On a Hardy and Littlewood imbedding theorem
type: Item
instance of: scholarly article
title: On a Hardy and Littlewood imbedding theorem
zbMATH Open document ID: 0593.42008
author: Alberto Torchinsky
author: Björn D. Jawerth
published in: Michigan Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The Hardy-Littlewood imbedding theorem, cited by the authors, is the following: \[ \{\int^{1}_{0}[(1-\rho)^{1/p- 1/r}(\int^{2\pi}_{0}| f(\rho e^{i\theta})| \quad^ rd\theta)^{1/r}]^ q(1-\rho)^{-1}d\rho \}^{1/q}\leq C\| f\|_{H^ p}, \] where \(0<p\leq r\leq \infty\), \(p\leq q\leq \infty\). The authors give general versions of this theorem in the following setting. Let (X,\(\mu)\), (T,\(\nu)\) be positive measure spaces. Assume that to each (x,t)\(\in X\times T\) one associates a \(\mu\)-measurable set B(x,t)\(\subset X\) so that \({\mathcal B}=\{B(x,t)\}\) satisfies (i) \(x\in B(x,t)\), \(t\in T\); (ii) if \(y\in B(x,t)\), then \(x\in B(y,t)\); and (iii) \(0<\mu (P(x,t))<\infty\). Now for functions f on \(X\times T\) one sets \(M_{{\mathcal B}}f(x)=\sup_{t\in T} \sup_{y\in B(x,t)}| f(y,t)|.\) Using this maximal function, they give general versions of the above theorem, one of which contains the above theorem as an easy corollary. Their proofs are simple. Several applications are given to the real Hardy spaces of Fefferman-Stein, Calderón-Torchinsky, the complex Hardy spaces on smooth domains in \({\mathbb{C}}^ n\), and the Hardy spaces in the poly half-space.
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 3953497
zbMATH Keywords: interpolation
zbMATH Keywords: Hardy-Littlewood imbedding theorem
zbMATH Keywords: maximal function
zbMATH Keywords: Hardy spaces
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003018
OpenAlex ID: W2086251584
DOI: 10.1307/MMJ/1029003018
description: scientific article; zbMATH DE number 3953497