Hardy-Littlewood theorems for \(A\)-harmonic tensors
type: Item
instance of: scholarly article
title: Hardy-Littlewood theorems for \(A\)-harmonic tensors
zbMATH Open document ID: 0957.35046
author: Craig A. Nolder
published in: Illinois Journal of Mathematics
publication date: 2001-03-22T00:00:00Z
review text: The theory of conjugate harmonic functions plays a central role in such areas of mathematics as potential theory, harmonic analysis and the theory of \(H^p\)-spaces. Conjugate harmonic functions have many analytical properties in common, among which are global \(L^p\)-integrability and Hölder continuity. These discoveries essentially began with the work of Hardy and Littlewood in the 1930's. Here we mention three specific results. Theorem A. For each \(p>0\), there is a constant \(C\) such that \[ \int_\mathbb{D}\bigl|u-u(0) \bigr|^p dx dy\leq C\int_\mathbb{D} \bigl|v-v(0) \bigr|^p dx dy \] for all analytic functions \(u+iv\) in the unit disk \(\mathbb{D}\). Theorem B. For each \(0<k\leq 1\), there is a constant \(C\) such that \[ \|u\|_{\text{Lip}_k, \mathbb{D}} \leq C\|v\|_{\text{Lip}_k,\mathbb{D}} \] for all analytic functions \(u+iv\) in \(\mathbb{D}\). Here \(\|\cdot \|_{\text{Lip}_k, \mathbb{D}}\) is the usual Lipschitz norm over \(\mathbb{D}\). Theorem C. There is a constant \(C\) such that \[ \|u\|_{\text{BMO},\mathbb{D}}\leq C\|v\|_{\text{BMO},\mathbb{D}} \] for all analytic functions \(u+iv\) in \(\mathbb{D}\). Here \(\|\cdot \|_{\text{BMO}, \mathbb{D}}\) is the usual BMO norm over \(\mathbb{D}\). Conjugate \(A\)-harmonic tensors are interesting and important generalizations of conjugate harmonic functions and \(p\)-harmonic functions, \(p>1\). The main results of this paper generalize Theorems A, B and C to conjugate \(A\)-harmonic tensors defined in domains in \(\mathbb{R}^n\) which possess an appropriate geometry. Examples show that in many ways the results are the fest possible. In particular we prove the local integrability result for conjugate \(A\)-harmonic tensors. This is the basic estimate, a result of the conjugacy, which is used throughout the paper to obtain global results. Then we investigate the global \(L^2\)-integrability of conjugate \(A\)-harmonic tensors. Here the global geometry of the domains of integration is important. The integrabi…
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1361048
zbMATH Keywords: conjugate \(A\)-harmonic tensors
zbMATH Keywords: quasiregular mappings
zbMATH Keywords: global \(L^s\)-integrability
zbMATH Keywords: local integrability
zbMATH Keywords: John domains
zbMATH Keywords: local Lipschitz spaces
zbMATH Keywords: BMO spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1361048