Fractional and hypersingular operators in variable exponent spaces on metric measure spaces
type: Item
instance of: scholarly article
title: Fractional and hypersingular operators in variable exponent spaces on metric measure spaces
zbMATH Open document ID: 1182.43011
author: Alexandre Almeida
author: S. G. Samko
published in: Mediterranean Journal of Mathematics
publication date: 2009-08-03T00:00:00Z
review text: The authors study fractional integrals of variable order in variable exponent Lebesgue spaces on doubling and non-doubling spaces. Let \((X,d,\mu)\) be a (quasi-) metric measure space and \(\Omega\) be an open set in \(X\). The authors study two variants of fractional integral that are defined by \[ \begin{aligned} I^{\alpha(\cdot)}_m f(x) & = \int_\Omega {f(y)d\mu(y)\over d(x,y)^{m-\alpha(x)}},\quad m> 0,\\ J^{\alpha(\cdot)} f(x) & = \int_\Omega {d(x,y)^{\alpha(x)}\over\mu B(x,d(x, y))} f(y) \,d\mu(y),\end{aligned} \] where \(B(x,r)\) is the ball centered at \(x\) with radius \(r\). For the above defined operators, the authors establish certain Sobolev-type theorems on Lebesgue spaces with variable exponents. More precisely, they obtain the boundedness of operators from \(L^{p(\cdot)}(\Omega)\) to \(L^{q(\cdot)}(\Omega)\) with \(p(\cdot)\) and \(q(\cdot): X\to [0,\infty)\) satisfying a certain relationship.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 5589424
zbMATH Keywords: fractional integrals
zbMATH Keywords: hypersingular integral
zbMATH Keywords: maximal functions
zbMATH Keywords: variable exponents
zbMATH Keywords: metric measure space
zbMATH Keywords: Ahlfors regularity
reviewed by: Dashan Fan
MaRDI profile type: Publication
full work available at URL: s00009-009-0006-7
OpenAlex ID: W2123921154
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DOI: 10.1007/S00009-009-0006-7
description: scientific article