Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

Details Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces

2008-05-14

arxiv.org/abs/0805.2028v1

Mathematics

Functional Analysis

29 pages

Scientific paper

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces $L^{p(\cdot)}(\varrho)$ with variable exponent $p(x)$, mainly in the Euclidean setting and dwell on a new result of the boundedness of the Hardy-Littlewood maximal operator in the space $L^{p(\cdot)}(X,\varrho)$ over a metric measure space $X$ satisfying the doubling condition. In the case where $X$ is bounded, the weight function satisfies a certain version of a general Muckenhoupt-type condition For a bounded or unbounded $X$ we also consider a class of weights of the form $\varrho(x)=[1+d(x_0,x)]^{\bt_\infty}\prod_{k=1}^m w_k(d(x,x_k))$, $x_k\in X$, where the functions $w_k(r)$ have finite upper and lower indices $m(w_k)$ and $M(w_k)$. Some of the results are new even in the case of constant $p$.

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