Weighted norm inequalities for conjugate \(A\)-harmonic tensors
type: Item
instance of: scholarly article
title: Weighted norm inequalities for conjugate \(A\)-harmonic tensors
zbMATH Open document ID: 0873.30012
author: Shusen Ding
author: Yi Ling
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-12-09T00:00:00Z
review text: \noindent Conjugate harmonic tensors generalize conjugate harmonic functions and \(p\)-harmonic functions which arise in non-linear potential theory. In this paper the authors extend two recent results of Nolder (to appear in Illinois J. Math) to analogous weighted versions. Let \(u\) and \(v\) denote conjugate \(A\)-harmonic tensors in a domain \(\Omega \subset R^{n}\) and \(s\) and \(t\) be any constants, \(0<s,t <\infty\). The first inequality is a local result which holds for all cubes \(\sigma Q \subset \Omega , \sigma >1\) and relates the weighted \(L^{s}\)-norm of \(u-u_{Q}\) over \(Q\) ( or \(|v- v_{Q}|_{s, Q}\) ) to the \(L^{t}\)-norm of \(v -c_{1}\) over \(\sigma Q\) (or respectively \(|u -c _{2} |_{t,\sigma Q}\)). Here \(c_{1}\) is any form in the Sobolev space \(W_{q,loc}^{1} (\Omega , \Lambda) \) with the Hodge star operator satisfying \(d\star c_{1} = 0\). The second inequality holds globally over John domains \(\Omega\). Here \(s\) and \(t\) are related by a specific algebraic formula in terms of the conjugate exponents \(p\) and \(q\) tied to \(u\) and \(v\). In that setting the authors prove a weighted form of \(|u- u_{Q}|_{s, \Omega} \leq C |v -c_{1} |_{t, \Omega} ^{q/p}\), with \(c_{1}\) as above . The weights the authors use in each of their theorems bear some resemblance to Muckenhoupt's \(A^{p}\)-weights.
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 944388
zbMATH Keywords: \(A\)-harmonic tensors
zbMATH Keywords: John domains
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0380
OpenAlex ID: W1528035462
DOI: 10.1006/JMAA.1996.0380
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description: scientific article; zbMATH DE number 944388