Reverse estimates in growth spaces
type: Item
instance of: scholarly article
title: Reverse estimates in growth spaces
zbMATH Open document ID: 1251.30064
author: Evgueni Doubtsov
author: Evgeny Abakumov
published in: Mathematische Zeitschrift
publication date: 2012-06-26T00:00:00Z
review text: Recall that for for a bounded circular domain \(D\) which is strictly convex and has \(\mathcal C^{2}\) smooth boundary, the growth space \(A^{w}(D)\) is defined as follows. It consists of all holomorphic in \( D\) functions \(f\) such that \(|f(z)|<Cw(r(z))\) for all \(z\in D\). Here \(w\) is the so-called weight function which is a general non decreasing continuous and unbounded on the interval \([0,1)\) function and \(r(z)\) is the Minkowski (or gauge) function of \(D\). The main theorem is that when \(w\) is a doubling function then for some \(M\in\mathbb N\), which depends only on \(D\), one can find \(M+1\) functions from the space \(A^{w}(D)\) such that the sum of their absolute values at every point \(z\in D\) dominates the quantity \(w(r(z))\). The exact value of the smallest such \(M\) is unknown in general (and probably an interesting quantity to study). One obviously has that \(M>0\), which is true even without the assumption of \(w\) being doubling. The authors manage to prove that, when \(D\) is the unit disc in the complex plane, \(M\) is \(1\) (this generalizes a result of Ramey and Ullrich from 1991), but even for such particular domains as the unit balls in \(\mathbb C^n\), \(M\) is unknown (and far from clear). It is also interesting what happens when the property of \(w\) being doubling is replaced by some weaker assumption. The authors provide some corollaries to their result. In particular they apply it in the study of composition and multiplication operators in various spaces of holomorphic functions.
reviewed by: Żywomir Dinew
Mathematics Subject Classification ID: 30H99
Mathematics Subject Classification ID: 30H30
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 6050604
zbMATH Keywords: growth space
zbMATH Keywords: circular domain
zbMATH Keywords: composition operator
zbMATH Keywords: Bloch space
MaRDI profile type: Publication
full work available at URL: s00209-011-0869-8
OpenAlex ID: W1985421290
cites work: Q3737768
cites work: Q5385115
cites work: Q3032807
cites work: Bloch-to-BMOA compositions in several complex variables
cites work: The Polynomial Approximation Property in Fock-type Spaces
cites work: Radial growth of functions in the Korenblum space
cites work: Fractional derivatives of Bloch functions, growth rate, and interpolation
cites work: Q4888816
cites work: Growth spaces on circular domains: Composition operators and Carleson measures
cites work: Radial growth of harmonic functions in the unit ball
cites work: On \(B_{\log}\) to \(Q^p_{\log}\) pullbacks
cites work: BiBloch-type Maps: Existence and Beyond
cites work: Carleson measures for the Bloch space
cites work: Extended Cesàro operators on mixed norm spaces
cites work: Homogeneous polynomials on strictly convex domains
cites work: On composition operators in \(Q_K\) type spaces
cites work: On the iterated logarithmic Bloch space on the unit ball
cites work: Hyperbolic mean growth of bounded holomorphic functions in the ball
cites work: The Essential Norm of a Bloch-to-Qp Composition Operator
cites work: Compact Composition Operators on the Bloch Space
cites work: Composition operators between Bloch-type spaces and Möbius invariant \(\mathcal Q_k\) spaces
cites work: Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights
cites work: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation
cites work: On Homogeneous Polynomials on a Complex Ball
cites work: Composition operators and classical function theory
cites work: A note on the growth of functions in \(H^ p\)
cites work: RIEMANN–STIELTJES OPERATORS ON WEIGHTED BLOCH AND BERGMAN SPACES OF THE UNIT BALL
DOI: 10.1007/S00209-011-0869-8
description: scientific article; zbMATH DE number 6050604