Embedding theorems for spaces of analytic functions via Khinchine's inequality
type: Item
instance of: scholarly article
title: Embedding theorems for spaces of analytic functions via Khinchine's inequality
zbMATH Open document ID: 0801.46019
author: Daniel H. Luecking
published in: Michigan Mathematical Journal
publication date: 1994-11-20T00:00:00Z
review text: Let \(D\) denote the open unit disk of the complex plane, and suppose we are given a space of functions analytic on \(D\) with norm \(\| \cdot \|\). Fix an integer \(n \geq 0\) and a positive number \(s\). The problem is then (for various spaces) to classify those positive measures \(\mu\) for which there exists a positive constant \(C_ \mu\) such that \[ \Bigl( \int_ D \bigl | f^{(n)} (z) \bigr |^ sd \mu (z) \Bigr)^{1/s} \leq C_ \mu \| f \| \] for all \(f\) in the space under discussion; such an embedding problem is inspired by \textit{L. Carleson}'s famous theorem for the \(H^ p\) spaces (then \(n=0\) and \(s=p)\) [Ann. of Math., II. Ser. 76, 547-559 (1962; Zbl 0112.297)]. The paper under review is part of a series of papers by the same author on similar problems [see, e.g., Proc. Edinb. Math. Soc., II. Ser. 29, 125-131 (1986; Zbl 0587.30048); Proc. Lond. Math. Soc., III. Ser. 63, 595-619 (1991; Zbl 0774.42011)]. The main purpose of the paper is to introduce a new method for handling such problems. The method consists in combining known embeddings of sequence spaces into spaces of analytic functions with Khinchine's inequality in order to obtain a necessary condition on \(\mu\). In each of the cases discussed in the paper, it is quite easy to prove that this condition is sufficient as well. The method is first illustrated in the case of unweighted Bergman spaces (here the solution is already known); then it is used to obtain embedding theorems for the mixed norm spaces, i.e., \[ \| f \| = \| f \|_{p,q} = \left( \int^ 1_ 0 \left( \int^{2 \pi}_ 0 \bigl | f(re^{i \theta}) \bigr |^ p \right)^{q/p} dr \right)^{1/q}, \] \((p,q\) positive numbers) and certain analytic tent spaces [inspired by the tent spaces introduced by \textit{R. Coifman}, \textit{Y. Meyer} and \textit{E. M. Stein}, J. Funct. Anal. 62, 304-335 (1985; Zbl 0569.42016)]. The two latter situations seem to require the full power of the method, and the author says: ``It is possible to use the same technique…
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 474974
zbMATH Keywords: Bergman spaces
zbMATH Keywords: embeddings of sequence spaces into spaces of analytic functions with Khinchine's inequality
zbMATH Keywords: unweighted Bergman spaces
zbMATH Keywords: tent spaces
reviewed by: Kristian Seip
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004756
OpenAlex ID: W1970629427
DOI: 10.1307/MMJ/1029004756
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description: scientific article; zbMATH DE number 474974