# Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces - Mathematics > Functional Analysis

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Abstract: We present a brief survey of recent results on boundedness of some classicaloperators within the frameworks of weighted spaces $L^{p\cdot}\varrho$ withvariable exponent $px$, mainly in the Euclidean setting and dwell on a newresult of the boundedness of the Hardy-Littlewood maximal operator in the space$L^{p\cdot}X,\varrho$ over a metric measure space $X$ satisfying thedoubling condition. In the case where $X$ is bounded, the weight functionsatisfies a certain version of a general Muckenhoupt-type condition For abounded or unbounded $X$ we also consider a class of weights of the form$\varrhox=1+dx 0,x^{\bt \infty}\prod {k=1}^m w kdx,x k$, $x k\in X$,where the functions $w kr$ have finite upper and lower indices $mw k$ and$Mw k$.Some of the results are new even in the case of constant $p$.

Author: V.Kokilashvili, S.Samko

Source: https://arxiv.org/