\(L^{s}(\mu)\)-averaging domains
type: Item
instance of: scholarly article
title: \(L^{s}(\mu)\)-averaging domains
zbMATH Open document ID: 1027.30053
DOI: 10.1016/S0022-247X(03)00216-6
author: Shusen Ding
author: Craig A. Nolder
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-08-25T00:00:00Z
review text: After introducing \(L^s(\mu)\)-averaging domains the authors obtain a similar characterization of these domains as in the special case \(\mu=m =n\)-dimensional Lebesgue measure which are given by \textit{S. G. Staples} [Ann. Acad. Sci. Fenn., Ser. A I 14, 103--127 (1989; Zbl 0706.26010)]. As an application of their results they prove norm inequalities for conjugate \(A\)-harmonic tensors which are generalizations of the following theorem being due to \textit{G. H. Hardy} and \textit{J. E. Littlewood} [J. Reine Angew. Math. 167, 405--423 (1932; Zbl 0003.20203)]. Theorem (G. H. Hardy and J. E. Littlewood). For each \(p>0\) there exists a constant \(C=C(p)\) such that \[ \iint_D\bigl|u(x,y)- u(0,0)\bigr|^p dx dy\leq C\iint_D\bigr|v(x,y)-v(0,0)\bigr |^p dx dy \] holds for all real functions \(u=u(x,y)\), \(v=v(x,y)\), where \(f=f(z)= f(x+iy)= u(x,y)+iv(x,y)\) is analytic in the unit disk \(D\). Furthermore, it should be noted that the consideration of \(L^s(\mu)\)-averaging domains allows the choice \(d\mu=J_fdm\), where \(J_f\) denotes the Jacobian determinant of a quasiconformal mapping \(f\) in \(\mathbb{R}^n\). This provides the results for quasiconformal mappings and quasiregular functions at the end of the paper.
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 26D20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1970509
zbMATH Keywords: Poincaré inequality
zbMATH Keywords: \(L^p\)-averaging domains
zbMATH Keywords: \(A\)-harmonic tensor
zbMATH Keywords: quasiregular function
reviewed by: Erich Hoy
MaRDI profile type: Publication
cites work: Convexity conditions and existence theorems in nonlinear elasticity
cites work: \(W^{1,p}\)-quasiconvexity and variational problems for multiple integrals
cites work: Pointwise multipliers for reverse Holder spaces
cites work: Weighted Hardy-Littlewood inequality for 𝐴-harmonic tensors
cites work: Uniform domains and the quasi-hyperbolic metric
cites work: Q4693180
cites work: Q4479842
cites work: \(p\)-harmonic tensors and quasiregular mappings
cites work: Integral estimates for null Lagrangians
cites work: Quasiregular mappings in even dimensions
cites work: Hardy-Littlewood inequality for quasiregular mappings in certain domains in R^n
cites work: Q3797452
cites work: Hardy-Littlewood theorems for \(A\)-harmonic tensors
cites work: Exponential integrability of the quasi-hyperbolic metric on Hölder domains
cites work: L^p-averaging domains and the Poincaré inequality
cites work: Fractional Integrals on Weighted H p and L p Spaces
full work available at URL: s0022-247x(03)00216-6
OpenAlex ID: W1514746766
description: scientific article; zbMATH DE number 1970509