞.
(11.8)
Proof 1◦ . Let the representation (11.5) be true. Then, the integral in (11.5) uniformly converges inside .G+ due to the estimate (11.1). Hence, the function u is harmonic in .G+ . Further,
.
−
.
f +∞ eizt dt ∂ ∂ Cω (z) = − f +∞ ∂y 0 ∂y t 0 e−tx dω1 (x) f +∞ eizt dt = = Cω1 (z), f +∞ e−tx dω1 (x) 0 0
z = x + iy ∈ G+ ,
(11.9)
fx where .ω1 (x) = 0 ω(λ)dλ is a nondecreasing function such that .ω1 (x) x x 1+α as .x → +∞, and .ω1 satisfies the conditions of Lemma 5.3. Therefore, by (11.1), Fubini’s theorem, and the estimate (5.19), we conclude that for any .z ∈ G+ ρ with a fixed .ρ > 0 and any .ε ∈ (0, 1) f
| | | f +∞ | f | | ∂ 1 +∞ | | | ∂ u(z + is)|dω1 (s) = Cω (z + is − t)dμ(t)|| dω1 (s) | | ∂y π | ∂y 0 −∞ | f +∞ |f +∞ | | 1 | Cω1 (z + is − t)dμ(t)|| ω(s)ds = | π 0 −∞ ( ) +∞ f A0 f +∞ V Mρ,ω,ε ' −2+ε ≤ ω(s)ds + M s ds μ < +∞, ρ,ω,ε 2+α−ε ρ 0 A0 −∞
+∞ |
.
0
' where .Mρ,ω,ε andMρ,ω,ε > 0 are some constants depending only on .ρ, .ω, and .ε. Hence .u ∈ Mω .
11 Subclasses of Harmonic in .G+ Functions
252
Further, using the simple two-sided inequality .(a + b)λ x a λ + bλ (a, b, λ ≥ 0), in the same way, we conclude that for any .y > ρ > 0 and .ε ∈ (0, 1 + α) there exists a constant .Mρ,ω > 0 such that f
| | | f f +∞ | | | | ∂ 1 +∞ | ∂ u(x + iy)|dx = | Cω (z − t)dμ(t)|| dx | | ∂y | ∂y π −∞ −∞ ) f +∞ f +∞ f +∞(f +∞ | | |dμ(t)| 1 | | Cω1 (z − t) |dμ(t)| dx ≤ Mρ,ω dx ≤ π −∞ |z − t|1+ε −∞ −∞ −∞ f +∞ f +∞ 1 ' ≤ Mρ,ω |dμ(t)| ds 1+ε + ρ01+ε −∞ −∞ |s| f +∞ 1 ≤ Mρ,ρ0 ,ω du < +∞, 1+ε −∞ (|u| + 1)
+∞ |
.
−∞
' where .Mρ,ω > 0 and .Mρ,ρ0 ,ω > 0 are some constants. Thus, (11.3) is true. Now, observe that for any .z = x + iy ∈ G+
1 .Lω u(z) = Re π 1 = Re π 1 = Re π
+∞ (
f
∂ − ∂y 0 f +∞ ( f +∞ f
−∞
0
f
+∞
−∞
) Cω (z + is − t)dμ(t) dω1 (s) )
Cω1 (z + is − t)dμ(t) dω1 (s)
+∞ ( −∞
) 1 Lω1 Cω1 (z − t) dμ(t) = Re πi
f
+∞
−∞
dμ(t) , t −z
since the above integrals are absolutely and uniformly convergent inside .G+ . Thus, f dμ(t) y +∞ .Lω u(z) = , z = x + iy ∈ G+ , (11.10) π −∞ (x − t)2 + y 2 which implies the condition (11.4): f .
+∞
sup
y>0 −∞
f |Lω u(x + iy)| dx ≤
+∞
−∞
|dμ(t)| =
+∞ V
μ < +∞.
−∞
Thus, .u(z) ∈ Nm ω. Conversely, let .u ∈ Nm ω . Then .u ∈ Mω , and hence the function .Lω u is harmonic + in .G by Lemma 11.2. Besides, .Lω u satisfies the condition (11.4), and therefore by V Theorem 10.5 a representation of the form (11.10) is true, where . +∞ −∞ μ < +∞. Hence Lω u(z) = Re
.
1 πi
f
+∞
−∞
dμ(t) 1 = Re t −z π
f
+∞ ( f +∞
−∞
0
) ei(z−t)s ds dμ(t),
z ∈ G+ ,
11.3 Boundary Values
253
and at any point .z ∈ G+ 1 .Lω u(z) = Re π
+∞ ( f +∞
f
−∞
f
0
f
+∞
f +∞ −λs ) e dω1 (λ) i(z−t)s 0 e ds dμ(t) f +∞ e−σ t dω1 (σ ) 0 f
+∞
+∞
ei(z−t+iλ)s ds f +∞ e−σ s ω(σ )dσ −∞ 0 0 0 [ ( ) ] f +∞ f +∞ f 1 +∞ ei(z−t+iλ)s ds ∂ = Re − dμ(t) dω1 (λ) f +∞ ∂y 0 π −∞ s 0 e−σ s ω(σ )dσ 0 [ ]) f +∞ ( f ∂ 1 +∞ − Re = Cω (z + iλ − t)dμ(t) dω1 (λ) ≡ Lω u∗ (z), ∂y π −∞ 0
= Re
1 π
dμ(t)
dω1 (λ)
where 1 .u (z) = Re π ∗
f
+∞ −∞
Cω (z + iλ − t)dμ(t) ∈ Nm ω.
∗ + Besides, it is obvious that .u − u∗ ∈ Nm ω and .Lω (u − u ) ≡ 0, .z ∈ G . Consequently, Lemma 11.2 implies the representation (11.5). ◦ .2 . The representation (11.6) is a consequence of (11.5). The relation (11.7) follows from (11.6) and (11.1), while (11.8) is the classical Stieltjes inversion formula for u n the measure .μ in (11.10).
11.3
Boundary Values
We start by some preliminary definitions and lemmas. Definition 11.4 Let .E ⊆ (−∞, +∞) be a Borel measurable set (B-set) and .ω ∈ oαN (G+ ). Then we say that E is of positive .ω-capacity, or .Cω (E) > 0, if there exists a Borel measure (B-measure) .τ ≥ 0 supported on .E(τ ≺ E) and such that f +∞ . dτ (t) = 1 (11.11) −∞
and f S1 ≡ sup
+∞
.
z∈G+ −∞
|Cω (z − t)|dτ (t) < +∞.
(11.12)
If there is no such a measure, i.e., .S1 = +∞ for any nonnegative B-measure .τ ≺ E satisfying (11.11), then we say that E is of zero .ω-capacity, i.e., .Cω (E) = 0. Below, we give some auxiliary lemmas related to .ω-capacities.
11 Subclasses of Harmonic in .G+ Functions
254
Lemma 11.3 Let .E1 and .E2 ⊂ (−∞, +∞) be any B-sets such that .Cω (E1 ) = Cω (E2 ) = 0 for some .ω ∈ oαN (G+ ). Then .Cω (E1 ∪ E2 ) = 0. Proof Suppose .τ ≺ E1 ∪ E2 is a nonnegative B-measure satisfying (11.11). Taken over E1 or .E2 , the integral (11.12) equals to a number .M ∈ (0, 1].| For instance, let this be true for the integral over .E1 . Then also the measure .τ1 = M −1 τ |E supported on .E1 satisfies 1 (11.11). But .Cω (E1 ) = 0, and hence
.
f .
+∞
sup
z∈G+ −∞
f
+∞
|Cω (z − t)|dτ (t) ≥ M sup
z∈G+ −∞
|Cω (z − t)|dτ1 (t) = +∞.
Lemma 11.4 If .ω ∈ oαN (G+ ), then the Volterra equation f x . ω(x − t)d∼ ω(t) ≡ 1, 0 < x < +∞, 0
ω such that .∼ ω(0) = 0 and .∼ ω(x) ≤ [ω(x)]−1 for .0 < x < has a nondecreasing solution .∼ +∞. Besides, Cω (z) = L∼ ω
.
for the operator .L∼ ω f (z) ≡
f +∞ 0
( 1 ) , −iz
z ∈ G+ ,
(11.13)
f (z + iσ )d∼ ω(σ ).
Proof By Theorem 1.9 with .ω1∗ ≡ 1, .ω2∗ ≡ ω, we conclude that the function .α ≡ ∼ ω with the desired properties exists. Hence, for any .t > 0 f
+∞
e
.
0
−tσ
f
+∞
d∼ ω(σ )
e
−tλ
0
f ω(λ)dλ =
f =
+∞
d∼ ω(σ ) 0
+∞
f
+∞
e−tμ dμ
σ
f
0
e−tμ ω(μ − σ )dμ
μ
ω(μ − σ )d∼ ω(σ ) =
0
1 , t
(11.14)
due to the absolute convergence of these integrals, which is obvious for those with .ω(λ)dλ and .d∼ ω(σ ), since for any .t ∈ (−∞, +∞) there are some numbers .M1 > 0 and .α ∈ (−1, 0) such that f .
0
+∞
|+∞ f | e−tσ d∼ ω(σ ) = e−tσ ∼ ω(σ )|| +t f ≤t
σ =0
+∞
0
≤
t ω(A0 )
e−tσ dσ ω(σ ) f
A0 0
+∞
e−tσ ∼ ω(σ )dσ
0
(f
=t
A0
+∞ ) e−tσ dσ
f +
0
e−tσ dσ + M1 t
f
A0 +∞
A0
ω(σ ) e−tσ σ −α dσ < +∞.
11.3 Boundary Values
255
By (11.14), )−1 f +∞ ( f +∞ 1 −tλ e ω(λ)dλ = e−tσ d∼ ω(σ ) . = t Iω (t) 0 0 for any .t ∈ (0, +∞), and consequently ) (f +∞ f +∞ f +∞ dt eizt e−tσ d∼ ω(σ ) dt. .Cω (z) = eizt = Iω (t) 0 0 0 For completing the proof, it suffices to see that for any .z ∈ G+ (f +∞ ( ) f +∞ ) f +∞ f +∞ 1 i(z+σ )t izt −tσ = .L∼ d∼ ω(σ ) e dt = e e d∼ ω(σ ) dt, ω −iz 0 0 0 0 where the integrals are absolutely convergent, since f +∞ f +∞ | f +∞ | d∼ ω(σ ) | i(z+iσ )t | < +∞, . d∼ ω(σ ) |e | dt ≤ y +σ 0 0 0
z = x + iy ∈ G+ ,
due to the inequalities f
+∞
.
0
(f A0 f +∞ ) dσ ∼ ω(σ )dσ ≤ + 2 (1 + σ ) (1 + σ )2 ω(σ ) 0 0 A0 f +∞ f A0 dσ dσ 1 + M < +∞ ≤ 2 2 ω(A0 ) 0 (1 + σ ) (1 + σ )2 σ α A0
d∼ ω(σ ) = 1+σ
f
+∞
which are true for some numbers .M2 > 0 and .α ∈ (−1, 0).
u n
The next statement follows by Fatou’s lemma. + Lemma 11.5 Let .ω ∈ oN α (G ), let .E ⊆ (−∞, +∞) be a B-set such that .Cω (E) > 0, and let a B-measure .τ ≺ E satisfy (11.11) and (11.12). Then for any .x ∈ (−∞, +∞)
f
+∞
.
−∞
f |Cω (x − t)|dτ (t) ≤ S1 ≡ sup
+∞
z∈G+ −∞
|Cω (z − t)|dτ (t) < +∞.
Proof In virtue of (11.13), .Re Cω ≥ 0 in .G+ , and hence the function .Cω has nontangential boundary values at almost all points .−∞ < x < +∞. Further, by Fatou’s lemma f .
+ ∞ > lim inf y→+0
+∞
−∞
f |Cω (x + iy − t)|dτ (t) ≥ f ≥
+∞ −∞ +∞ −∞
lim inf |Cω (x + iy − t)|dτ (t) y→+0
|Cω (x − t)|dτ (t)
11 Subclasses of Harmonic in .G+ Functions
256
for any .x ∈ (−∞, ∞), and hence f .
+∞
sup
y>0 −∞
f |Cω (z − t)|dτ (t) ≥
+∞
−∞
|Cω (x − t)|dτ (t).
Taking the supremum over .x ∈ (−∞, +∞) in the right-hand side, we finish the proof. n u The next lemma is necessary for proving the main theorem of this section. Lemma 11.6 Let .ω ∈ oαN (G+ ) and let f be a holomorphic in .G+ function of the form f f (z) =
+∞
z ∈ G+ ,
Cω (z − t)dμ(t),
.
−∞
where .μ is a function of bounded variation on .(−∞, +∞). Then, the set of those .x ∈ (−∞, +∞), where the nontangential boundary value .f (x) does not exist as a finite limit, is of zero .ω-capacity. Proof By (11.13), .Re Cω ≥ 0 in .G+ , and hence f is a difference of two holomorphic in + + .G functions with nonnegative real parts. Consequently, f is of bounded type in .G , and hence it has nontangential boundary values at least at almost all points .−∞ < x < +∞. In particular, .
lim f (x + iy) = f (x) /= ∞
a.e. in (−∞, +∞).
y→+0
(11.15)
For proving the existence of the latter limit out of a set of zero .ω-capacity, we write f
λ
f (x + iy) = f (x + iλ) − i
.
f ' (x + iσ )dσ,
0 < y < λ < +∞.
(11.16)
y
As .ω ∈ oαN (G+ ), by the estimate (11.9) and a calculation similar to that in the beginning of the proof of Theorem 11.1, we conclude that for any .y > ρ > 0 and .ε ∈ (0, 1 + α) f .
y
+∞ |
| |f ' (x + iσ )|dσ =
f y
| f | ∂ | ∂σ
+∞ |
f
≤
+∞ (f
y
f
≤ Mρ,ω
+∞
−∞
| | Cω (x + iσ − t)dμ(t)|| dσ ) | |Cω (x + iσ − t)||dμ(t)| dσ 1
+∞ |
−∞ +∞
dσ y
f
+∞
−∞
|dμ(t)| |x + iσ − t|1+ε
11.3 Boundary Values
257
' ≤ Mρ,ω
f
+∞
|dμ(t)| (|x − t| + y)ε
−∞
'' ≤ Mρ,ω y −ε
+∞ V
f
+∞ 0
dτ (1 + τ )1+ε
μ(t) < +∞,
−∞ ' , M '' where .Mρ,ω , Mρ,ω ρ,ω > 0 are some constants. Hence, the passage .λ → +∞ in (11.16) gives
f
+∞
f (x+iy) = f (x+i∞)−i
.
f ' (x+iσ )dσ,
y > 0,
−∞ < x < +∞,
(11.17)
y
where the integral is absolutely convergent and .f (x + i∞) is a finite limit. Now, let .E0 be the set of those .x ∈ (−∞, +∞) for which |f | | |
+∞
.
0
| | f ' (x + iσ )dσ || = +∞,
and let E be the set of those .x ∈ (−∞, +∞), for which (11.15) is not a finite limit. Then E ⊆ E0 by (11.17), and .Cω (E0 ) = 0 implies .Cω (E) = 0. Indeed, if .Cω (E) > 0, then there exists a nonnegative B-measure .τ ≺ E for which (11.11) and (11.12) are true. But .E ⊆ E0 , and therefore, .τ ≺ E0 , and .Cω (E0 ) > 0 by Definition 11.4. Thus, for completing the proof, it suffices to show that .Cω (E0 ) = 0. In contrary, suppose .Cω (E0 ) > 0. Then there must be a nonnegative B-measure .τ0 ≺ E0 for which (11.11) and (11.12) are true. By (11.11) .
f
+∞
.
−∞
f dτ0 (x) =
dτ0 (x) = 1. E0
Besides, it is clear that f
| | |
+∞ |f +∞
.
−∞
0
| | f ' (x + iσ )dσ || dτ0 (x) = +∞.
On the other hand, by (11.13) Cω' (z) = −
f
∞
.
0
d∼ ω(t) . (z + it)2
(11.18)
11 Subclasses of Harmonic in .G+ Functions
258
Therefore, using Fubini’s theorem and (11.13), we obtain f
+∞
.
f
'
f (x + iσ )dσ =
y
y
+∞ (f +∞ −∞
Cω' (x
) + iσ − t)dμ(t) dσ
) dσ =− d∼ ω(λ) dμ(t) (x + iσ − t + iλ)2 −∞ 0 y ) f +∞ f +∞ ( f +∞ 1 d∼ ω(λ) d dμ(t) =− i (x − t + is) −∞ 0 y+λ ) f +∞ (f +∞ d∼ ω(λ) = −i dμ(t) z − t + iλ −∞ 0 ) ( f +∞ 1 L∼ dμ(t) = −i ω z−t −∞ f +∞ Cω (z − t)dμ(t). =− f
+∞ (f +∞
f
+∞
−∞
Consequently, for any .z = x + iy ∈ G+ |f | | . |
+∞
y
| f | f (x + iσ )dσ || ≤ '
+∞ |
−∞
| |Cω (z − t)||dμ(t)|,
and by the definition (1.22) of the kernel .Cω f
| | |
+∞ |f +∞
.
−∞
| f | f ' (x + iσ )dσ || dτ0 (x) ≤
−∞
y
f ≤
+∞
−∞
f =
f
+∞
+∞
−∞
≤ S1
+∞ V
dτ0 (x)
+∞ | −∞
f |dμ(t)|
| | | |Cω (−t + iy + x)| |dτ0 (x)|
+∞ |
−∞
f |dμ(t)|
| |Cω (x + iy − t)||dμ(t)|
+∞ |
−∞
| |Cω (t + iy − x)||dτ0 (x)|
μ < +∞.
−∞
Hence, by Fatou’s lemma f
| | |
+∞ |f +∞
.
−∞
0
| f | f ' (x + iy)dy || dτ0 (t) =
| | f +∞ | ' | dτ0 (t) | lim f (x + iy)dy | |y→0 −∞ y | |f +∞ f +∞ | | = lim || f ' (x + iy)dy || dτ0 (t) y→0 +∞ |
−∞
y
11.3 Boundary Values
259
f ≤ lim inf y→0
≤ S1
+∞ V
| | |
+∞ |f +∞
−∞
y
| | f (x + iy)dy || dτ0 (t) '
μ < +∞
−∞
u n
which contradicts (11.18).
The below theorem describes the boundary behavior of the functions from .Nm ω in the terms of .ω-capacity. m + Theorem 11.2 Let .ω ∈ oN α (G ). Then any function .u ∈ Nω has nonzero, finite perpendicular boundary values at all points .x ∈ (−∞, +∞), with a possible exception of a set of zero .ω-capacity.
Proof The statement is obvious by Lemma 11.6 and the representation (11.5) of .u ∈ Nm ω, where .a0 and .a1 are some real constants and .μ is a function of bounded variation in .(−∞, +∞). u n Below, we give one more theorem on the boundary values of functions from .Nm ω. Theorem 11.3 Let .ω ∈ oαN (G+ ); let .E ⊆ (−∞, +∞) be a B-set with positive .ωcapacity, i.e. .Cω (E) > 0; and let .a1 = 0 in the representation (11.5) of a function .u ∈ Nm ω. Then f
+∞
.
−∞
|u(x)|dτ (x) < +∞
(11.19)
for the same B-measure .τ as in (11.12), for which .Cω (E) > 0. Proof Obviously |u(z)| ≤
.
1 π
f
+∞ −∞
|Cω (z − t)|dμ(t) + a0 ,
−∞ < t < +∞,
and hence, by Fubini’s theorem and the inequality (11.12), we get f
+∞
.
−∞
|u(x + iy)|dτ (x) ≤
1 π
≤
1 π
f
+∞
−∞
f
+∞
−∞
f |dμ(t)|
−∞
f |dμ(t)|
+∞
+∞
−∞
|Cω (x + iy − t)|dτ (x) + Ma0 |Cω (x + iy − t)|dτ (x) + Ma0
11 Subclasses of Harmonic in .G+ Functions
260
≤
1 π
f
+∞
−∞
f |dμ(t)|
+∞
−∞
|Cω (t + iy − x)|dτ (x) + Ma0
+∞ S1 V ≤ μ + Ma0 < +∞. 2π −∞
Consequently, (11.19) holds by Theorem 11.2 and Fatou’s lemma.
11.4
Notes
The results of the chapter are published in [52].
u n
Subclasses of Delta-subharmonic Functions of Bounded Type in the Halfplane
12
This chapter extends to delta-subharmonic functions the results of the previous chapter on the descriptive representations of some subclasses of harmonic functions with nonnegative harmonic majorants in a halfplane. We often use the classes .oαN (G+ )(−1 < α < 0) of nonincreasing parameter-functions introduced by Definition 1.3, the Blaschke-type factors .∼ bω of the form (10.2), the .Cω kernels of the form (1.22), and the fractional differentiation operators .Lω defined by the second equality in (1.21). Also, we use the following equalities ∂ . Cω (z) = −Cω1 (z) and Cω (z) = ∂y
12.1
f
+∞
Cω1 (x + it)dt, z = x + iy ∈ G+ .
y
(12.1)
Blaschke-Type Factors
Assuming that .ω ∈ oαN (G+ )(−1 < α < 0) and .ζ = ξ + iη ∈ G+ are fixed points, we consider the Blaschke-type factors { f ∼ .bω (z, ζ ) = exp −
η
−η
} Cω (z − ξ − it)ω(η − |t|)dt ,
Im z > η.
and note that the following representation is true for the principal branch of the logarithm of the ordinary Blaschke factor: .
log b0 (z, ζ ) = log
z−ζ z−ζ
f =
η
−η
dt , t + i(z − ξ )
z /= ζ.
(12.2)
We start by the following theorem. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_12
261
12 Subclasses of Functions Delta-Subharmonic in .G+
262
+ + Theorem 12.1 For any .ω ∈ oN α (G )(−1 < α < 0) and any fixed point .ζ ∈ G , the + function .∼ bω is holomorphic in the halfplane .G , where it has a unique, first-order zero at the point .z = ζ .
Proof The proof is obvious, since Lemma 10.1 and its proof remain in force for .ω ∈ oαN (G+ ). n u Now, we study some properties of .∼ bω , which are revealed by an application of .Lω . Lemma 12.1 If .ω ∈ oαN (G+ )(−1 < α < 0), then for any .ζ = ξ + iη ∈ G+ , .ρ > η and .ε ∈ (0, 1 + α) | | bω (z, ζ )| ≤ .| log ∼
Mρ,ε (ρ − η)2+α−ε
f
η
ω(t)dt,
Im z > 2ρ,
(12.3)
0
where .Mρ,ε > 0 is a constant depending only on .ρ and .ε. Proof It is not difficult to verify that in view of the second equality in (12.1) and the estimate (5.19) |Cω1 (z)| ≤
.
Mω,ρ,ε , |z|2+α−ε
Im z > ρ.
This inequality and the representation (10.2) imply the estimate (12.3).
(12.4) u n
Lemma 12.2 If .ω ∈ oαN (G+ )(−1 < α < 0) and .ζ = ξ + iη ∈ G+ are fixed points, then .Lω log |∼ bω (z, ζ )| has a harmonic continuation to the whole finite complex plane .C, except the closed intercept .[ζ, ζ ] with the endpoints .ζ and .ζ , and the following representations are true at any point .z = x + iy ∈ / [ζ, ζ ]: Lω log |∼ bω (z, ζ )| = Re
f
η
.
Lω log |∼ bω (z, ζ )| = −2y
−η
f
ω(η − |t|) dt, . t + i(z − ξ ) η
0
y 2 + (x − ξ )2 − t 2 ω(η − t)dt. [t 2 + y 2 + (x − ξ )2 ]2 − 4t 2 y 2
(12.5) (12.6)
bω | is In addition, if .ω satisfies Hölder’s condition on .(0, η], then the function .Lω log |∼ continuous at all points of the open interval .(ζ, ζ ). Proof The proofs of the representations (10.9) and (10.10) remain in force, and the proof of the continuity of .Lω log |∼ bω | in all points of the open interval .(ζ, ζ ), provided .ω satisfies Hölder’s condition on .(0, η], is the same as that given after the proof of Lemma 10.3. n u
12.1 Blaschke-Type Factors
263
Lemma 12.3 For any .ω ∈ oαN (G+ )(−1 < α < 0) and any fixed .ζ = ξ + iη ∈ G+ , ⎧ f η ⎪ ⎪ ω(x)dx, ⎨− +∞ 1 0 ∼ f Lω log |bω (x + iy, ζ )|dx = . η ⎪ 2π −∞ ⎪ ω(x)dx, ⎩− f
y ≥ η, (12.7) 0 < y < η,
η−y
which is a continuous function of .y ∈ (0, +∞). Besides, .
1 2π
f
| |Lω log |∼ bω (x + iy, ζ )||dx ≤ 3
+∞ | −∞
f
η
0 < y < +∞.
ω(x)dx,
(12.8)
0
Proof The proof of (12.7) is the same as that of (10.20). To prove (12.8), observe that by (12.6), .Lω log |∼ bω (z, ζ )| ≤ 0, if .|x − ξ | ≥ η or .y ≥ η. Therefore, by (12.7) f
| |Lω log |∼ bω (x + iy, ζ )||dx =
+∞ |
.
−∞
f
η
ω(x)dx,
η ≤ y < +∞.
0
To prove (12.8) for .0 < y < η, observe that by (12.6) the set .Sy+ of those x, at which .Lω log |∼ bω (x + iy, ζ )| ≥ 0, lies in the interval .(ξ − η, ξ + η), and by (12.5) .
f
1 2π
| |Lω log |∼ bω (x + iy, ζ )||dx
ξ +η |
ξ −η
≤
1 2
f
η
(
−η
|t − y| π
f
ξ +η ξ −η
dx (t − y)2 + (x − ξ )2
)
f
η
ω(η − |t|)dt
0 depends only on .ω and .η. Indeed, for .z ∈ G+ , .|z| > 2η, ∗ .|ϕω (z, ζ )|
f
f η 2ηω(η)dt 2|z + iη|ω(t)dt | | | | ≤ + | | | [z + i(t − η)][z + i(t + η)] [z + i(η − t)][z + i(t + η)]| η 0 f η f +∞ dt ω(t)dt 3/2 ≤ 4ηω(η) +2 (|z| + t − η)(|z| + t + η) η 0 |z| + η − t f η f +∞ 3/2 Mω,η dt 2 + ω(t)dt ≤ . < 4ηω(η) 2 |z| 0 |z| (|z| + t − η) η +∞
In virtue of this estimate, the nonpositivity of .Re ϕω∗ (z, ζ ) will be extended to .G+ by an application of Theorem 1.1 in Chapter 7 of [44], if we prove that f .
lim inf y→+0
1
−1
| | |Re ϕ ∗ (x + iy, ζ )|dx < +∞. ω
To this end, observe that in (12.12) f
1
.
−1
| | |Re I2 (x + iy)|dx ≤
f
f
η
ω(t)dt 0
f
= 2π
η
+∞ {
−∞
} t +y+η t −y−η + dx (t − y − η)2 + x 2 (t + y + η)2 + x 2
ω(t)dt < +∞.
0
Besides, the variable changes .λ = (t + y − η)/x and .t = 2η/x give f
1
.
−1
| | |Re I1 (x + iy)|dx ≤ 16ηω(η)
f
+∞ 2η
dt t
f
+∞
0
[(λ + 1)2 + λt]dλ ≡ K1 + K2 . (λ + 1)2 (λ + t + 1)2
Here, f K1 = 16ηω(η)
+∞
.
2η
f
dt t
+∞ 0
dλ < 16ηω(η) (λ + t + 1)2
f
+∞
2η
dt < 8ω(η) < +∞, t (t + 1)
and by the variable change .λ2 = x we get f K2 = 16ηω(η) f < 8ηω(η)
f
+∞
+∞
dt
.
2η +∞
2η
0
f
f +∞ dx dt = 16ηω(η) log(t + 1) 2 dt < +∞. (x + 1)[x + (t + 1)2 ] t 2η
+∞
dt 0
λdλ (λ + 1)2 (λ + t + 1)2
12.1 Blaschke-Type Factors
267
Lemma 12.7 If .ω ∈ oαN (G+ ) with some .α ∈ (−1, 0), then for any .ζ = ξ + iη ∈ G+ f ∼ 1 +∞ bω (z, ζ ) =− Cω (z − t)Lω log |b0 (t, ζ )|dt ≡ −Jω (z, ζ ), . log b0 (z, ζ ) π −∞
z ∈ G+ , (12.14)
and for any .ε ∈ (0, 1 + α) f
η
|Jω (z, ζ )| ≤ Aω,ρ (1 + η)
Im z > ρ > 0,
ω(t)dt,
.
(12.15)
0
where the constant .Aω,ρ > 0 depends only on .ρ and .ω. Proof By the Herglotz-Riesz theorem, Re ϕω (z, ζ ) = py + Re
.
1 π
f
+∞
−∞
Re ϕω (t, ζ ) dt, −i(z − t)
z = x + iy ∈ G+ ,
where .p = lim y −1 Re ϕω (iy, ζ ) = 0 by (12.13). On the other hand, one can see that y→+∞
{ Lω
.
1 π
f
+∞
−∞
}
1 Cω (z − t)Re ϕω (t, ζ )dt = π
f
+∞ −∞
Re ϕω (t, ζ ) dt, −i(z − t)
z ∈ G+ .
| | | ∼ (z,ζ ) | Besides, it is easy to verify that the function .log | bbω0 (z,ζ ) | satisfies all requirements of Theorem 11.1, and therefore | | | | f |∼ |∼ bω (t, ζ ) || bω (z, ζ ) || 1 +∞ | | dt, . log Re Cω (z − t)Lω log | | b (z, ζ ) | = a0 + a1 x + π b0 (t, ζ ) | 0 −∞
z ∈ G+ .
Considering the behavior of the left-hand side and the integral in the right of this formula when .|z| → +∞ on the rays .z = reiϑ (|ϑ − π/2| < π/2),| we see| that the left-hand | ∼ (t,ζ ) | 1 side obviously vanishes as .|z| → +∞. In the integral, .Lω log | bbω0 (t,ζ ) | ∈ L (−∞, +∞), while the estimate (12.4) is true for the kernel. Hence, also the integral vanishes as .|z| → bω (t, ζ ) = 0 for any .x ∈ +∞, and therefore .a0 = a1 = 0. Further, by (12.6) .Lω log |∼ (−∞, +∞), and hence we come to formula (12.14). To prove the estimate (12.15), by the representation (12.2), one can calculate f Lω log |b0 (t, ζ )| = Re
f
+∞
ω(σ )dσ
.
0
η −η
dλ . (iσ − iλ + t − ξ )2
12 Subclasses of Functions Delta-Subharmonic in .G+
268
Hence f .
f
+∞
− Jω (z, ζ ) = −
ω(σ )dσ
η
−η
0
[I1 (λ) + I2 (λ)]dλ,
where I1 (λ) =
.
I2 (λ) =
1 2π 1 2π
f
+∞ −∞
f
+∞ −∞
⎧ ⎨−C (z − ξ − iλ + iσ ), Cω (z + t)dt ω1 = ⎩0, [t − (−ξ − iλ + iσ )]2 ⎧ ⎨0,
Cω (z + t)dt = ⎩−Cω (z − ξ + iλ − iσ ), [t − (−ξ + iλ − iσ )]2 1
λ < σ, λ > σ, λ < σ, λ>σ
by the residue theorem and formula (12.1). Consequently, f − J (z, ζ ) =
.
f
η
ω(σ )dσ 0
f
σ
Cω1 (z − ξ − iλ + iσ )dλ
−η
f
+∞
+
ω(σ )dσ η
f
f
η
+
−η η
ω(σ )dσ 0
η
σ
Cω1 (z − ξ − iλ + iσ )dλ
Cω1 (z − ξ + iλ − iσ )dλ ≡ K1 + K2 + K3 .
One can verify that an application of the estimate (12.4) gives that for any .ε ∈ (0, 1 + α) and .Im z = y > ρ > 0 f |K2 | ≤ Mω,ρ,ε
+∞
.
{ ω(λ + η)
0
} 1 1 − dλ, (y + λ)1+α−ε (y + λ + 2η)1+α−ε
where the constant .Mω,ρ,ε > 0 depends only on .ω, .ρ, and .ε. Further, evaluating the above integrand, we get |K2 | ≤
.
' Mω,ρ,ε
η 1+η
f 0
+∞
ω(λ)dλ η '' ''' < Mω,ρ,ε ≤ Mω,ρ,ε 2+α−ε 1+η (ρ + λ)
f
η
ω(t)dt. 0
In the same way, but much easier, we come to the following estimates with similar constants: f η '''' .|K1,3 | ≤ Mω,ρ,ε η ω(t)dt. 0
The proved estimates imply (12.15).
u n
12.2 Green-Type Potentials
12.2
269
Green-Type Potentials
We start by the convergence of Green-type potentials. We give the following theorem with a complete proof, since there are some differences with the proof of the similar Theorem 10.2. + Theorem 12.2 Let .ω ∈ oN α (G )(−1 < α < 0), and let .ν ≥ 0 be a Borel measure in the + halfplane .G supported in a strip, i.e., .sup{Im(support ν)} = D0 < +∞, and such that
(f
ff .
G+
Im ζ
) ω(t)dt dν(ζ ) < +∞.
(12.16)
0
Then, the Green-type potential ∼ω (z) = .P
ff G+
| | bω (z, ζ )|dν(ζ ) log |∼
converges in .G+ , where it is a subharmonic function with the Riesz-associated measure .ν. Proof In any halfplane .G+ ρ = {z : Im z > ρ} with .0 < ρ < A, we define the Green-type potential by the formula Pω (z) = P0 (z, ρ) + Uω (z, ρ),
.
z ∈ G+ ρ,
where ff P0 (z, ρ) =
.
| | ff |z − ζ | | dν(ζ ) = log |b0 (z, ζ )|dν(ζ ) log || z−ζ| G+ G+ ρ ρ
is an ordinary Green potential, which is well-known to be convergent in .G+ under a weaker than the Blaschke condition (12.16), while ff Uω (z, ρ) =
.
G+ \G+ ρ
log |∼ bω (z, ζ )|dν(ζ ) +
ff
| | |∼ bω (z, ζ ) || | dν(ζ ) log | b0 (z, ζ ) | G+ ρ
= Uω(1) (z, ρ) + Uω(2) (z, ρ). We shall prove that .Pω converges in .G+ in the sense that for any .ρ ∈ (0, A) the potential + + .P0 (z, ρ) converges in .G , while .Uω (z, ρ) is harmonic in .Gρ .
12 Subclasses of Functions Delta-Subharmonic in .G+
270
(1) Note that by Theorem 12.1 the integrand .log |∼ bω | of .Uω is a harmonic function in .G+ ρ. Therefore, assuming that .z = x + iy, .ζ = ξ + iη, and .y > ρ1 , where .ρ1 > ρ is a fixed number, and then using the definition (10.2) of .∼ bω , we get
| | bω (z, ζ )|| = .| log |∼
f
f η | | |Cω (z − ξ − it)|ω(η − |t|)dt ≤ 2Cω (i(ρ1 − ρ)) ω(t)dt.
η
−η
0
Thus, ff (1) .|Uω (z, ρ)|
≤
G+ \G+ ρ
| log |∼ bω (z, ζ )||dν(ζ )
ff ≤M
G+ \G+ ρ
(f
η
) ω(t)dt dν(ζ ) < +∞
0 (1)
with .M = 2Cω (i(ρ1 − ρ)), i.e., the modulus of the integrand in .Uω , which is a + harmonic function in .G+ ρ , possesses an independent of .z ∈ Gρ1 , integrable majorant,
and consequently .Uω is a harmonic function in .G+ ρ . For proving that .Uω is a harmonic function in .G+ , observe that its integrand is a harmonic function in .G+ for any .ζ ∈ G+ , while the integral is uniformly convergent inside .G+ due to the estimate (12.15). (1)
(2)
Theorem 12.3 Let .ω ∈ oαN (G+ )(−1 < α < 0), let the Borel measure .ν ≥ 0 of a Green∼ω satisfy the condition (12.16), and let .sup{Im(support ν)} = D0 < +∞. type potential .P } { ∩ Further, let the closure of the set of limit points of .Re (support ν) G+ ρ be of zero Lebesgue measure for any .ρ > 0. ∼ω is a harmonic function in the domain .D = G+ \ ∪ζ ∈supp ν [ζ, Re ζ ] and is Then .Lω P representable by the absolutely and uniformly convergent inside .D integral ff
∼ω (z) = .Lω P
G+
Lω log |∼ bω (z, ζ )|dν(ζ ),
z ∈ D.
(12.17)
Proof First, we show that the function ff F (z) =
.
G+
Lω log |∼ bω (z, ζ )|dν(ζ )
(12.18)
is harmonic in .D. Indeed, any compact .K ⊂ D is of some distance .d > 0 from the set ∪ ζ ∈ supp ν [ζ, Re ζ ]. Hence, by formula (12.5), we get a suitable majorant for the modulus of the integrand in (12.18) at any point .z ∈ K:
.
ff |F (z)| ≤
.
| | 2 |Lω log |∼ bω (z, ζ )||dν(ζ ) ≤ d G+
(f
ff G+
η 0
) ω(t)dt dν(ζ ) < +∞.
12.2 Green-Type Potentials
271
Thus, the integral in (12.18) is absolutely and uniformly convergent inside .K, where it bω (z, ζ )| is represents a harmonic function, since by Lemma 12.2 the function .Lω log |∼ ω(t) ≤ harmonic in .G+ \ [ζ, Re ζ ]. Besides, by (12.5) and (12.16) and the inequality .∼ −1 ∼ = ω(t) -, we obtain that for .A [ω(t)] (0 < t < +∞) of Lemma 4.2 in [52], where .∼ max{A0 , D0 } and any .z ∈ K f L∼ ω |F (z)| =
+∞
|F (z + iσ )|d∼ ω(σ )
.
0
( f ∼ ) (f ) f +∞ η 2 2A d∼ ω(σ ) = ω(t)dt dν(ζ ) d∼ ω(σ ) + 4 σ ∼ G+ d 0 0 2A ) (f ( | ) ff f +∞ η ∼ ω(σ ) ||+∞ ∼ ω(σ ) 2 ∼) + 4 ∼ ω(2A ≤ + 4 dσ ω(t)dt dν(ζ ) d σ |2 A σ2 ∼ ∼ G+ 2A 0 ) (f | ) ff ( f +∞ η dσ 4 ||+∞ 2 ≤ + +4 ω(t)dt dν(ζ ) ∼ ) σ ω(σ ) |2A σ 2 ω(σ ) ∼ dω(2A ∼ G+ 2A 0 ff
) (f η ) f +∞ 2 4 dσ + ω(t)dt dν(ζ ) ∼ ) C 1 2A σ 2+α ∼ 0 G+ dω(2A ) f f (f η ω(t)dt dν(ζ ) < +∞, =M ff
.
(
≤
G+
0
where .M > 0 is a constant. Further, it follows by (10.2) and formula (11.13) of Lemma 11.4 that for any z with .Im z > D0 f L∼ ω F (z) =
+∞
0
ff =
G+
(f
ff
F (z + iσ )d∼ ω(σ ) =
.
G+
0
| | |∼ | L∼ ω Lω log bω (z, ζ ) dν(ζ ) =
) | | ω(σ ) dν(ζ ) bω (z + iσ, ζ )|d∼ Lω log |∼
+∞
ff G+
| | ∼ω (z), bω (z, ζ )|dν(ζ ) = P log |∼
and hence, this formula is true for all .z ∈ D by the uniqueness of a harmonic function. ∪ ∼ω is Moreover, .L∼ ω F has a harmonic continuation to the set . ζ ∈supp ν [Re ζ, ζ ), and .P + subharmonic in .G . Further, due to the absolute and uniform convergence of the integrals representing the function F in .Im z > D0 , one can easily prove that .Lω L∼ ω F = F in the halfplane .Im z > ∼ ∼ D0 and verify that .Pω ∈ Mω . Consequently, .Lω Pω is harmonic in .D by Lemma 11.2. Thus, .Lω L∼ ω F = F in the halfplane .Im z > D0 , where .L∼ ω F (z) = Pω (z) for all .z ∈ D.
12 Subclasses of Functions Delta-Subharmonic in .G+
272
∼ω (z) = F (z) for any .Im z > D0 , where both sides of the equality are functions Hence, .Lω P harmonic in .D. These functions coincide in the domain .D by the uniqueness of harmonic function, which proves formula (12.17). u n Theorem 12.4 Let a function .ω ∈ oαN (G+ )(−1 < α < 0) satisfy Hölder’s condition in .(0, +∞), let the Borel measure .ν ≥ 0 satisfy the condition (12.16), and let .sup{Im(support ν)} = D0 < +∞. Further, let the closure of the set of limit points of } { ∩ .Re (support ν) G+ ρ be of zero Lebesgue measure for any .ρ > 0. Then, the following relations are true: f .
sup
+∞ |
y>0 −∞
f
lim
| |L ω P ∼ω (x + iy)|dx < +∞, .
(12.19)
+∞ |
| |Lω P ∼ω (x + iy)|dx = 0, .
y→+0 −∞
f
(12.20)
| | | |∂ P ∼ω (z + iσ )| ω(σ )dσ < +∞. | | ∂y
+∞ |
sup
Im z>D0 +1 0
(12.21)
Proof By (12.17), (12.8), and Fubini’s theorem, for any .y > 0 f
+∞ |
| |Lω P ∼ω (x + iy)|dx ≤
.
−∞
ff
=
G+
f dν(ζ )
f
ff
+∞ −∞
dx
| | |Lω log |∼ bω (x + iy, ζ )||dν(ζ )
ff | |Lω log |∼ | bω (x + iy, ζ )| dx ≤ 2π
+∞|
−∞
G+
(f G+
η
) ω(t)dt dν(ζ ) < +∞.
0
Hence, (12.20) holds by Lebesgue’s theorem on dominated convergence and (12.10). To ∼ prove (12.21), observe that the function .log |∼ bω | is harmonic in .G+ D0 and the integral of .Pω + is uniformly convergent in .GD0 +1 , since by (10.2) | | | log |∼ bω (x + iy, ζ )||
D0 + 1. Evidently, in .G+ D0 +1 the modulus of the derivative of .log |bω | admits the same estimate with .Cω (i) replaced by .Cω1 (i). Consequently, for .y > D0 + 1 .
∂ ∼ Pω (x + iy) = ∂y
ff G+
| | ∂ log |∼ bω (x + iy, ζ )|dν(ζ ). ∂y
12.3 Weighted Classes of Delta-Subharmonic Functions
273
Further, by the estimate (12.4), we conclude that | | | ff | |∂ |∂ | || | | | | | | |∼ ∼ . | log bω (x + iy, ζ ) | dν(ζ ) | ∂y Pω (x + iy)| ≤ G+ ∂y ff f η | | |Cω (x − ξ + i(y − t))|ω(η − |t|)dt ≤2 dν(ζ ) 1 G+
−η
ff
≤ 2Mε,1
G+
dν(ζ )
f
η −η
ω(η − |t|)dt |x − ξ + i(y − t)|2+α−ε
for any .ε ∈ (0, 1 + α) and .y > D0 + 1. Hence, the relation (12.21) follows.
12.3
u n
Weighted Classes of Delta-Subharmonic Functions
We assume that U is a delta-subharmonic function in the upper halfplane .G+ , i.e., it is a difference .U = U1 −U2 of two functions subharmonic in .G+ . Besides, we assume that the Riesz-associated signed measure of U , i.e., its charge .ν, is minimally decomposed in the Jordan sense, i.e., .ν = ν+ −ν− , where .ν± are the positive and the negative variations of the measure .ν, which are some nonnegative Borel measures with non-overlapping supports in + .G . Below, we use the Tsuji-type characteristics of the form L(ρ, ±Lω U ) =
.
1 2π
f
+∞ { −∞
}+ ± Lω U (x + iρ) dx +
(f
ff G+ ρ
Im ζ 0
) ω(t)dt dν∓ (ζ ), (12.22)
where .0 < ρ < +∞ and the notation .a + = max{a, 0}, .a = a + − a − is used. Note that this definition is of sense. Indeed, if U has a sufficiently rapid rate of decrease at .∞ and its charge .ν is good enough to provide the convergence of the corresponding Green-type ∼ω is harmonic in .G+ , and potential, then the function .u = U − P ∼ω (z), Lω U (z) = Lω u(z) + Lω P
.
∼ω are well defined above. where .Lω u and .Lω P Definition 12.1 Let a function .ω ∈ oαN (G+ )(−1 < α < 0) satisfy Hölder’s condition on m + + functions U such .(0, +∞), and then .Nω (G ) is the class of all delta-subharmonic in .G that: (i) The associated charge .ν of U is such that .Im {supp ν} ≤ D0 < +∞, and for any + .ρ > 0, the closure of the set of limit points of .Re {(supp ν)∩Gρ } is of zero Lebesgue measure.
12 Subclasses of Functions Delta-Subharmonic in .G+
274
(ii) .U ∈ Mω , i.e., there exists an angular domain .A(δ0 , R0 ) = {z : |π/2 − arg z| ≤ δ0 , |z| ≥ R0 } with some .0 < δ0 ≤ π/2 and .0 < R0 < +∞, such that f .
sup
| | | | ∂ U (z + iσ )| ω(σ )dσ < +∞ | | ∂y
+∞ |
z∈K 0
(12.23)
for any compact .K ⊂ G ∩ A(δ0 , R0 ). (iii) The following relation is true: .
[ ] sup L(ρ, Lω U ) + L(ρ, −Lω U ) = S < +∞.
(12.24)
ρ>0 + Remark 12.1 The above definition extends Definition 11.3 of the class .Nm ω (G ) of harmonic functions to the case of delta-subharmonic functions.
Remark 12.2 The condition (12.24) provides the validity of the restriction (12.16) on .ν± ∼ over any halfplane .G+ ρ . Thus, by Theorem 12.2 the potential .Pω , the charge .ν of which + + is restricted to .Gρ , converges in .G , and hence the functions .[Lω U ]± participating in (12.24) are well defined in .G+ . The following theorem gives the descriptive representations of the classes .Nm ω. Theorem 12.5 For any .ω ∈ oαN (G+ )(−1 < α < 0), the class .Nm ω coincides with the set of functions representable in the form 1 .U (z) = a0 +a1 x + π
f
+∞ −∞
{ } Re Cω (z−t) dμ(t)+
ff G+
| | log |∼ bω (z, ζ )|dν(ζ ),
(12.25)
where .z = x + iy ∈ G+ , .a0 and .a1 are some real numbers, .μ is of bounded variation on .(−∞, +∞) and .ν = ν+ − ν− , where .ν± ≥ 0 are Borel measures in .G+ , such that .Im {supp ν± } ≤ D0 < +∞, for any .ρ > 0 the closure of the set of limit points of + .Re {(supp ν± ) ∩ Gρ } is of zero Lebesgue measure and (f
ff .
G+
0
Im ζ
) ω(x)dx dν± (ζ ) < +∞.
(12.26)
The measure .μ in (12.25) can be revealed by the Stieltjes inversion formula: f μ(x) = lim
.
y→+0 0
x
Lω U (t + iy)dt
a.e.
x ∈ (−∞, +∞),
(12.27)
12.3 Weighted Classes of Delta-Subharmonic Functions
275
and the following relation is true for the numbers .a0 and .a1 : .
lim U (x + iy) = a0 + a1 x,
y→+∞
−∞ < x < +∞.
(12.28)
Proof Let .U ∈ Nm ω . Then, for any .ρ > 0, the Green-type potentials ff .
G+
| | log |∼ bω (z − iρ, ζ )|dν± (ζ + iρ) =
ff G+ ρ
| | log |∼ bω (z − iρ, ζ − iρ)|dν± (ζ )
are convergent in .G+ ρ by (12.24) and Theorem 12.2, and the function ff U0 (z) = U (z) −
.
G+ ρ
| | log |∼ bω (z − iρ, ζ − iρ)|dν(ζ )
is harmonic in .G+ ρ . By (12.21), the integral in the right-hand side of this formula belongs to .Mω . Hence, also .U0 ∈ Mω , and by Theorem 12.3 ff Lω U0 (z) = Lω U (z) −
.
G+ ρ
| | Lω log |∼ bω (z − iρ, ζ − iρ)|dν(ζ ),
z ∈ D,
where .D = G+ \ ∪ζ ∈supp ν [ζ, Re ζ ], while the function .Lω U0 is harmonic in the halfplane + .Gρ by Lemma 11.2. Besides, f .
sup
| |Lω U0 (x + iy)|dx ≤ sup
+∞ |
y>0 −∞
f
| |Lω U (x + iy)|dx
+∞ |
y>0 −∞
| | | +∞ |f f | | | | + sup bω (x + iy − iρ, ζ − iρ)|dν(ζ )| dx ≡ A + B, Lω log |∼ | + | | Gρ y>0 −∞ f
where [ ] A ≤ sup L(y, Lω U ) + L(y, −Lω U ) < +∞
.
y>0
by (12.24) and .B < +∞ by (12.19). Thus, by a theorem of E. D. Solomentsev [83] (see also Theorem 10.5) y .Lω U0 (z + iρ) = π
f
+∞
−∞
dμρ (t) , (x − t)2 + y 2
z = x + iy ∈ G+ ,
12 Subclasses of Functions Delta-Subharmonic in .G+
276
where .μρ is of bounded variation on .(−∞, +∞). Similar to Lemma 1.3, Chapter 3 of [44], one can show that .dμρ (t) = Lω U0 (t + iρ)dt. Hence, for any .z = x + iy ∈ G+ ff Lω U (z + iρ) =
.
G+ ρ
Lω log |∼ bω (z, ζ − iρ)|dν(ζ ) +
y π
f
+∞ −∞
Lω U (t + iρ)dt dt (x − t)2 + y 2 (12.29)
because of (12.20) and the inclusion .Lω U (t + iρ) ∈ L1 (−∞, +∞) which follows from (12.24). Further, .
y y→+∞ 2
ff
lim
G+ ρ
| | Lω log |∼ bω (iy, ζ − iρ)|dν(ζ ) = −
(f
ff G+ ρ
) ω(t)dt dν(ζ )
η−ρ 0
by (12.28), and hence we came to a B. Ya. Levin-type formula: for any .ρ > 0 y 1 . lim Lω U (iy) = y→+∞ 2 2π
f
−∞
(f
ff
+∞
Lω U (t +iρ)dt −
η−ρ
) (12.30)
ω(t)dt dν(ζ ),
G+ ρ
0
where the limit exists and is finite. By the way, note that in the terms of the Tsuji-type characteristics (12.22) formula (12.30) is the equilibrium .
y Lω U (iy) + L(ρ, −Lω U ) = L(ρ, Lω U ). y→+∞ 2 lim
Now, observe that the condition (12.24) implies (12.26). Indeed, by (12.24) ff .
sup ρ >0
G+ ρ0
(f
η−ρ
) ω(t)dt dν(ζ ) ≤ S
0
for any fixed .ρ0 > 0 and any .ρ ∈ (0, ρ0 ). Hence (f
ff .
G+ ρ0
0
η
) ff ω(t)dt dν(ζ ) ≤ lim inf ρ→0
(f
G+ ρ0
η−ρ
) ω(t)dt dν(ζ ) ≤ S
0
which implies (12.26). Consequently, the Green-type potentials with .ν± converge and ff .
lim
ρ→+0
G+ ρ
Lω log |∼ bω (z, ζ − iρ)|dν(ζ ) =
ff G+
Lω log |∼ bω (z, ζ )|dν(ζ ),
z ∈ D,
since the potential over .G+ \G+ ρ vanishes, while in view of (12.5), the modules of integrands in both sides of the above relation have integrable majorants at any point .z ∈ D.
12.3 Weighted Classes of Delta-Subharmonic Functions
277
Further, by some well-known theorems of real analysis, there exist a sequence .ρk ↓ 0 and a function .μ of bounded variation in .(−∞, +∞), such that .
y k→∞ π lim
f
+∞ −∞
Lω U (t + iρk )dt y dt = 2 2 π (x − t) + y
f
+∞ −∞
dμ(t) dt, (x − t)2 + y 2
z = x + iy ∈ G+ .
Thus, for any point .z = x + iy ∈ D, a limit passage in (12.29) gives ff Lω U (z) =
.
Lω log |∼ bω (z, ζ )|dν(ζ ) +
G+
y π
f
+∞
−∞
dμ(t) . (x − t)2 + y 2
(12.31)
Now, observe that due to the convergence of the Green-type potential, the function ff V (z) = U (z) −
.
G+
| | log |∼ bω (z, ζ )|dν(ζ )
is harmonic in .G+ . Besides, .V ∈ Mω , since .U ∈ Mω and also the potential is of .Mω in view of (12.21). Further, due to Definition 12.1 and the inequality (12.21), the function V satisfies the relation (12.23). Thus, .V ∈ Nm ω of harmonic functions and by Theorem 3.1 of [52] V (z) = a0 + a1 x +
.
1 π
f
+∞ −∞
Re Cω (z − t)dμ1 (t),
z = x + iy ∈ G+ ,
where .a0 and .a1 are some real numbers and .μ1 (t) is a function of bounded variation in (−∞, +∞). Applying the operator .Lω to both sides of the above equality, by (12.31) we conclude that the measures .μ1 and .μ are the same. Thus, the representation (12.25) is proved. The relation (12.28) for the numbers .a0 and .a1 follows from the same relation for harmonic functions in Theorem 11.1, since the corresponding limit is zero for the Greentype potential. The Stieltjes inversion formula (12.27) is obvious in view of the relation (12.20) for the Green-type potential. Conversely, if U is representable in the form (12.25) with the required parameters, then m .U ∈ Nω in virtue of Theorem 11.1, and the properties (12.21) and (12.19) of the Green type potentials. u n .
Remark 12.3 For the particular case of discrete measure, the above theorem becomes the statement that the class of those functions f meromorphic in .G+ , for which .log |f | ∈ Nm ω with some .ω ∈ oαN (G+ )(−1 < α < 0), coincides with the set of functions representable in the form f (z) =
.
} { f Bω (z, {ak }) 1 +∞ Cω (z − t)dμ(t) + iC , z ∈ G+ , exp a0 + a1 z + Bω (z, {bn }) π −∞ (12.32)
12 Subclasses of Functions Delta-Subharmonic in .G+
278
where .μ is of bounded variation in .(−∞, +∞), .a0 ; .a1 and C are real numbers; and the sequences .{ak }, {bn } ⊂ G+ satisfy the density condition Σf
Im ak
.
ω(t)dt < +∞,
Σf
0
k
Im bn
ω(t)dt < +∞.
0
n
If the factorization (12.32) is true, then f μ(x) = lim
x
.
y→+0 0
Lω log |f (t + iy)|dt
lim log |f (x + iy)| = a0 + a1 x,
y→+∞
a.e.
x ∈ (−∞, +∞),
−∞ < x < +∞.
Remark 12.4 The additional to the inclusion in .Nm ω conditions .
lim U (x1,2 + iy) = 0
y→+∞
lim log |f (x1,2 + iy)| = 0 for some x1 /= x2
and
y→+∞
shrinks the considered classes of delta-subharmonic and meromorphic functions to some subclasses of functions of bounded type.
12.4
Boundary Property of Subclasses of Functions of Bounded Type in the Halfplane
We start by some statement on the perpendicular boundary values of the ordinary Blaschke product, where we use the notion of omega-capacity introduced by Definition 11.4. It is well-known that, if a sequence of numbers .{ζk }k = {ξk + iηk }k ⊂ G+ satisfies the condition Σ .
ηk < +∞,
(12.33)
k
then the ordinary Blaschke product B0 (z) ≡
||
.
k
b0 (z, ζk ) =
|| z − ζk k
z − ζk
,
z ∈ G+ ,
uniformly converges everywhere in .C, except the closure of the set .{ζ k }, and represents a meromorphic in .C function with zeros .{ζk } and poles .{ζ k }. Also, the following theorem is true.
12.4 Boundary Property of Subclasses in G+
279
+ Theorem 12.6 Let a sequence .{ζk }∞ k=1 ⊂ G satisfy (12.33) and the additional condition ∞ Σ .
k=1
ηk < +∞ |ζk − x|
for some point .x ∈ (−∞, +∞). Then, at this point there exists the perpendicular boundary value B0 (x) = lim B0 (x + iy)
.
y→+0
and
|B0 (x)| = 1.
(12.34)
Proof The relations (12.34) are true for all factors of the product .B0 . For extending (12.34) to the infinite product, it suffices to observe that
.
| ∞ | ∞ ∞ Σ Σ Σ | z − ζk | ηk ηk | |=2 < +∞, ≤ 2 − 1 |z − ζ | |x − ζk | |z − ζ | k k k=1 k=1 k=1
for any .z = x + iy with .0 ≤ y ≤ R0 < +∞. Hence. the product .B0 is uniformly convergent on the closed interval .{z = x + iy : 0 ≤ y ≤ R0 } perpendicular to the real axis at the point x, and the property of its factors pass to the desired property of the product. u n Lemma 12.8 Let .ω ∈ oαN (G+ ), and let ∞ f Σ .
ηk
ω(x)dx < +∞
(12.35)
k=1 0 + for a sequence .{ζk }∞ k=1 ⊂ G . Then
.
(f ∞ Σ | | |Cω (ζk − t)| k=1
)
ηk
ω(x)dx
< +∞
(12.36)
0
for all .t ∈ (−∞, +∞) except, perhaps, a set E of zero omega-capacity, i.e., with Cω (E) = 0.
.
12 Subclasses of Functions Delta-Subharmonic in .G+
280
Proof On the contrary, suppose .Cω (E) > 0 for a set E, where the sum (12.36) diverges. Then, the relations (11.11) and (11.12) are true for some nonnegative B-measure .τ ≺ E, and by (12.35) and (11.12), we come to a contradiction: f +∞=
.
=
[∞ (f Σ| | |Cω (ζk − t)|
+∞
−∞
)f
ηk
ω(x)dx 0
k=1
{f ≤ sup
ζ ∈G+
ω(x)dx
dτ (t)
0
k=1
∞ [(f Σ
)]
ηk
] | |Cω (ζk − t)|dτ (t)
+∞ |
−∞
}Σ ∞ f | |Cω (ζ − t)|dτ (t)
+∞ |
−∞
ηk
ω(x)dx < +∞.
k=1 0
Lemma 12.9 Let .ω ∈ oαN (G+ ), and let f .
lim o(y) = +∞ for
y→+0
+∞
o(y) ≡
h
y
fh 0
dh
(12.37)
.
ω(x)dx
Then f
Cω (iy) . lim inf ≥J ≡ y→+0 o(y)
+∞
0
e−x dx . 1 + x −1 e−x
(12.38)
Proof The function .ω is nonincreasing on .(0, +∞), and hence ω(h) ≤
.
1 h
f
h
(12.39)
ω(x)dx 0
for any .h > 0. Further, for any .t > 0 f
+∞
e
.
−tx
f
h
ω(x)dx ≤
0
f ω(x)dx + ω(h)
0
+∞
e
−tx
f dx =
h
ω(x)dx +
0
h
ω(h) −th e . t
Therefore, by (12.39) f
+∞
.
0
( )f h e−th e−tx ω(x)dx ≤ 1 + ω(x)dx. th 0
Hence, .
−
∂ Cω (ih) = ∂h
f 0
+∞
f +∞ 0
e−ht dt e−tx ω(x)dx
f
+∞
≥ 0
e−ht dt 1 + (ht)−1 e−ht
/f
h
ω(x)dx, 0
12.4 Boundary Property of Subclasses in G+
281
i.e., [ f J h
]−1
h
≤−
ω(x)dx
.
0
∂ Cω (ih). ∂h
(12.40)
Now, note that due to the estimate (11.1) for any fixed .0 < M < +∞ and .0 < y < M f
M
Cω (iy) = −
.
y
where .A0 ≡ −
f +∞ M
∂ ∂h Cω (ih)dh
∂ Cω (ih)dh + A0 , ∂h
> 0 is a constant. Therefore, by (12.40) f
Cω (iy) ≥ A0 + J
[ f h
M
.
]−1
h
ω(x)dx
dh.
0
y
u n
Hence, (12.38) holds by (12.37). Lemma 12.10 Let a function .ω ∈ oαN (G+ ) be such that (12.37) is true. Then .
[ f lim inf Cω (iy) y→+0
y
] ω(x)dx ≥ J > 0.
(12.41)
0
Proof Along with .o defined by (12.37), consider the function [f G(y) =
]−1
y
ω(x)dx
.
,
0 < y < +∞,
0
and observe that .limy→+0 o(y) = limy→+0 G(y) = +∞. Further, applying the Cauchy mean value theorem and (12.39), we conclude that for any fixed points .0 < y < y0 < +∞, y (y < ∼ y < y0 ) such that there is some .∼
.
o' (∼ y) o(y0 ) − o(y) = = ' G (∼ y) G(y0 ) − G(y)
f∼ y 0
ω(x)dx ≥ 1. ∼ y ω(∼ y)
Hence .
o(y0 ) G(y0 ) o(y) − = 1 + o(1) ≥1+ G(y) G(y) G(y) [
Thus, .lim infy→+0
o(y) G(y)
]
as
≥ 1, and (12.41) follows by (12.38).
Now, we proceed to the main theorem of this section.
y → +0. u n
12 Subclasses of Functions Delta-Subharmonic in .G+
282
+ Theorem 12.7 Let .ω ∈ oαN (G+ ), let (12.37) be true, and let .{ζk }∞ k=1 ⊂ G be sequence + satisfying (12.35). Then, the product .B0 is convergent in .G , and the relations (12.34) are true for all .x ∈ (−∞, +∞) except, perhaps, a set E with .Cω (E) = 0.
Proof It is easy to verify that the statement of Lemma 11.3 is true for any countable unions of zero .ω-capacity sets. Hence, it suffices to prove that for any .0 < R < +∞ the sets .ER = E ∩ (−R, R), where (12.34) is not true, are of zero .ω-capacity, i.e., .Cω (ER ) = 0. To this end, we decompose ⎛ B0 (z) = ⎝
||
.
|ξk | 0 is a constant depending only on R and .R0 . Indeed, (12.43) is obvious for σ ≥ 1. For the case .0 < σ < 1, observe that the function
.
f (ϕ) ≡
.
|1 − reiϕ | , |1 − τ reiϕ |2
0 ≤ τ, r < 1, 0 ≤ ϕ ≤ 2π,
attains its minimum at .ϕ = 0, i.e., .f (ϕ) ≥ f (0) = (1 − r)/(1 − τ r)2 , and hence .
|1 − w| 1 − |w| , ≥ (1 − |w|τ )2 |1 − wτ |2
|w| < 1, 0 ≤ τ < 1.
The change of variables .w = eiz , .τ = e−σ in this inequality gives .
|1 − eiz | 1 − e−y ≥ , |1 − ei(z+iσ ) |2 (1 − e−(y+σ ) )2
z ∈ G+ , 0 < σ < +∞.
For completing the proof of (12.43), it remains to see that expanding the considered only| functions in their Taylor series we can find some constants .C1,2,3 > 0 depending | on R and .R0 and such that .C1 |z| ≥ |1 − eiz | ≥ 1 − e−y ≥ C2 y, .|z + iσ | ≤ C3 |1 − ei(z+iσ ) | and .1 − e−(y+σ ) ≤ y + σ . By (12.42) and (12.43), we obtain that for .|x| < R and .0 < y < R0 |Cω (x + iy)| ≥
.
My |x + iy|
f
+∞
0
My d∼ ω(σ ) = Cω (iy). y+σ |x + iy|
(12.44)
Further, by (12.41) there exists a number .δ > 0 such that f
y
Cω (iy)
.
ω(x)dx ≥ δ > 0,
0 < y ≤ R0 .
0
Consequently, by (12.44) and (12.39), we conclude that Mδ
.
η ≤ |Cω (ζ − t)| |ζ − t|
f
η
ω(x)dx,
ζ = ξ + iη ∈ G+ , −R < t < R.
0
Hence, the desired statement for the function A follows by Theorem 12.6. Now, we proceed to the boundary values of the Green-type potentials.
u n
12 Subclasses of Functions Delta-Subharmonic in .G+
284
Theorem 12.8 Let .ω ∈ oαN (G+ ), and let the Borel measure .ν ≥ 0 of a Green-type ∼ω satisfy (12.35) and .sup{Im(supp ν)} = R0 < +∞. Further, let for any .ρ > 0 potential .P } { ∩ the set .Re (supp ν) G+ ρ be of zero measure in .(−∞, +∞). Then, the function ff oω (z) ≡
.
G+
Lω log
bω (z, ζ ) dν(ζ ) b0 (z, ζ )
(12.45)
is holomorphic in .G+ , and for any .z ∈ G+ ∼ω (z) − P0 (z) = a0 + a1 x − 1 Re L∼ ω oω (z) = P π
f
+∞
.
−∞
Re Cω (z − t)dσ (t),
(12.46)
where .L∼ ω is the operator of (12.42), .a0 , a1 ∈ (−∞, +∞) are some numbers, and .σ is of bounded variation on .(−∞, +∞). (z,ζ ) Proof Note that by Lemma 12.5 the integrand .ϕω (z, ζ ) ≡ Lω log bbω0 (z,ζ ) in (12.45) is holomorphic in .G+ . Besides, by formulas (12.11) and (12.39), we obtain that for any .0 < ρ0 < +∞
f
+∞
|ϕω (z, ζ )| ≤
.
η
f
f
2ηω(t)dt
|z − ζ + it||z − ζ + it|
η
+ 0
+∞
f
ω(t)dt |z − ζ − it|
0
f
η
ω(t)dt |z − ζ + it|
η ω(t)dt dt ω(t)dt + + (y − η + t)(y + η + t) η 0 y+η−t 0 y+η+t f η f f +∞ dt 2 2ηω(η) 2 η + ω(t)dt = ω(t)dt + ≤ 2ηω(η) y 0 y y 0 (y − η + t)2 η f η f 4 η 4 ≤ ω(t)dt ≤ ω(t)dt, z = x + iy ∈ G+ (12.47) ρ0 . y 0 ρ0 0
≤ 2ηω(η)
η
f +
Hence, by (12.35) we conclude that for any .z ∈ G+ ρ0 ff .
|oω (z)| ≤
| | ) f f (f η | bω (z, ζ ) | | dν(ζ ) ≤ 4 |Lω ω(t)dt dν(ζ ) < +∞. | b (z, ζ ) | ρ0 0 G+ G+ 0
Thus, the integral of .oω is uniformly convergent in any halfplane .G+ ρ0 (ρ0 > 0), and .oω + + is holomorphic in .G . Besides, .Re oω ≤ 0 in .G by Lemma 12.6, and hence by the Herglotz-Riesz theorem: y .Re oω (z) = py − π
f
+∞ −∞
dσ (t) , (x − t)2 + y 2
z = x + iy ∈ G+ ,
12.4 Boundary Property of Subclasses in G+
285
where .p = lim y −1 Re oω (iy) ≤ 0, while .σ is a nondecreasing function such that y→+∞
f
+∞
.
−∞
dσ (t) < +∞. 1 + t2
Further, by (12.47) and (12.35) .sup y |oω (iy)| < +∞ and by Fatou’s lemma y>0
f
+∞
.
−∞
f dσ (t) ≤ lim inf
+∞
y→+∞ −∞
y2 |dσ (t)| t 2 + y2
= lim inf y |Re oω (iy)| ≤ sup y |o(iy)| < +∞. y→+∞
y>0
Consequently, for any .z = x + iy ∈ G+ Re oω (z) = −
.
y π
f
+∞
−∞
dσ (t) = −Lω (x − t)2 + y 2
( Re
1 π
f
+∞ −∞
) Cω (z − t)dσ (t) , (12.48)
∼ω − Lω P0 in the where .σ is nondecreasing and bounded. On the other hand, .Re oω = Lω P whole .G+ . To prove this, first note that ff Lω P0 (z) =
.
G+
Lω log |b0 (z, ζ )|dν(ζ ),
y ≥ 2η.
Indeed, by (12.2) f
f
+∞
η
dt 2 0 −η [z + iσ − ξ − it] (f η f A0 f +∞ ) f η dt ω(σ )dσ = Re + + 2 η 0 A0 −η [z + iσ − ξ − it]
Lω log |b0 (z, ζ )| = Re
.
ω(σ )dσ
≡ K1 + K2 + K3 . Evidently, for any .z = x + iy ∈ G+ 2R0 , .η ≤ R0 and .y + σ − t ≥ y/2 f
f
f η f η dt dt ≤ ω(σ )dσ 2 (y + σ − t)2 |z + iσ − ξ − it| 0 −η 0 −η f η f η f η 4 ≤ 2 ω(σ )dσ dt ≤ MR0 ω(σ )dσ. R0 0 −η 0
|K1 | ≤
.
η
ω(σ )dσ
η
12 Subclasses of Functions Delta-Subharmonic in .G+
286
Besides, by (12.39) f
f
A0
|K2 | ≤
ω(σ )dσ
.
η
f
' ≤ MA 0 ,R0
η
−η
dt 8ηω(η)A0 ≤ 2 |z + iσ − ξ − it| y2
η
ω(σ )dσ, 0
and by the inequalities .y + σ − t ≥ σ and .ω(η) ≥ ω(R0 ), we get f
f +∞ f η dt α−2 ≤ σ dσ dt 2 A0 −η |z + iσ − ξ − it| A0 −η f η 2ηω(η)Aα−1 0 '' ≤ ω(σ )dσ, ≤ MA0 ,R0 ,α (1 − α)ω(R0 ) 0
|K3 | ≤
.
f
+∞
σ α dσ
η
' '' where .MR0 , MA , andMA > 0 depend only on .α, .A0 , and .R0 . Thus, 0 ,R0 0 ,R0 ,α
| | |Lω log |b0 (z, ζ )|| ≤ M ''' A0 ,R0 ,α
f
η
ω(σ )dσ
.
0
''' ' '' = MR0 + MA + MA , and by (12.35) with .MA 0 ,R0 ,α 0 ,R0 0 ,R0 ,α
ff |Lω P0 (z)| ≤
.
≤
G+
| | |Lω log |b0 (z, ζ )||dν(ζ )
''' MA 0 ,R0 ,α
ff G+
(f
η
) ω(σ )dσ dν(ζ ).
(12.49)
0
So, the integrand of .Lω P0 has an independent of .z ∈ G+ 2R0 , integrable majorant, and hence + .Lω P0 is harmonic in .G . Besides, by Theorem 12.3 2R0 ∼ω (z) = Lω P
.
ff G+
| | Lω log |∼ bω (z, ζ )|dν(ζ )
{ } ∪ is harmonic in the domain .D = z ∈ G+ : z /∈ ζ ∈supp ν [ζ, Re ζ ] . Thus, . Re oω = ∼ω − Lω P0 in the whole .G+ by the uniqueness of harmonic function. Lω P
12.4 Boundary Property of Subclasses in G+
287
∼ω ∈ Mω Now observe that by (12.49) .P0 ∈ Mω with the domain .A(π/2, 2R0 ), while .P for .A(π/2, R0 + 1) by (12.21). Therefore, by (12.35) f
| | f | | ∂ P0 (x + iy)|dx ≤ | ∂y |
+∞ |
.
−∞
ff
f
ff
+∞
−∞
η
dx f
f
G+
dν(ζ )
η −η
dt |z − ξ − it|2
+∞
dx π = dν(ζ ) dt = 2 + (y − t)2 + y −t (x − ξ ) G −η −∞ ff 4π η dν(ζ ) < +∞, y ≥ 2R0 . ≤ y G+
f
ff G+
dν(ζ )
η −η
dt
Thus, .P0 satisfies the condition (11.3). Besides, for any .z = x + iy with .y ≥ 2R0 ∼ω (z) = Re P
.
(f
ff G+
η −η
) Cω (z − ξ − it)ω(η − |t|)dt dν(ζ ).
Hence, using the estimate (5.19) with .β = ε−1, .ε ∈ (0, 1), and .y ≥ 2R0 (then .y−t ≥ y/2) and the two-sided inequality .(a + b)λ x a λ + bλ (a, b, λ ≥ 0), we get f
| | f | |∂ P ∼ω (x + iy)|dx ≤ | ∂y |
+∞ |
.
−∞
+∞ −∞
≤ CR0 ,ε ≤ CR' 0 ,ε ≤ CR'' 0 ,ε
ff |Cω1 (z − ξ − it)|dx f
+∞
−∞
f
−∞
f
+∞
−∞
G+
G+
|s|1+ε
+ R0
1+ε
ds (|s| + 1)1+ε
ff
G+
ω(η − |t|)dt η
ω(η − t)dt
0
f
η
dν(ζ )
(f
G+
−η
dν(ζ )
ff
ds
η
f
ff
dx |z − ξ − it|1+ε
+∞
f dν(ζ )
ω(t)dt 0
η
) ω(t)dt dν(ζ ) < +∞,
0
∼ω and where the constants .CR0 ,ε , CR' 0 ,ε , andCR'' 0 ,ε > 0 depend only on .R0 and .ε. Thus, .P .P0 satisfy the condition (11.3). Finally, by (12.48) ∼ω (z) − P0 (z)) = −Lω Re oω (z) = Lω (P
.
(
1 π
f
+∞
−∞
) Re Cω (z − t)dσ (t)
for all .z ∈ G+ , and by Lemma 11.2, we come to formula (12.46).
u n
Now, we are ready to prove the main theorem of this chapter, which describes the boundary + behavior of meromorphic functions f for which .log |f | ∈ Nm ω (G ) (see Definition 12.1).
12 Subclasses of Functions Delta-Subharmonic in .G+
288
Theorem 12.9 If .ω ∈ oαN (G+ ) and (12.37) is true, then any meromorphic in .G+ function + f , such that .log |f | ∈ Nm ω (G ), has non-zero, finite perpendicular boundary values at all points .x ∈ (−∞, +∞), except, perhaps, a set of zero .ω-capacity. + Proof If .log |f | ∈ Nm ω (G ) and (12.37) is true, then by Remark 12.3, the factorization (12.32) is true. Further, by Theorem 12.7 the limits of the Blaschke products in the numerator and denominator of the factorization (12.32) by perpendiculars to the real axis exist are finite and non-zero everywhere, except some sets .E1,2 with .Cω (E1,2 ) = 0. By Lemma 11.6, the exponential factor in the factorization (12.32) has the same property everywhere, except a set .E3 with .Cω (E3 ) = 0. Hence, the function f has the same property everywhere, except the set .E4 = E1 ∪ E2 ∪ E3 and .Cω (E4 ) = 0. u n
Remark 12.5 If for some .x ∈ (−∞, +∞) c1 = lim y −1 log |f (x + iy)| = 0
.
y→+∞
+ in the factorization (12.32) of a meromorphic in .G+ function f with .log |f | ∈ Nm ω (G ) and .ω ∈ oαN (G+ ), then the function f is of bounded type in .G+ . Thus, in particular Theorem 12.9 establishes a boundary property of a subclass of functions of bounded type in .G+ .
12.5
Notes
The results of the chapter are published in [51, 53]. The work from [53] was done within the frames of University of Antioquia CIEN Project 2016-11126.
Banach Spaces of Functions Delta-subharmonic in the Halfplane
13
In this chapter, we introduce and investigate several Banach spaces of Green-type potentials in .G+ , the squares of the generalized fractional integrals of which possess bounded integral means. In the first sections of this chapter, we use the class of functional parameters defined below, while in the last section, we use the class .QA (G+ ) (see Definition 10.1). + Definition 13.1 The class .QB α (G ) .(−1 ≤ α < +∞) is the set of functions containing .ω ≡ 1 and all functions .ω satisfying the conditions:
(i) .ω is strictly increasing, continuously differentiable in .[0, +∞), .ω(0) = 0. (ii) .ω(t) X t 1+α for .A0 ≤ t < +∞ and some .A0 ≥ 0. Besides, we use the integral operator formally defined as follows: f Lω u(z) ≡
.
0
+∞
u(z + iσ )dω(σ ),
| | Lω u(z)|
ω≡1
≡ u(z),
z ∈ G+ .
+ Note that the above definition of the classes .QB α (G ) differs from Definitions 1.2 and 10.1 + + + + of the classes .Qα (G ) and .QA (G ). Anyway, .QB α (G ) \ {ω ≡ 1} ⊂ Qα (G ), and the restriction that .ω(t) ∈ QA (G+ ) is constant for .t ≥ A does not change the properties of + the considered in this chapter potentials, which are true for any .ω ∈ QB α (G ) .(−1 ≤ α < +∞).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5_13
289
290
13 Banach Spaces in the Halfplane
13.1
Additional Statements on Green-Type Potentials
In this section, we give some necessary in future statements on the Green-type potentials formed by means of the Blaschke-type factors {
.bω (z, ζ ) = exp
f −
{ bω (z, ζ ) = exp
−η
f −
} Cω (z − ξ − it)ω(η − |t|)dt ,
η
Im z > Im ζ > 0, .
(13.1)
}
η
Cω (z − ξ + it)ω(η + t)dt
−η
Im z > Im ζ > 0.
(13.2)
We start by some additional properties of the factors .bω and the corresponding potentials. If .ω ∈ QB . (−1 ≤ α < +∞), then by Lemma 10.2 and Theorem 10.1, the function α .Lω log |bω | is harmonic in .G+ , except the intercept .{z = ξ + iλ : 0 ≤ λ ≤ η}, and also it is subharmonic in .G+ , nonpositive and continuous in .G+ , and .Lω log |bω (x, ζ )| = 0 .(−∞ < x < +∞). Besides, the following representation is true: f f (z, ζ ) ≡ Lω log bω (z, ζ ) =
η
.
−η
ω(η − |τ |)dτ , z ∈ G+ , τ + i(z − ξ )
(13.3)
+ where .ζ = ξ + iη ∈ G+ . Further, if .ω ∈ QB α (G ) .(−1 ≤ α < +∞) and a Borel-signed + measure .ν in .G satisfies the Blaschke-type condition
(f
ff .
G+
η
) ω(t)dt |dν(ζ )| < +∞,
ζ = ξ + iη ∈ G+ ,
(13.4)
0
where .|dν| = dν+ + dν− , then by Theorem 10.2, the Green-type potential -ω (z) ≡ P
ff
.
G+
| | log |bω (z, ζ )||dν(ζ )|
(13.5)
is convergent in .G+ , where it represents a delta-subharmonic function with the charge .ν. + At last, Theorem 10.3 states that, if .ω ∈ QB α (G ) .(−1 ≤ α < +∞) and a Borel-signed -ω is a continuous delta-subharmonic measure .ν satisfies the condition (13.4), then .Lω P + function in .G , -ω (z) = Lω P
.
ff G+
| | bω (z, ζ )|dν(ζ ), Lω log |-
z ∈ G+ ,
and the following inequality is true by Theorem 10.4: f .
sup
| |Lω P -ω (x + iy)|dx < 4π
+∞ |
y>0 −∞
(f
ff G+
η 0
) ω(t)dt |dν(ζ )| < +∞.
(13.6)
13.1 Additional Statements on Green-Type Potentials
291
-ω is subharmonic in In particular, if the measure .ν is nonnegative, then the function .Lω P + .G . Now, we proceed to proving some additional to those in Chap. 10 statements on the Green-type potentials ff Pω (z) ≡
.
G+
| | log |bω (z, ζ )|dν(ζ )
(13.7)
formed by the factors .bω , which are analogous to those in Chap. 12. Note that, as it is + proved in Chap. 7, if .ω ∈ Qα (G+ ) .(⊃ QB α (G ) \ {ω ≡ 1}) .(−1 ≤ α < +∞) and a Borel-signed measure .ν satisfies the Blaschke -type condition (f
ff .
G+
2η
) ω(t) dt |dν(ζ )| < +∞,
ζ = ξ + iη,
(13.8)
0
then the Green-type potential (13.7) is convergent in .G+ . We continue by the following additional statements where we use some results with .ω ∈ QA (G+ ) from Chap. 10. It is easy to verify that all results on the Blaschke-type factors and Green-type potentials of + Chap. 10 are true for the class of functional parameters .QB α (G ) \ {ω ≡ 1} .(−1 ≤ α < +∞). Hence, we shall refer to several statements of Chap. 10 under this assumption. + + Lemma 13.1 If .ω ∈ QB α (G ) \ {ω ≡ 1} and .ζ = ξ + iη ∈ G is a fixed point, then
f Lω log |bω (z, ζ )| = −Re
η
.
−η
ω(η + τ )dτ , τ − i(z − ξ )
z ∈ G+ ,
(13.9)
this is a nonpositive, continuous, subharmonic function in .G+ = {z ∈ C : Im z ≥ 0}. Proof Due to (13.2), it is easy to see that formula (13.9) is true, and obviously the Cauchytype integral is to be understood as its principal value on the intercept .{z = ξ + ih : −η ≤ h ≤ η}. Further, by (13.2) and (13.1) .
log |bω (z, ζ )| = log |bω (z, ζ )| − Uω (z, ζ ),
z ∈ G+ , z /= ζ,
where f
η
Uω (z, ζ ) = − Re
.
f
[ ] Cω (z − ζ + iτ ) + Cω (z − ζ − iτ ) ω(τ )dτ
0 2η
+ Re 0
Cω (z − ζ + iτ )ω(τ )dτ
(13.10)
292
13 Banach Spaces in the Halfplane
f = − Re
η −η
f + Re
η
−η
Cω (z − ξ − iτ )ω(η − |τ |)dτ Cω (z − ξ + iτ )ω(η + τ )dτ.
Hence, we get f
η
Uω (z, ζ ) = Re
.
Cω (z − ξ + iτ ) [ω(η + τ ) − ω(η − τ )] dτ
(13.11)
0
for .−η < Im z < +∞, and consequently Lω log |bω (z, ζ )| = Lω log |bω (z, ζ )| − Lω Uω (z, ζ ),
.
z ∈ G+ ,
(13.12)
where, as it is proved in Theorem 10.1, the function .Lω log |bω | is continuous, nonpositive in .G+ and subharmonic in .G+ , while obviously the function f η τ +y (13.13) .Lω Uω (z, ζ ) = [ω(η + τ ) − ω(η − τ )] dτ, 2 2 0 (x − ξ ) + (τ + y) is positive and harmonic even in .−η < y < +∞.
u n
Remark 13.1 If .ω ≡ 1, then .Lω is the identical operator, and we simply have | | .Lω log |bω (z, ζ )|| |
ω≡1
| | f η |z − ζ | dτ | | = −Re , = log | | τ − i(z − ξ) z−ζ −η
z ∈ G+ \ {ζ }.
Thus, the above lemma remains true, except the continuity at the point .z = ζ , where .log |b0 | = −∞. + + Lemma 13.2 If .ω ∈ QB α (G ) and .ζ = ξ + iη ∈ G is a fixed point, then
f 0>
+∞
.
−∞
f
2η
Lω log |bω (x + iy, ζ )|dx ≥ −π
ω(t)dt > −∞,
y > 0.
(13.14)
0
Proof Using (13.9) and changing the order of integration, we get f
+∞
.
−∞
f Lω log |bω (x + iy, ζ )|dx = −
η −η
f
= −π f
f ω(η + τ )dτ η
−η
0
−∞
τ +y dx (x − ξ )2 + (τ + y)2
ω(η + τ )sign(y + τ )dτ
2η
= −π
+∞
ω(t)sign(t − η + y)dt ≡ Dω (η, y).
13.1 Additional Statements on Green-Type Potentials
293
Obviously f
2η
Dω (η, y) = −π
ω(t)dt,
.
y ≥ η,
0
and {f Dω (η, y) = −π
2η
.
f
η−y
ω(t)dt −
}
f
0
η−y
2η
ω(t)dt > −π
ω(t)dt,
0 < y < η.
0
+ Theorem 13.1 If .ω ∈ QB α (G ) \ {ω ≡ 1} .(−1 ≤ α < +∞) and the Blaschke-type condition (13.8) is satisfied, then .Lω Pω is a continuous delta-subharmonic function in + .G , and
ff Lω Pω (z) =
.
G+
Lω log |bω (z, ζ )|dν(ζ ),
z ∈ G+ ,
(13.15)
where the integral is absolutely and uniformly convergent inside .G+ . Besides, the following inequality is true: f .
sup
| |Lω Pω (x + iy)|dx ≤ π
+∞ |
y>0 −∞
ff G+
(f
)
2η
ω(t) dt
|dν(ζ )| < +∞.
(13.16)
0
If .ω(t) ≡ 1, then all above statements are true again, except the continuity of .Lω Pω and the nature of convergence of the integral (13.15). Proof If .ω(t) ≡ 1, then .Lω is the identical operator and .Lω Pω ≡ P0 , i.e., is the ordinary Green potential. Hence, its convergence is well-known, and formula (13.15) is obvious. Now, let .ω /≡ 1 and .Im z ≥ ρ > 0. For proving (13.15), where the right-hand side integral is absolutely and uniformly convergent in .G+ ρ = {ζ : Im ζ > ρ}, we use formula (13.12), and note that the same type formula with the same kind convergence of the integral with .Lω log |bω | is proved in Theorem 10.3. So, by (13.12) it suffices to prove the uniform convergence of the integral ff .
G+
Lω Uω (z, ζ )|dν(ζ )|
in .G+ ρ = {z ∈ C : Im z ≥ ρ}. To this end, observe that by (13.13) for any .z ∈ K, where + .K ⊂ Gρ is any compact, 1 .0 < Lω Uω (z, ζ ) ≤ ρ
f 0
η
1 [ω(η + τ ) − ω(η − τ )] dτ < ρ
f
2η
ω(t)dt, 0
294
13 Banach Spaces in the Halfplane
and the uniform convergence holds by the Blaschke-type condition, and the desired statements will hold by the equality ff Lω Pω (z) =
.
G+
ff Lω log |bω (z, ζ )|dν(ζ ) −
Lω log |Uω (z, ζ )|dν(ζ ),
G+
z ∈ G+ .
For proving (13.16), even for .ω ≡ 1, observe that by (13.15) and (13.14) and the nonpositivity of the function .Lω log |bω |, we get f
| |Lω Pω (x + iy)|dx =
+∞ |
.
−∞
ff
f
G+
|dν(ζ )| (f
ff ≤π
G+
2η
| |Lω log |bω (x + iy, ζ )||dx
+∞ | −∞
) ω(t)dt |dν(ζ )| < +∞.
0
+ Noting that for any .ω ∈ QB α (G ) .(−1 ≤ α < +∞) the inequalities (13.6) and (13.16) are true; in the next three sections, we shall exactly calculate the similar quadratic integral means which obviously can be bounded only by some expressions greater than the righthand side Blaschke-type integrals in (13.6) and (13.16).
Banach Spaces of Potentials P-ω
13.2
In this section, some Banach spaces of Green-type potentials formed by the Blaschketype factors .bω are introduced, the regularizations of which by means of the operator .Lω possess bounded square integral means. Note that the following equality is true for any function .ϕ and any measure .μ, for which the below integrals exist:
.
( f )2 f f [ ] 1 Re ϕ(ζ )dμ(ζ ) = Re dμ(ζ1 ) ϕ(ζ1 )ϕ(ζ2 ) + ϕ(ζ1 )ϕ(ζ2 ) dμ(ζ2 ). 2 E E E (13.17)
Thus, we come to the following representation: 1 . 2π
f
+∞ [
−∞
]2
1 Lω Pω (x + iy) dx ≡ 2π ff =
f
+∞ [f f
−∞
G+
dν(ζ1 )
G+
]2 Re f (x + iy, ζ ) dν(ζ ) dx
ff
G+
J-(y, ζ1 , ζ2 ) dν(ζ2 ),
(13.18)
-ω 13.2 Banach Spaces of Potentials P
295
where 1 J-(y, ζ1 ,ζ2 ) = 2π
f
+∞
Lω log |bω (x + iy, ζ1 )|Lω log |bω (x + iy, ζ2 )| dx
.
= Re
1 4π
−∞
f
+∞ [
] f-(x + iy, ζ1 )f-(x + iy, ζ2 ) + f-(x + iy, ζ1 )f-(x + iy, ζ2 ) dx,
−∞
(13.19) provided these integrals exist and the change of integration order is true in (13.18). + Lemma 13.3 If .ω ∈ QB α (G ) .(−1 ≤ α < +∞), then the function .J (y, ζ1 , ζ2 ) is + nonnegative in .0 ≤ y < +∞ and continuous for any .ζ1,2 ∈ G .
Proof The desired statements are obvious, since the function .log |b0 (x + iy, ζ )| ≤ 0 is continuous in .G+ , except the point .ζ ∈ G+ , where it has a logarithmic singularity, and, at the same time, for any .y > 0, it has second order of decrease as .x → ±∞. The same is + true for any .ω ∈ QB α (G ) .(α ≥ −1), since .Lω log |bω | ≤ 0, in addition, is continuous in u n the whole .G+ . If a Borel-signed measure .ν is finite in any compact in .G+ , then by the Jordan theorem, it can be decomposed in .G+ as the difference of its positive and negative variations, .ν = ν+ − ν− , where both .ν± are nonnegative Borel measures in .G+ , the supports of which do not intersect. Hence, several conditions on both .ν± , which provide some properties of the Green-type potentials by this measures shall provide similar properties of the whole Green-type potential. + Definition 13.2 Assuming that .ω ∈ QB α (G ) .(−1 ≤ α < +∞) and .0 < h < +∞, -ω associated with Borel-signed we define .Pω,h as the class of Green-type potentials .P measures .ν satisfying the Blaschke-type condition (13.4), the supports of the measures + .ν± are located in the strip .G (0,h] = {ζ = ξ + iη : 0 < η ≤ h}. In .Pω,h , we define the norm
|| ||2 ||P -ω || ≡ sup
f
+∞
.
y>0 −∞
-ω (x + iy)|2 dx |Lω P
{f f
= sup y>0
G+
ff G+
} J-(y, ζ1 , ζ2 )|dν(ζ1 )||dν(ζ2 )| < +∞,
(13.20)
where J is the integral (13.19) and .|dν| = dν+ + dν− is the differential of the complete variation of the measure .ν.
296
13 Banach Spaces in the Halfplane
Remark 13.2 In the same way as Remark 9.1, it can be proved that the quantity (13.20) satisfies all the norm axioms including the triangle inequality. In particular, we have the following inequalities: |ν1± − ν2± | ≤ |ν1 − ν2 |.
.
(13.21)
+ Theorem 13.2 For any .ω ∈ QB α (G ) .(−1 ≤ α < +∞) and .0 < h < +∞, the set .Pω,h is a Banach space with the norm (13.20).
Pω,h is to be proved. To this Proof It is obvious that only the completeness of the space .{ (n) }∞ -ω Pω,h , which satisfy the is a sequence of Green-type potentials from .end, suppose . P 1 Cauchy condition, i.e., for any .ε > 0 || (n+m) || ||P -ω(n) ||2 -ω −P ff = sup
.
ff
G+
y>0
G+
| || | J-(y, ζ1 , ζ2 )|d(νn+m − νn )(ζ1 )||d(νn+m − νn )(ζ2 )| < ε,
provided .n ≥ Nε for some .Nε ≥ 1 large enough and || (n) || any .m ≥ 1. We shall prove that || there is a potential .Pω ∈ Pω,h such that . Pω − Pω ||ω → 0 as .n → ∞. Note that by the triangle inequalities (13.21) the same Cauchy condition is true for the positive, negative, and complete variations of the measures .νn , and hence it suffices to consider only one of these variations, which we denote again by .νn . By Lemma 13.3, the function .J (y, ζ1 , ζ2 ) is positive and continuous in .0 < y < +∞. Further, we fix an arbitrary .ρ ∈ (0, h] and suppose that .|supp ν| < h < +∞ and .ζ1,2 = ξ1,2 + η1,2 ∈ supp ν are such that .η1,2 ≥ ρ > 0. Then, by (13.3) we get f .
sup
+∞
y>0 −∞
Lω log |bω (x + iy, ζ1 )|Lω log |bω (x + iy, ζ2 )|dx ω2 (ρ)h3 > 4
f |x|>2h
1 [ ]4 dx ≡ Kω,ρ,h > 0. 2 |x| + 9h2
+ Denoting .G+ ρ = {ζ = ξ + iη ∈ G : ρ < η < +∞}, observe that for any .ε > 0 there is some .Nε ≥ 1 such that
( ff .
) | 2 | || || |d(νn+m − νn )(ζ )| ≤ (Kω,ρ,h )−1 ||P (n+m) − P (n) || < (Kω,ρ,h )−1 ε ω ω ω +
Gρ
-ω 13.2 Banach Spaces of Potentials P
297
for any .n ≥ Nε and .m ≥ 1. If .Dρ is the disc centered at the point .i(ρ + 1/ρ), with the ' radius .1/ρ. Then obviously .Dρ ⊂ G+ ρ , and there is some .Nε ≥ 1 such that | | |νn+m (Dρ ) − νn (Dρ )| ≤
ff
.
Dρ
| | |d(νn+m − νn )(ζ )| ≤
ff + Gρ
| | |d(νn+m − νn )(ζ )| < ε
for any .n ≥ Nε' and .m ≥ 1. Observe that the Cauchy sequence of numbers .νn (Dρ ) is bounded: .0 ≤ νn (Dρ ) ≤ Aρ < +∞ .(n ≥ 1). Besides, it is easy to see that the sequence of Borel measures .{νn }∞ n=1 satisfies the Cauchy condition in the linear manifold .oρ of real, finite, continuous functions in the closed disc .Dρ , which vanish outside .Dρ . Hence, by Theorem .0.4' in [62], there is a Borel measure .ν ρ ≥ 0 such that the weak convergence ρ .νn ⇒ ν .(n → ∞) is true in .oρ , i.e., for any function .g ∈ oρ ff
ff .
G+
g(ζ )dνn (ζ ) →
G+
g(ζ )dν ρ (ζ )
as
n → ∞.
Since this is true for any .ρ ∈ (0, h] and the embedded discs .Dρ exhaust .G+ as .ρ → 0, with .ν ρ in any .Dρ . there is a limit Borel measure .ν over the whole| .G+ , which coincides | | | Obviously, for any fixed .ρ ∈ (0, h], we have . (νn − ν)(Dρ ) < ε, if n is large enough. Hence .νn (Dρ ) → ν(Dρ ) and .νn ⇒ ν as .n → ∞ on .oρ for any fixed .ρ ∈ (0, h]. -(n) || satisfies the Cauchy condition because of the Also the| sequence of numbers | .||Pω(m+n) (m+n) (n) | | -ω -ω -ω || ≤ ||P -ω(n) ||, and hence || − ||P −P inequality . ||P -ω(n) ||2 → b = ||P / +∞ as n → ∞ and
.
-ω(n) ||2 ≤ B < +∞ for n ≥ 1. ||P
Further, assuming that .{ρk }∞ 0 ⊂ (0, h] is a sequence such that .ρ0 = h and .ρk ↓ 0, we conclude that { ∞ ff ∞ E E B ≥ lim inf sup
.
n→∞
≥ sup
y>0 k=1 m=1
ff Dρk \Dρk−1
ff
∞ E ∞ ff E
y>0 k=1 m=1
Dρm \Dρm−1
Dρk \Dρk−1
Dρm \Dρm−1
J-(y, ζ1 , ζ2 )|dνn (ζ1 )||dνn (ζ2 )|
}
-ω ||2 , J-(y, ζ1 , ζ2 )|dν(ζ1 )||dν(ζ2 )| = ||P
-ω || < +∞. Then, -ω is the Green-type potential generated by the measure .ν, and .||P where .P introducing the sequence of nonincreasing in .0 < ρ ≤ h functions ff ϕn (ρ) ≡ ϕn (ρ, y) ≡
.
Dρ ×Dρ
J-(y, ζ1 , ζ2 )|d(ν − νn )(ζ1 )||d(ν − νn )(ζ2 )|,
298
13 Banach Spaces in the Halfplane
where .0 < y < +∞ is fixed, we see that -ω − P -ω(n) ||2 ≤ 2B + 1 ϕn (ρ) → ϕn (0) ≤ ||P
.
as
ρ → 0,
-ω − P -ω(n) ||2 ≤ 2B + 1 for any .ρ ∈ (0, h], provided n is large and evidently .ϕn (ρ) ≤ ||P enough. Hence, by a Helly theorem, there is a subsequence of natural numbers .nk ↑ ∞ such that at all points .0 < ρ ≤ h there exists a limit function ϕ(ρ) ≡ lim ϕnk (ρ).
.
k→∞
On the other hand, for any .ρ ∈ (0, h], the complete variation of the measure .ν − νn in .Dρ weakly tends to zero. Consequently, due to the continuity of .Jω (y, ζ1 , ζ2 ) by .ζ1 , ζ2 ∈ Dρ , we conclude that .ϕn (ρ) → 0 as .n → ∞ for any .ρ ∈ (0, 1], and hence .ϕ(ρ) ≡ 0 .(0 < ρ ≤ h). Further, .ϕnk (ρ) is monotone by .0 < ρ ≤ h, and hence ( .
lim
k→∞
) lim ϕnk (ρ) = lim lim ϕnk (ρ) = 0.
ρ→0
ρ→0 k→∞
-ω − P -ω(nk ) || = 0, since Remind that this is true for any .0 < y < +∞, and hence .limk→∞ ||P ( ) the function .supy ' 0
{f ff
+
+∞ −∞
}1/2 |Lω u(x + iy)| dx 2
(f
G+
η
0
dt ω(η + t) √ t
) |dν(ζ )| < +∞.
(13.39)
Definition 13.7 .D'ω,h .(0 < h < +∞) is the set of those delta-subharmonic in .G+ functions u with associated Borel-signed measures .ν, the supports of which are located + in the strip .G+ (0,h] = {ζ = ξ + iη ∈ G : 0 < η ≤ h}, for which 1 2π y>0
{f
||u||ω = sup
.
ff +
+∞
−∞
}1/2 |Lω u(x + iy)|2 dx
(f
G+
2η 0
dt ω(η + t) √ t
) |dν(ζ )| < +∞.
(13.40)
- ' and .D' Theorem 13.7 For any .0 < h < +∞, the sets .D ω,h ω,h are Banach spaces.
13.5 Banach Spaces of Delta-subharmonic Functions
307
Proof Note that, if a Borel-signed measure .ν is such that (13.39) is true, then it satisfies also the Blaschke-type condition (10.21) providing its convergence. Hence, for any .u ∈ -ω and obviously also .Lω U are harmonic in .G+ . Further, the - ' , the function .U ≡ u − P D ω,h quantity (13.39) evidently satisfies all the norm axioms including the triangle inequality. -ω by Theorem 10.6. Hence, .Lω U belongs Besides, the “sup” in (13.39) is bounded for .P 2 + to the ordinary harmonic Hardy space .h (G ). This means that U is of the Hilbert space 2 + .h ∗ (G ) of harmonic functions, which is defined by the condition ω { ff ||U ||h2 ∗ =
.
G+
ω
[ ]2 U (z) dμω∗ (z)
}1/2 < +∞,
where .dμω∗ (x + iy) = dxdω∗ (2y) and .ω∗ is the Volterra square of .ω, i.e., ω∗ (t) ≡
f
t
.
ω(t − λ)dω(λ), if 0 < t < +∞ and
ω∗ (0) ≡ 0.
0
Besides, the following representation is true: U (z) = Re
.
= Re
1 π
ff
1 2π
f
G+
U (w)Cω∗ (z − w)dμω (w)
+∞
−∞
Cω (z − t)Lω U (t)dt,
z ∈ G+ .
(13.41)
' - ' is the direct sum of the Hilbert space .h2 ∗ (G+ ) and the Banach space .Pω,h So, .D ω ω,h ' of Green-type potentials, i.e., any delta-subharmonic function .u ∈ D ω,h has a unique representation of the form
-ω , u=U +P
.
U ∈ h2ω∗ (G+ ),
-ω ∈ P P'ω,h .
(13.42)
In the same way, we obtain that .D'ω,h is the direct sum of the Hilbert space .h2ω∗ (G+ ) of ' , i.e., any harmonic in .G+ functions and the Banach space of Green-type potentials .Pω,h delta-subharmonic function .u ∈ Dω,h has a unique representation of the form u = U + Pω ,
.
U ∈ h2ω∗ (G+ ),
' -ω ∈ Pω,h P .
(13.43)
Remark 13.6 Obviously, the analog of Theorem 13.7 is true for the large Banach spaces - ω,h of delta-subharmonic functions defined as direct sums of the harmonic .Dω,h and .D Pω,h of Hilbert space .h2ω∗ (G+ ) and, correspondingly, the large Banach spaces .Pω,h and .Green-type potentials (see Definitions 13.2 and 13.4 and Theorems 13.2 and 13.4). Remark 13.7 Since .log |f | of a meromorphic function f is a particular case of deltasubharmonic function, all the considered above spaces can be understood also as Banach
308
13 Banach Spaces in the Halfplane
spaces of functions meromorphic in the upper halfplane .G+ . Then, the representations (13.42) and (13.43) become factorizations in the corresponding four Banach spaces of functions meromorphic in .G+ . Namely, the function U turns to the exponent of any of the integrals in (13.41) without the real part, which is multiplied by some quotients of Blaschke-type products with zeros and poles of the considered meromorphic function, -ω . which replaces the potentials .Pω and .P
13.6
Notes
This section gives the results of [55] in a somehow modified form. This work was done in the frames of the University of Antioquia Investigation Project 2020-33252.
A
Addendum
This chapter contains the English translation of the detailed and complemented version [9] (1948) of the earliest paper of M.M. Djrbashian [8] (1945) which was mainly aimed at improving R. Nevanlinna’s result of 1936 (see [66], Sec. 216) on the density of zeros and poles of his weighted classes of functions meromorphic in .D and where the theory of the p weighted spaces .Aα was introduced and developed. Academy of Sciences of Armenian SSR Soobscheniya instituta matematiki i mekhaniki Vipusk 2, 1948, pp. 3–40 M. M. Djrbashian On the Representability Problem of Analytic Functions The investigations of F. Riesz.(1) , R. Nevanlinna.(2) , I.I. Privalov.(3) , and other authors, which are devoted to the study of several classes of holomorphic and meromorphic functions in the unit disc, are well-known at present. There are also numerous works devoted to the completeness of different systems of analytic functions. The present paper is devoted to the investigation of some classes of analytic functions. In Sect. A.1, a class .Hp (α) .(p > 0, .α > −1) of functions holomorphic in the interior of the unit disc is introduced and some theorems on parametric representation1
1 Translator’s note. The term “parametric representation” is used for a representation which
completely describes the considered class of functions, i.e., a function is of that class, if and only if it is representable in that way. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5
309
310
A
Addendum
and uniqueness are proved for functions of this class. The completeness of several systems of rational functions in .H2 (α) is established. In Sect. B.1, a generalization of the well-known Jensen–Nevanlinna formula for meromorphic functions is proved. This permits to prove a theorem on the canonical representation of meromorphic functions which have unbounded characteristics in the unit disc. Besides, it is shown that formula (B.2.3) of Sect. B.1 and Theorem XII1 are natural generalizations of several results of R.Nevanlinna.(2) . Section C.1 gives an apparatus for the integral representation of several classes of entire functions. A theorem on the completeness of a Mittag-Leffler.(4) type entire functions system is proved in the whole complex plane, in a class of entire functions of finite order and type. Some of the results of this paper were presented in two reports.(14,15) .
A.1
On Representation of Some Classes of Functions Holomorphic in the Unit Disc
1. We define .Hp (α) .(p > 0, .α > −1) as the set of all those functions .f (z) which are holomorphic in .|z| < 1 and for which the integral α+1 . π
f 1f 0
2π
(1 − ρ 2 )α |f (ρeiϑ )|p ρdρdϑ
(A.1.1)
0
exists.2 It is obvious that, if .α1 < α2 , then .Hp (α1 ) ⊂ Hp (α1 ), and .Hp2 (α) ⊂ Hp1 (α), if .p1 < p2 . We prove the following lemma. Lemma A.1 If .f (z) belongs to .Hp (α) .(p ≥ 1, .α ≥ 0), then f 1f .
2π
lim
x→1−0 0
(1 − ρ 2 )α |f (ρeiϑ ) − f (xρeiϑ )|ρdρdϑ = 0.
(A.1.2)
0
Proof For an arbitrary .ε > 0, we choose some .r1.(0 < r1 < 1) to provide that f 1f
2π
.
0
0
(1 − ρ 2 )α |f (ρeiϑ )|ρdρdϑ
0), if the integral
.
1 2π
f 0
2π
|f (reiϑ )|p dϑ,
0 < r < 1,
(A.2.1)
A
Addendum
313
remains bounded as .r → 1 − 0. Therefore it is obvious that .Hp ⊂ Hp (α) for any .α > −1, and the next theorem is true. Theorem II A function .f (z) holomorphic in the unit disc belongs to .Hp , if and only if the following integral is bounded as .α → −1: α+1 . π
f 1f 0
2π
(1 − ρ 2 )α |f (ρeiϑ )|p ρdρdϑ.
(A.2.2)
0
Proof If .f (z) ∈ Hp , then the integral (A.2.1) remains bounded as .r → 1 − 0. Consequently, the integral α+1 . π
f 1f 0
2π
1 (1 − r ) |f (re )| rdrdϑ < 2π 2 α
0
iϑ
f
2π
p
|f (reiϑ )|p dϑ
(A.2.3)
0
is uniformly bounded as .α → −1, since (A.2.1) is nondecreasing as .r → 1. Conversely, let the integral (A.2.2) be bounded from above by a number M as .α → −1. As (A.2.1) is nondecreasing as .r → 1 − 0, for any .0 < ρ < 1 we get (1 − ρ 2 )α+1
.
1 2π
f
2π
|f (ρeiϑ )|p dϑ
0, .α > −1). Let .{αn } be the sequence of zeros of a function .f (z) ∈ Hp (α) in .0 < |z| < 1. We assume that .{αn } are numerated in the order of increasing modulus and according their multiplicities, and by .n(t) we denote the number of occurrences of .αn in the disc .|z| ≤ t. If .f (z) /≡ 0, then by Jensen’s.(7) formula f N (r) ≡
.
0
r
1 n(t) dt < t 2π
f
2π
log |f (reiϑ )|dϑ + A
(A.3.1)
0
(henceforth A is for a constant independent of r). Using the well-known inequality
.
1 2π
f
2π 0
{ log |f (reiϑ )|p dϑ ≤ log
1 2π
f 0
2π
} |f (reiϑ )|p dϑ ,
(A.3.2)
314
A
Addendum
we obtain f .
1
(1 − r)α epN (r) dr < A
f 1f
0
0
2π
(1 − r)α |f (reiϑ )|p rdrdϑ.
(A.3.3)
0
The right-hand side integral in (A.3.3) converges since .f (z) ∈ Hp (α). Thus, we can state the following theorems. Theorem III1 If a function .f (z) ∈ Hp (α) is not identically zero, then f
1
.
(1 − r)α epN (r) dr < +∞.
(A.3.4)
0
Theorem III2 If .f (z) ∈ Hp (α) and f
1
.
(1 − r)α epN (r) dr = +∞,
(A.3.5)
0
then .f (z) ≡ 0. In particular, Theorem III2 implies the following statement. Theorem III3 Let .f (z) ∈ Hp (α) and .n(r), which counts the zeros of .f (z), satisfies one of the conditions (a) lim inf (1 − r)n(r) = lim inf (1 − |αn |)n >
.
n→∞
r→1−0
α+1 , p
(A.3.6)
(b) lim sup
.
r→1−0
1−r log
1 1−r
n(r) = lim sup n→∞
1 − |αn | log
1 1−|αn |
n>e
α+1 . p
(A.3.7)
Then .f (z) ≡ 0. Proof (a) Let lim inf (1 − r)n(r) > A >
.
r→1−0
α+1 . p
(A.3.8)
A
Addendum
315
Then .(1 − r)n(r) ≥ A for .r ≥ r0 . Hence f
r0
N(r) =
.
0
n(t) dt + t
f
r
(1 − t)n(t)
r0
1 dt > A + A log . t (1 − t) 1−r
(A.3.9)
By (A.3.8) and (A.3.9) we obtain (1 − r)α epN (r) >
.
A . 1−r
(A.3.10)
Consequently, the integral (A.3.5) diverges, and .f (z) ≡ 0 by Theorem III2 . (b) If .f (z) ∈ Hp (α) and .f (z) /≡ 0, then (A.3.4) in particular implies √ .
√ r(1 − r)α+1 epN (r)
dt > (1 − r )n(r) t (1 − t) t r r } { 1−r 1 . = (1 − r λ )n(r) log 1−λ + log 1 − rλ r f
λ
rλ
(A.3.12)
The inequalities (A.3.11) and (A.3.12) imply that lim sup
.
r→1−0
1−r log
1 1−r
n(r) ≤
1 λ log
1 λ
α+1 . p
For .λ < 1 the expression .λ log λ1 takes its maximal value at .λ = 1/e. Hence lim sup
.
r→1−0
1−r log
1 1−r
n(r) ≤ e
α+1 . p
n(r) > e
α+1 , p
So, if lim sup
.
r→1−0
1−r 1 log 1−r
(A.3.13)
316
A
Addendum
then necessarily .f (z) ≡ 0. Our theorem holds since the proof of the equalities (A.3.6) and (A.3.7) is simple.(8) . 4. If the zeros .{αn } are located in some special way, then the uniqueness property remains true for several classes of functions holomorphic in the unit disc under less density of the sequence .{αn }. Let the sequence .{αn } satisfy the conditions 1. lim αn = eiϑ0
.
n→∞
(0 ≤ ϑ0 < 2π )
(A.4.1)
2. | ( )| | | |arg αn e−iϑ0 − 1 | ≥ π/2 + δ
.
(δ > 0).
(A.4.2)
Then it is obvious that the following series diverge simultaneously: ∞ Σ .
|1 − αn | and
n=1
∞ Σ
(1 − |αn |).
(A.4.3)
n=1
Theorem IV If .f (z) is holomorphic in .|z| < 1 and |f (z)|
0)
(A.4.6)
in a neighborhood of a point .z = eiϑ0 , then the condition .f (αn ) = 0 .(n = 1, 2, . . .) implies .f (z) ≡ 0 only when the following series is divergent: ∞ Σ .
(1 − |αn |).
(A.4.7)
n=1
Proof The function .F (z) = (1−z)2p f (z) is holomorphic in the disc .|z −1/2| < 1/2, and .F (αn ) = 0 .(n = 1, 2, . . .). Besides, by (A.4.6) .F (z) is bounded in the disc .|z−1/2| < 1/2. The mapping .w = 2z − 1 transforms this disc to .|w| < 1, and it is obvious that the function .ϕ(w) = F ( w+1 2 ) is holomorphic and bounded in .|w| < 1, and .ϕ(βn ) = 0 for .βn = 2αn − 1 .(n = 1, 2, . . .). Suppose the series (A.4.7) diverges. Then obviously also the series ∞ Σ .
n=1
∞
|1 − αn | =
1Σ |1 − βn | 2 n=1
A
Addendum
317
is divergent. As also the sequence .{βn } satisfies the conditions (A.4.1) and (A.4.2), the Σ series . ∞ 1 (1 − |βn |) is divergent. Hence .ϕ(w) = F (z) = f (z) ≡ 0 by a well-known theorem. The example of the Blaschke product shows that in general .f (z) /≡ 0, if the series (A.4.7) is convergent. Corollary In virtue of the inequality (A.1.12), Theorem IV is true for any function .f (z) ∈ Hp (α) .(p ≥ 1). 5. Let us consider the class .H2 (α) .(α > −1). Let a function f (z) =
∞ Σ
.
an zn
(A.5.1)
n=0
belong to .H2 (α) .(α > −1). Then, the integral α+1 . π
f rf 0
2π
(1 − ρ ) |f (ρe )| ρdρdϑ = (α + 1) 2 α
iϑ
2
0
∞ Σ
f |an |
2
r
(1 − t)α t n dt
0
n=0
(A.5.2) is uniformly bounded for .0 < r < 1. Hence, by (A.1.9)
.
α+1 π
f 1f 0
2π
(1 − ρ 2 )α |f (ρeiϑ )|2 ρdρdϑ = (α + 1)
0
∞ Σ r(α + 2)r(1 + n) n=0
r(α + 2 + n)
|an |2 . (A.5.3)
The difference .f (z) −
Σ∞ 0
ak zk obviously belongs to .H2 (α). Hence, by (A.5.3)
| |2 f f n Σ | | α + 1 1 2π 2 α| iϑ iϑ k | . (1 − ρ ) |f (ρe ) − ak (ρe ) | ρdρdϑ π 0 0 k=0
=
∞ Σ k=n+1
r(α + 2)r(1 + k) |ak |2 . r(α + 2 + k)
(A.5.4)
Thus, by the convergence of the series (A.5.3), | |2 f f n Σ | | α + 1 1 2π (1 − ρ 2 )α ||f (ρeiϑ ) − ak (ρeiϑ )k || ρdρdϑ = 0. n→∞ π 0 0
.
lim
k=0
Now, we give two theorems on parametric representations of the class .H2 (α).
(A.5.5)
318
A
Addendum
Theorem V If a function .f (z) belongs to .H2 (α) .(α > −1), then the function f
α+1 .ϕ0 (z) = 2
1
(1 − ρ)
α−1 2
|z| < 1,
f (ρz)dϑ,
(A.5.6)
0
belongs to the class .H2 of Riesz. Besides, the following representation is true: 1 .f (z) = 2π
f
ϕ0 (t) (1 − tz)
|t|=1
α+3 2
|dt|,
|z| < 1.
(A.5.7)
Proof By (A.5.1),
ϕ0 (z) =
.
∞ α+1Σ
2
f
1
(1 − ρ)
an zn
α−1 2
ρ n dρ =
0
n=0
∞ r Σ n=0
(
)
r (1 + n) ) an zn α+3 r 2 +n α+3 2
(
(A.5.8)
in .|z| < 1. By Stirling’s formula
.
lim
n→∞
r(1 + n) r(α + 2 + n)
r2
(
α+3 2
) +n
r 2 (1 + n)
= 1.
(A.5.9)
Further, by the convergence of the series (A.5.3) and by (A.5.9) ∞ Σ .
n=0
r 2 (1 + n) ( ) |an |2 < +∞. α+3 2 r 2 +n
(A.5.10)
But (A.5.10) means that .ϕ0 (z) ∈ H2 in .|z| < 1. Consequently ( r an
.
)
f r (1 + n) ϕ0 (t) 1 ) |dt|, = 2π |t|=1 t n r α+3 + n 2 α+3 2
(
n = 0, 1, 2, . . .
(A.5.11)
Thus, by (A.5.1) and (A.5.11) 1 .f (z) = 2π
f |t|=1
ϕ0 (t)
(
{Σ ∞ n=0
r ( r
α+3 2
α+3 2
)
) +n
r(1 + n)
} (zt)n |dt|,
|z| < 1.
(A.5.12)
A
Addendum
319
By (A.1.8) 1 .
(1 − z)
α+3 2
=
(
∞ Σ
r ( r
n=0
) +n
α+3 2
α+3 2
)
r(1 + n)
|z| < 1.
zn ,
(A.5.13)
Therefore, our theorem follows from (A.5.12).3 Theorem VI The functions of .H2 (α) are representable in the following parametric form: f (z) =
.
1 2π
f
ϕ(t)
|t|=1
(1 − tz)
α+3 2
|dt|,
(A.5.14)
where .ϕ(t) is an arbitrary function of .L2 over .|t| = 1. The function .ϕ0 (t) .(|t| = 1), which is the boundary function of .ϕ0 (z), minimizes the integral f .
|t|=1
|ϕ(t)|2 |dt|
in the class of all functions .ϕ(t), which represent .f (z) by the integral (A.5.14). Proof Denote bn =
.
1 2π
f
ϕ(t)t n |dt|,
|t|=1
n = 0, 1, 2, . . . ,
(A.5.15)
where .ϕ(t) ∈ L2 over .|t| = 1. By the equalities 1 . 2π
f |t|=1
t n t m |dt| =
⎧ ⎨0,
m /= n
⎩1,
m=n
and Bessel’s inequality, the series ∞ Σ .
|bn |2
n=0
3 The case .α = 0 of Theorem V was proved by M.V. Keldysch.
(A.5.16)
320
A
Addendum
is convergent. Further, by (A.5.14), (A.5.13), and (A.5.12)
f (z) =
(
∞ Σ
r (
.
n=0
r
α+3 2
α+3 2
)
) +n
r(1 + n)
bn zn .
(A.5.17)
By the relation (A.5.9) and the convergence of the series (A.5.16) we conclude that f 1f
2π
.
0
(1 − ρ)α |f (ρeiϑ )|2 ρdρdϑ < +∞,
0
i.e., .f (z) ∈ H2 (α). Further, by Theorem V the function α+1 .ϕ0 (z) = 2
f
1
(1 − ρ)
α−1 2
f (ρz)dρ =
∞ Σ
bn zn
(A.5.18)
| |2 n Σ | | n| |ϕ0 (t) − bn t | |dt| = 0. |
(A.5.19)
0
n=0
belongs to the class .H2 of Riesz. Consequently f .
lim
n→∞ |t|=1
k=0
By (A.5.15) and (A.5.18) f .
|t|=1
[ϕ0 (t) − ϕ(t)]
(Σ n
bk
tk
) |dt| = 0
(A.5.20)
k=0
for any .n ≥ 0. Hence, using the Schwarz inequality we conclude that f
f .
|t|=1
[ϕ0 (t) − ϕ(t)]ϕ0 (t)|dt| =
|t|=1
[ϕ0 (t) − ϕ(t)]ϕ0 (t)|dt| = 0.
(A.5.21)
By (A.5.21) f
f .
|t|=1
|ϕ(t)|2 |dt| −
f |t|=1
|ϕ0 (t)|2 |dt| =
|t|=1
Hence the desired extremal property of .ϕ0 (t) holds.
|ϕ(t) − ϕ0 (t)|2 |dt|.
(A.5.22)
A
Addendum
321
One can find the particular case .α = 0 of the below theorem in .(5,6) . Theorem VII If .F (ρ, ϑ) is an arbitrary function in .|z| < 1, such that f 1f
2π
(1 − ρ 2 )α |F (ρ, ϑ)|2 ρdρdϑ < +∞,
.
0
(A.5.23)
0
then the function α+1 π
fF (z) =
.
f 1f 0
2π
(1 − ρ 2 )α
0
F (ρ, ϑ) ρdρdϑ (1 − zρe−iϑ )α+2
(A.5.24)
belongs to .H2 (α) and minimizes the integral f 1f
2π
.
0
(1 − ρ 2 )α |F (ρ, ϑ) − f (ρeiϑ )|2 ρdρdϑ
(A.5.25)
0
in the class of functions .f (z) ∈ H2 (α). Proof Denote an =
.
r(α + 2 + n) 1 r(α + 1)r(1 + n) π
f 1f 0
2π
(1 − ρ 2 )α F (ρ, ϑ)(ρeiϑ )n ρdρdϑ.
(A.5.26)
0
Then, by (A.1.8) fF (z) =
∞ Σ
.
an zn ,
|z| < 1.
(A.5.27)
n=0
Further, we have α+1 . π =
f 1f 0
2π
0
α+1 π
| |2 n Σ | | iϑ k | (1 − ρ ) |F (ρ, ϑ) − ak (ρe ) | ρdρdϑ 2 α|
k=0
f 1f 0
2π
(1 − ρ 2 )α |F (ρ, ϑ)|2 ρdρdϑ −
0
n Σ r(α + 2)r(1 + k) k=0
Hence, the series n Σ r(α + 2)r(1 + k)
.
k=0
r(α + 2 + k)
|ak |2
r(α + 2 + k)
|ak |2 ≥ 0.
322
A
Addendum
is convergent, and consequently .fF (z) ∈ H2 (α). Besides, by (A.5.26) and (A.5.27) f 1f
2π
(1 − ρ 2 )α [F (ρ, ϑ) − fF (ρeiϑ )](ρeiϑ )n ρdρdϑ = 0,
.
0
n = 0, 1, 2, . . .
0
Using the Schwarz inequality, we obtain that for any .f (z) ∈ H2 (α) f 1f
2π
.
0
(1 − ρ 2 )α [F (ρ, ϑ) − fF (ρeiϑ )][f (ρeiϑ ) − fF (ρeiϑ )]ρdρdϑ = 0.
0
Hence f 1f
2π .
0
0
=
(1 − ρ 2 )α |F (ρ, ϑ) − f (ρeiϑ )|2 ρdρdϑ
f 1f 0
+
2π
(1 − ρ 2 )α |F (ρ, ϑ) − fF (ρeiϑ )|2 ρdρdϑ
0
f 1f 0
2π
(1 − ρ 2 )α |fF (ρeiϑ ) − f (ρeiϑ )|2 ρdρdϑ.
0
Thus, the integral (A.5.25) takes its minimal value only when .f (z) ≡ fF (z). 6. Now we use the above results for proving the completeness of some systems of rational functions in the unit disc. Let .{αn }∞ 0 be a sequence of pairwise different complex numbers, such that .0 ≤ |α0 | ≤ |α1 | ≤ · · · ≤ |αn | ≤ · · · < 1. Consider the system of rational functions ϕn (z) =
.
1 , (1 − α n z)α+2
α > −1,
n = 0, 1, 2, . . .
(A.6.1)
It is obvious that .ϕn (z) ∈ H2 (α). Hence, by Theorem I α+1 .ϕn (z) = π
f 1f 0
2π
(1 − ρ 2 )α
0
ϕn (ρeiϑ ) ρdρdϑ (1 − zρe−iϑ )α+2
(A.6.2)
for .n = 0, 1, 2, . . . In particular, for .z = αm ϕn (αm ) =
.
α+1 π
f 1f 0
2π 0
(1 − ρ 2 )α ρdρdϑ . (1 − αm ρe−iϑ )α+2 (1 − α n ρeiϑ )α+2
(A.6.3)
A
Addendum
323
In other words, α+1 .[ϕn , ϕm ] ≡ π
f 1f 0
2π
= ϕn (αm ) =
.
(1 − ρ 2 )α ϕn (ρeiϑ )ϕm (ρeiϑ )ρdρdϑ
0
1 , (1 − α n αm )α+2
n, m = 0, 1, 2, . . .
(A.6.4)
For orthogonalizing the system (A.6.1) over the surface of the unit disc with the measure α+1 2 α π (1 − ρ ) , we introduce the following well-known notation for .n = 0, 1, 2, . . .: | | |[ϕ 0 , ϕ0 ] [ϕ 0 , ϕ1 ] · · · [ϕ 0 , ϕn ] | | | | [ϕ , ϕ ] [ϕ , ϕ ] · · · [ϕ , ϕ ] | | 1 0 1 1 1 n | .An = | | ,. | ... ... ... ... | | | | [ϕ , ϕ0 ] [ϕ , ϕ1 ] · · · [ϕ , ϕn ] | n 1 n | | | | | [ϕ 0 , ϕn ] | | | | An−1 | 1 ··· | | , A−1 = 1. ϕn (z) = √ | An−1 An | [ϕ n−1 , ϕn ]|| | | |ϕ0 (z) ϕ1 (z) · · · ϕn (z) |
(A.6.5)
(A.6.6)
It is obvious that
.
α+1 π
f 1f 0
2π 0
⎧ ⎨0, (1 − ρ 2 )α ϕn (ρeiϑ )ϕm (ρeiϑ )ρdρdϑ = ⎩1,
if m /= n, if m = n.
(A.6.7)
Further, by (A.6.4) | | 1 | | (1−α0 a 0 )α+2 | | (1−α 1a )α+2 1 0 | 1 | ... .ϕn (z) = √ | An−1 An | | | (1−αn−11 a 0 )α+2 | 1 | | (1−za 0 )α+2
1 (1−α0 a 1 )α+2 1 (1−α1 a 1 )α+2
···
...
...
···
1 (1−αn−1 a 1 )α+2 1 (1−za 1 )α+2
··· ···
| | | | | | | | ... |. | | 1 | α+2 (1−αn−1 a n ) | 1 | (1−za n )α+2 | 1 (1−α0 a n )α+2 1 (1−α1 a n )α+2
(A.6.8)
By (A.6.8), for any .n = 0, 1, 2, . . . / ϕn (α0 ) = ϕn (α1 ) = · · · = ϕn (αn−1 ) = 0,
.
ϕn (αn ) =
An . An−1
(A.6.9)
324
A
Addendum
Now suppose .f (z) ∈ H2 (α) and set ak =
.
α+1 π
f 1f 0
2π
(1 − ρ 2 )α f (ρeiϑ )ϕk (ρeiϑ )ρdρdϑ.
(A.6.10)
0
By Bessel’s inequality ∞ Σ .
k=0
α+1 |ak | ≤ π 2
f 1f 0
2π
(1 − ρ 2 )α |f (ρeiϑ )|2 ρdρdϑ.
(A.6.11)
0
By the Riesz–Fisher theorem, there exists a function .f1 (z) ∈ H2 (α), such that ∞ Σ .
k=0
α+1 |ak | = π 2
f 1f 0
2π
(1 − ρ 2 )α |f1 (ρeiϑ )|2 ρdρdϑ
(A.6.12)
0
and α+1 .ak = π
f 1f 0
2π
(1 − ρ 2 )α f1 (ρeiϑ )ϕk (ρeiϑ )ρdρdϑ.
(A.6.13)
0
But it is obvious that .F (z) = f (z) − f1 (z) ∈ H2 (α), and by (A.6.10) and (A.6.13) .
α+1 π
f 1f 0
2π
(1 − ρ 2 )α F (ρeiϑ )ϕk (ρeiϑ )ρdρdϑ = 0,
k = 0, 1, 2, . . . ,
(A.6.14)
0
or, what is the same, for any .k = 0, 1, 2, . . . F (αk ) =
.
α+1 π
f 1f 0
2π
(1 − ρ 2 )α
0
F (ρeiϑ ) ρdρdϑ = 0. (1 − αk ρe−iϑ )α+2
(A.6.15)
The next theorem follows from Theorems III2 , III3 , IV and (A.6.15). Theorem VIII1 Let the sequence .{αn }∞ 0 satisfy one of the below conditions: (a) f .
1
(1 − r)α e2N (r) dr = +∞.
0
(b) lim inf(1 − |αn |)n >
.
n→∞
α+1 . 2
(A.6.16)
A
Addendum
325
(c) .
lim sup n→∞
1 − |αn | 1 log 1−|α n|
n>e
α+1 . 2
(d) π +δ 2
| arg(αn − 1)| ≥
.
(δ > 0)
and
∞ Σ (1 − |αn |) = +∞. n=0
Then, for any function .f (z) ∈ H2 (α) f 1f .
2π
lim
n→∞ 0
0
| |2 ∞ Σ | | (1 − ρ 2 )α ||f (ρeiϑ ) − ak ϕk (ρeiϑ )|| ρdρdϑ = 0
(A.6.17)
k=0
and uniformly inside .|z| < 1 | |2 ∞ Σ | | lim (1 − |z|)α+2 ||f (z) − ak ϕk (z)|| = 0,
.
n→∞
(A.6.18)
k=0
where ak =
.
α+1 π
f 1f 0
2π
(1 − ρ 2 )α f (ρeiϑ )ϕk (ρeiϑ )ρdρdϑ.
0
Besides, uniformly inside .|z| < 1 f (z) =
∞ Σ
.
ak ϕk (z).
(A.6.19)
k=0
Observe that in virtue of (A.6.9) the coefficients .{ak } can be found by means of the following recurrent formulas: f (α0 ) = a0 ϕ0 (α0 ),
.
f (α1 ) = a0 ϕ0 (α1 ) + a1 ϕ1 (α1 ), ..................................................... f (αn ) = a0 ϕ0 (αn ) + a1 ϕ1 (αn ) + · · · + an ϕn (αn ), .....................................................................................
(A.6.20)
326
A
Addendum
Theorem VIII2 For any fixed w .(|w| < 1) ∞
.
Σ 1 = ϕk (w)ϕk (a), α+2 (1 − wz)
(A.6.21)
k=0
where the series uniformly converges inside .|z| < 1, and in .|w| < 1 f 1f .
| |2 n Σ | | 1 iϑ) | (1 − ρ ) | ϕk (w)ϕk (ρe )| ρdρdϑ = 0. − iϑ α+2 (1 − wρe )
2π
lim
n→∞ 0 0
2 α|
k=0
(A.6.22) Proof Observe that for any .k = 0, 1, 2, . . . ϕk (z) =
k Σ
.
p=0
Ap . (1 − α p z)α+2
Hence, by (A.6.4) α+1 .ak = π =
k Σ p=0
=
k Σ p=0
f 1f 0
2π
(1 − ρ 2 )α
0
α+1 Ap π
f 1f 0
2π 0
ϕk (ρeiϑ ) ρdρdϑ (1 − wρeiϑ )α+2
(1 − ρ 2 )α ρdρdϑ (1 − wρeiϑ )α+2 (1 − αp ρe−iϑ )α+2
Ap = ϕk (w). (1 − wαp )α+2
(A.6.23)
The desired relations (A.6.21) and (A.6.22) follow from Theorem VIII1 and (A.6.23). The above Theorems VIII1 and VIII2 extend to .H2 (α) .(α > −1) some results of Takenaka.(9) which are proved for the Riesz’ class .H2 . 7. Let a sequence .{αn }∞ conditions (A.6.16). If a sequence of complex 0 satisfy one of the Σ∞ 2 is such that the series . numbers .{an }∞ 0 |an | converges, then by the Riesz - Fisher 0 theorem and Theorem VIII1 there exists a unique function .f (z) of .H2 (α) .(α > −1), such that (1) f 1f .
2π
lim
n→∞ 0
0
| |2 n Σ | | iϑ iϑ | (1 − ρ ) |f (ρe ) − ak ϕk (ρe )| ρdρdϑ = 0. 2 α|
k=0
(A.7.1)
A
Addendum
327
(2) The values of .f (z) at the points .{αn }∞ 0 are defined by the equalities f (αn ) = a0 ϕ0 (αn ) + a1 ϕ1 (αn ) + . . . + an ϕn (αn ),
.
n = 0, 1, 2, . . .
(A.7.2)
Thus, the following statement is true. Theorem IX Let a sequence .{αn }∞ 0 satisfy one of the conditions (A.6.16). Then the existence of a function .f (z) ∈ H2 (α), which takes the given values .f (αn ) at the points Σ∞ 2 .αn , is equivalent to the convergence of the series . 0 |ak | , where .ak are defined by the recurrent formulas (A.7.2). Σ∞ This function is given by the series .f (z) = k=0 ak ϕk (z) which is uniformly convergent inside .|z| < 1, and this is the unique function of the class .H2 (α), which takes the values .f (αn ) at the points .αn . The corresponding theorem in .H2 is proved by Takenaka.(9) . Using Theorem V, we prove the below statement by similar methods. Theorem X Let a sequence .{αn }∞ 0 satisfy one of the below conditions: (a) f
1
.
(1 − r)2β−1 e2N (r) dr = +∞,
0
(b) lim inf(1 − |αn |)n > β − 1,
.
n→∞
(c) lim sup
.
n→∞
1 − |αn | 1 log 1−|α n|
n > e(β − 1),
(d) | arg(αn − 1)| ≥
.
π +δ 2
(δ > 0),
∞ Σ (1 − |αn |) = +∞. n=0
328
A
Addendum
Then for any function .f (z) of .H2 in .|z| < 1 f inf
.
|z|=1
|f (z) − R(z)|2 |dz| = 0,
where .R(z) are all possible linear combinations of the functions .(1 − α n z)−β .(β > 1, .n = 0, 1, 2, . . .).
B.1
On a Generalization of the Jensen–Nevanlinna Formula. Canonical Representations of Meromorphic Functions of Unbounded Type4
1. Let .w = F (z) be a function meromorphic in .|z| < 1, and let .F (z) = Cλ zλ + . . . .(Cλ /= 0) be its Laurent expansion in the neighborhood of the origin. Further, let .{aμ } and .{bν }, respectively, be the sequences of zeros and poles of .F (z), which are in .0 < |z| < 1 and are numerated according their multiplicities and in the order of increasing modulus. It is well-known that for any .0 < r < 1 and any .|z| < r the following identity named Jensen–Nevanlinna.(2) formula is true: .
log |F (z)| =
1 2π −
f
2π
log |F (reiϑ )|Re
0
reiϑ + z dϑ + reiϑ − z
| | | r(z − bν ) | | | + λ log |z| . log | 2 r r − bν z | | b/α) or lim inf (ω ≤ b/α), > > n→∞ |an |α 2π B π cos ωα |an |α
where .B = b cos b is the maximal value of .x cos x in .(0, π/2). Any entire function, the class of which is less than .[α, σ ], belongs to .M2 (2σ, α). Therefore, the following statement holds by Theorem XV2 . Theorem XV3 If a sequence of points .{an } ∈ C satisfies one of the conditions of Theorems A or B, then for any entire function .f (z), the class of which is lower than .[α, σ ], 1. f (z) =
∞ Σ
.
Ak ψk (z),
z ∈ C,
(C.3.5)
k=0
where the series uniformly converges in any bounded part of .C. 2. Uniformly in the whole .C | | n Σ | α| lim e−σ |z| ||f (z) − Ak ψk (z)|| = 0, n→∞
.
(C.3.6)
k=0
where .ψk (z) and .Ak are those defined in subsection 2. 4. Now, we give an example of orthogonal system of entire functions. For .α = 2 and .an = n .(n = 0, 1, 2, . . .), from (C.1.22) and (C.2.1) we derive ϕn (z) = E2,σ (nz) = 2σ eσ nz ,
.
n = 0, 1, 2, . . .
(C.4.1)
350
A
Addendum
Hence, by (C.2.3) | |1 1 | | σ n+1 | 1 e .An = (2σ ) | |· · · · · · | | 1 enσ
... ... ··· ...
| 1 || 0,n || ( pσ ) enσ || e − eqσ | = (2σ )n+1 ··· | p>q | e2nσ |
(C.4.2)
and
.
n−1 ) || ( An enσ − ekσ . = 2σ An−1
(C.4.3)
k=0
Further, by (C.2.4) / ψn (z) = 2σ
.
n−1 ) An−1 || ( σ z e − eσ k . An
(C.4.4)
k=0
Thus, we have ψ0 (z) =
√
.
2σ ,
) ||n−1 ( σ z √ − eσ k k=0 e ψn (z) = 2σ /|| ), n−1 ( σ n σk e − e k=0
n = 1, 2, 3, . . . ,
(C.4.5)
and 1 . 2π
f
+∞f 2π
e 0
−σρ 2
ψn (ρeiϑ )ψm (ρeiϑ )ρdρdϑ =
0
⎧ ⎨1,
m = n,
⎩0,
m /= n.
(C.4.6)
It is to be noted that this system is not complete in .M2 (σ, 2). 1947, May Institute of Mathematics and Mechanics Academy of Sciences of Armenian SSR and Yerevan State University named after V.M.Molotov
A
Addendum
351
References 1. F. Riesz, Über die Randwerte einer analitichen Functionen. Mathematische Zetschrift, vol. 18, pp. 87–95. 2. R. Nevanlinna, Univalent analytic functions, Ch. VI, VII [in Russian], Moscow, (1941). 3. I.I. Privalov, Boundary properties of univalent analytic functions [in Russian], Moscow (1941). 4. L. Biberbach, Moderne Functionentheorie, pp. 265–269, Berlin (1927). 5. W. Wirtinger, Über eine Minimumaufgabe im Gebiet de analytischen Functionen. Monatschefte für Mathematik und Physik, vol. 39, pp. 377–384. 6. J.L. Walsh, Interpolation and approximation by rational functions (pub. by the American Mathematical Society), Sec. 10.6, 10.7, 10.12 (1935). 7. E.C. Titchmarsh, Theory of functions, p. 125, Oxford (1932). 8. G. Polya, G. Segö, Problems and theorems from analysis [in Russian], vol. I, Sec. II, Problem 148, Sec. IV, Problem 46. 9. S. Takenaka, On the orthogonal functions and a new formula of interpolation. Japanese Journal of Mathematics, vol. 2, pp. 129–145. 10. Reference by KING–ZAI HIONG, Journal de Liouville, 45, pp. 268–276 (1935). 11. A. Gelfond, Sur les systemes completes de functions analytiques, Mat. Sbornik, 4 (1936), 149–145. 12. A.I. Markushevich, On a basis in a space of analytic functions [in Russian], Mat. Sbornik, 17 (1945), 211–252. 13. R.P. Boas, Fundamental sets of entire functions, Annals of Mathematics, vol. 147, no. 1, pp. 21–32. 14. M.M. Djrbashian, Dokladi Armenian SSR, vol. III, no. 1, (1945). 15. M.M. Djrbashian, Dokladi Armenian SSR, vol. VI, no. 5, (1947).
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Index
Symbols ∼(D), 180 Nω with ω ∈ o
B Banach, 213 Banach space, xiv, 212, 213 Banach space ∼ Pω,d , 207 ∼ ω,d , 211, 212 Banach space D ∼ ω,h , 307 Banach space D ∼ ' , 306, 307 Banach space D ω,h Banach space ∼ P'ω,d , 207, 208 Banach space ∼ P'ω,h , 307 ' , 302 Banach space ∼ Pω,h Banach space ∼ Pω,d , 205, 212 Banach space ∼ Pω,h , 296, 307 p Banach space Aω (C), 65 p Banach space Aω,γ (G+ ), 95, 98 p Banach space Aω (D), 42, 61 p Banach space h (G+ ), 133 p Banach space Hγ , 97 p Banach space hω (G+ ), 133, 134 p Banach space Lω (G+ ), 134 p Banach space Lω (C), 65 p Banach space Lω (D), 42 p Banach space Lω,γ (G+ ), 98 Banach space Dω,h , 307 Banach space D'ω,h , 306, 307 Banach space Dω,d , 212 Banach space Dω,h , 212 Banach space P'ω,d , 211 Banach space P'ω,h , 212, 306, 307
Banach space Pω,d , 210, 212 Banach space Pω,h , 304, 307 Berberyan, 353 Bergman, xii, 353, 354 Bessel inequality, 49, 319, 324, 346 Biberbach, xi, 351, 353 Biorthogonal systems, 71, 72, 78, 128, 130, 353 Blaschke, xiii, xiv, 71, 80, 88, 89, 92, 93, 128, 129, 154, 159, 163, 169, 170, 174, 178, 183, 184, 187, 188, 193, 194, 197–199, 204, 208–211, 214–216, 220, 225, 241, 261, 264, 269, 278, 288, 290, 291, 293–295, 302–305, 307, 308, 317, 329, 336, 354, 355 Blaschke product, xiv, 214 Blaschke product, disc, 71, 92, 187 Blaschke product, halfplane, 129 Blasco, 353 Boas, 351
C Cauchy, xiii, 7–10, 12, 13, 37, 39, 44, 66, 78, 93, 99, 170, 191, 205, 206, 208, 209, 218, 220–222, 230, 245, 281, 291, 296, 297, 302, 303, 331, 353 Cauchy–Djrbashian kernel Cω for D, 10 Cauchy–Djrbashian kernel Cω for G+ , 12 Cauchy kernel, xiii, 13 Cauchy–Riemann polar equations, 7 Cauchy-type integral, 8 + Class Nm ω (G ), 273 + ), 158 Class NA (G ω Coifman, 353
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 A. M. Jerbashian, J. E. Restrepo, Functions of Omega-Bounded Type, Frontiers in Mathematics, https://doi.org/10.1007/978-3-031-49885-5
357
358 D Decomposition of Lω , 9 Decomposition orthogonal, disc, 87, 130 Decomposition orthogonal, halfplane, 133, 140 Delta-subharmonic function, xiv, 6 Descriptive representation, 92 p Dirichlet type space Aω (D), 56 Djrbashian, v, vii, xii–xiv, 3, 10–13, 27, 28, 39, 41, 71, 78, 79, 92, 143, 154, 161, 163, 172, 175, 177, 180, 183, 186, 187, 193–195, 197, 215, 309, 351, 353–356 Djrbashian, A.E., xiii, 354 Djrbashian factorization theory, xiv, 3, 163 Djrbashian omega-characteristics, 175, 176 Djrbashian operator Lω , disc, xiv, 3, 4, 6, 39 Djrbashian operator Lω , halfplane, 11 Djrbashian operator, united form, 3 Domain C, xiii Domain D, xi Domain G+ , xiii Domain starshaped, regular, 7 q Dual space A0 , xii q Dual space Aω (D), 55 q Dual space Aω,0 (G+ ), 127 Dunford, 354
E Erdélyi, vii, 5, 354
F Factor ∼ bω (z, ζ ), disc, 197 Factor bω (z, ζ ), disc, 84 Factor ∼ bω (z, ζ ) halfplane, 216 Factor bω (z, ζ ), halfplane, 154 Fatou lemma, 42, 98, 135, 172, 255, 258, 260, 285 Fichtenholz, 188, 354 Fisher, 324, 326 Fourier, 104, 114, 116, 117, 122, 137, 139, 214 Fourier series, 214 Fourier transform, 104, 114, 116, 117, 122, 137, 139 Frostman, 57, 187, 193, 245, 354
Index G Gakhov, 354 Garnett, 354 Gelfond, 351 Ghika, xii Goldberg, 354 Green function, 8 Green potential, 9, 85, 88, 154, 169, 170, 183, 208, 211, 225, 269, 293 Green-type potential, 84, 89–92, 140, 141, 146, 154, 163, 169, 172, 181, 182, 187, 194, 198, 204–212, 214, 215, 224, 225, 230, 232, 239, 241, 242, 269, 270, 273, 275–277, 283, 284, 289–291, 294–298, 300, 302–304, 306, 307, 356
H Hadamard, vii, 5, 354 Hahn-Banach theorem, 56, 127 Hardy space, 212 Hardy space h1 (G+ ), 247 Hardy space H 2 (G+ ), 115–118, 120, 122, 128, 130, 139 Hardy space h2 (G+ ), 307 Hardy space H 2 (D), 47, 69 Hardy space h2 (D), 212 Hardy space H p (G+ ), 133 Hardy space hp (G+ ), 133 Hardy space H p (D), 56, 71, 97 Hausdorff moment problem, 27, 36, 39 Hayman, 354 Hayrapetyan, 354 Hedenmalm, 354 Herglotz-Riesz theorem, 267, 284 Hilbert space, 133 Hilbert space A2ω (C), 65, 73 Hilbert space A2ω (D), 90 Hilbert space A2ω,0 (G+ ), 113 2 (G+ ), 120, 121 Hilbert space A∼ ω,0
Hilbert space h2ω (G+ ), 134 Hilbert space h2ω∗ (G+ ), 307 Hilbert space h2ω∗ (D), 212 Hilbert space of entire functions, 70 Horowitz, 354 Hunanyan, 356
Index I Inner product, 65, 73, 134 Integral mean Mp (r, f ), xiii, 42 Interpolation, 128, 129, 351
J Jensen, xiv, 9, 79, 173, 175, 182, 310, 313, 328–331 Jensen-Nevanlinna formula, xiv, 173 Jerbashian, vii, xiv, 354, 355 Jerbashian, V.A., 355 Jordan decomposition, 79, 144, 204, 237, 273, 295
K Karapetyan, 13, 355 Karapetyants, 355 Keldysch, xiii, 62, 319, 355 Kennedy, 354 Kilbas, xiv, 356 Kober, vii, 5, 355 Koosis, 355 Korenblum, 354 Kravchenko, 355
L Landkof, 355 Laplace transform, 11, 29, 34, 37, 156, 235–237, 243, 249 Lebesgue area measure, xi, xiii, 96, 146, 169, 170, 172, 176, 180, 182, 185 Lebesgue L2ω (D), 88 Lebesgue measure, 181, 270, 272–274 Lebesgue space, 65 Lebesgue space L20 (D), xi p Lebesgue space Lω,γ (G+ ), 96 p Lebesgue space Lω (D), 42 Lebesgue theorem, 34, 68, 87, 264, 272 Leontiev, 355 Levin, 144, 238, 242, 276, 355 Levin formula, 144 Liflyand, 355 Liouville, vii, xiii, xiv, 5, 11, 106, 353
359 M MacLane, 214, 355 Malonek, 355 Marichev, xiv, 356 Markushevich, 351 Mittag-Leffler, 70, 78, 310, 343
N Nevanlinna, xiii, xiv, 79, 92, 93, 95, 112, 117, 132, 143, 144, 154, 163, 172, 173, 175, 176, 178–180, 182, 186, 194, 195, 215, 238, 309, 310, 328–331, 340, 351, 355 Nevanlinna characteristics, 173 Nevanlinna characteristic T (r, f ), xiii Nevanlinna characteristic T (r, u), 79 Nevanlinna class N , 92, 173 Nevanlinna-Djrbashian characteristic Tω , xiv Nevanlinna–Djrbashian classes, G+ , 143 Nevanlinna-Djrbashian class N {ω}, xiv Nevanlinna-Djrbashian class Nω , xiv, 177 Nevanlinna factorization, xiv, 92, 173 Nevanlinna–M.M. Djrbashian class, disc, 79 Nevanlinna-M.M. Djrbashian class Nω◦ , 80 Nevanlinna weighted class, 92
O Orthogonal projection, xi, xiii, 46, 68, 113, 114, 135, 136 Ostrovskii, 27, 28, 39, 195, 354, 355
P Paley–Wiener theorem, 115, 117, 138, 139 Paley–Wiener type theorem, 117 Parameter class o, 3 Parameter class o(D), 3, 4 Parameter class o∗ (D), 27 + Parameter class oN α (G ), 12 Parameter class o0 (D), 56 Parameter class oα (G+ ), 12 Parameter class o∗α (G+ ), 134 + Parameter class oB α (G ), 289 Parameter class oA (G+ ), 215, 306 Parameter class oA (D), 42 Parameter class oB (D), 197 Parameter class oN (D), 79
360 ∼(D), 3, 4 Parameter class o ∼0 (D), 49 Parameter class o Perez-Esteva, 353 Petrosyan, 62, 355 Poisson, xiv, 9, 144, 171, 174, 181, 183, 199 Poisson formula, 183 Poisson-Jensen formula, xiv, 9 Poisson kernel, 199 Poisson representation, 144, 171, 174, 181 Polya-Segö, 351 Privalov, 92, 309, 351, 356
R Rafayelyan, 355 Restrepo, vii, xiv, 355 Ricci, 356 Riemann, vii, xiii, xiv, 5, 7, 106, 353 Riemann-Liouville fractional integrodifferentiation, xiv, 6 Riemann-Liouville fractional primitive, xiii Riesz, xiv, 7, 8, 79, 80, 86, 88, 89, 91, 92, 140, 144, 163, 169, 172, 173, 194, 215, 225, 237, 238, 241, 269, 273, 309, 312, 318, 320, 324, 326, 346, 351, 355 Riesz–Fisher theorem, 324, 326 Riesz type representation theory, xiv Rochberg, 353 Rubel, 214, 355 Rudin, xii, 356
S Samko, xiv, 356 Schwarz, 10, 178, 320, 322, 345, 347 Schwarz–Djrbashian kernel Sω for D, 10 Schwarz, J.I., 354 Schwarz kernel, disc, 92 Shamoyan, xiii, 93, 354, 356 Shapiro, 71, 356
Index Shields, 71, 356 Solomentsev, 229, 245, 275, 356 Space A20 , xi, xii Space A2α , xiii p Space Aα , xiii Space H p (α), xiii Stieltjes inversion formula, xiv, 174, 178, 184, 234, 238, 243, 251, 253, 274, 277
T Taibleson, 356 Takenaka, 351 Taylor series, 11 Titchmarsh, 351 Tsuji, 93, 144, 215, 237, 238, 242, 273, 276, 356 Tsuji characteristic L(y, U ), 144
V Volterra equation, 27, 28, 195, 254 Volterra square, 47, 120, 121, 138, 212, 307
W Walsh, xi, 351, 356 Widder, 356 Williams, 356 Wirtinger, xi–xiii, 312, 351, 356
Y Yudovich, xii, 356
Z Zakarian, 187, 193, 354–356 Zakaryuta, xii, 356 Zhu, 354 Zygmund problem, 214