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

2008/05/14 by Kokilashvili, V., Samko, S.
#42B25 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0805.2028

Abstract

We present a brief survey of recent results on boundedness of some classical operators within the frameworks of weighted spaces L^p(⋅)(\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(⋅)(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(x₀,x)]^\bt_∞∏_k=1^m w_k(d(x,x_k)), x_k∈ 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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