Weighted Boundedness of the Maximal, Singular and Potential Operators in Variable Exponent Spaces
Abstract
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 $p$.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2008
- DOI:
- 10.48550/arXiv.0805.2028
- arXiv:
- arXiv:0805.2028
- Bibcode:
- 2008arXiv0805.2028K
- Keywords:
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- Mathematics - Functional Analysis;
- 42B25;
- 47B38
- E-Print:
- 29 pages