On New Decomposition Theorems in some Analytic Function Spaces in Bounded Pseudoconvex Domains

Hadamard type operators on spaces of real analytic functions in several variables

Modulus of continuity with respect to semigroups of analytic functions and applications

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$.

Let f be an analytic function in the unit disc 𝔻. The Volterra integral operator If is defined as follows: If(h)(z)=∫0zf(w)h'(w)dw,h∈H(𝔻),z∈𝔻. In this paper, we compute the norm of If on some analytic function spaces.

Chapter 39 Spaces of analytic functions with integral norm

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.

This thesis deals with interpolation problems in spaces of analytic functions of Dirichlet type, that is spaces in which the corresponding norm is defined in terms of some integral involving (partial) derivatives of the function. These problems can be categorized roughly into two categories. The first one is problems of interpolation by sequences where one tries to find a function in a given space such that on prescribed points assumes prescribed values. The second is boundary interpolation where one prescribes the values of the function on a set that sits in the boundary of the natural domain of definition. In the first part we explore simply interpolating sequences for the Dirichlet space. We manage to give a partial answer to a problem first raised by Bishop. We prove that for sequences which satisfy the so called Shapiro–Shields condition there exists a potential theoretic characterization of when they are simply interpolating. In the second part we study random interpolating sequences for the standard weighted Dirichlet spaces. We are able to prove 0–1 Kolmogorov type characterizations for universally interpolating, zero and separated sequences for this class of spaces. Our results generalize and improve upon previous results of Rudowicz Bogdan and Cochran. Finally, in the last part we work with Hardy–Sobolev spaces in the unit ball of n complex variables. Our results concern the equivalence of two notions of “negligibility”of sets on the boundary of the ball. One notion is classical and is that of capacity zero sets and the second notion is that of a totally null set. Our main result shows that at least for compact sets these two notions coincide. Our result can be applied to improve a result on boundary interpolation of Cohn & Verbitsky.

In this paper, for conformal mapping f , we study the membership of log fto the QK (p, q)-type spaces of analytic functions. Moreover, geometric conditions and some important characterizations involving the Schwarzian derivative are also given.

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

The topic of this book is located at the intersection of complex analysis, operator theory and partial differential equations. It begins with results on the canonical solution operator to restricted to Bergman spaces of holomorphic d-bar functions in one and several complex variables.These operators are Hankel operators of special type. In the following the general complex is investigated ond-bar spaces over bounded pseudoconvex domains and on weightedd-bar spaces. The main part is devoted to the spectral analysis of the complex Laplacian and to compactness of the Neumann operator. The last part contains a detailed account of the application of the methods to Schrödinger operators, Pauli and Dirac operators and to Witten-Laplacians. It is assumed that the reader has a basic knowledge of complex analysis, functional analysis and topology. With minimal prerequisites required, this book provides a systematic introduction to an active area of research for both students at a bachelor level and mathematicians.

Resolvent estimates and decomposable extensions of generalized Cesàro operators

In this note we provide some analogues of our numerous recent results on traces and distances in terms of Martinelly — Bochner integrals and kernels. These are first results of this type in terms of such kernels. Some assertions for Martinelly — Bochner integrals related with Holder classes and Lebegues points will be also discussed. In recent years various new sharp results on traces and distances were provided in a big series of papers of the first author. In all these papers properties of Bergman-type kernels and Bergman-type integral representations are playing a critical role. The intension of this paper to find some analogues of these results in terms of or with the help of more general integral representations and more general kernels in analytic function spaces in higher dimension so-called Martinelly — Bochner integral representations and Martinelly — Bochner kernels in Cn. Our work consists of three parts. In the first part we partially generalize our results on traces. In the second part we provide estimates of distance function in terms of Martinelly — Bochner kernels and Martinelly — Bochner integrals. In the third part we present results on Martinelly — Bochner integrals related with Holder classes and Lebegues points. This type of issues arise naturally in view of recent series of papers and new results of the first author on multifunctional analytic spaces and related issues. In our proofs we modify the methods of earlier results and theorems for the case of Martinelli-Bochner integrals and kernels.

Automorphisms and generalized projections on spaces of analytic functions

Let a={am:m∈N} be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip {s=σ+it:1/2<σ<1} by shifts ζ(s+iτ;a) of the zeta-function defined, for σ>1, by ζ(s;a)=∑m=1∞amm−s and by analytic continuation elsewhere. Using stochastic techniques, we obtain that the set of the above shifts approximating a given analytic function has a positive lower density (or density with at most countably many exceptions) in the interval [T,T+V] with T23/70⩽V⩽T1/2 as T→∞. The proofs are based on a limit theorem with an explicitly given limit probability measure in the space of analytic functions.