The Boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type

TL;DR

This paper establishes boundedness conditions for fractional maximal and integral operators on variable Lebesgue spaces over spaces of homogeneous type, generalizing Euclidean results and simplifying proofs using dyadic cube theory.

Contribution

It provides new sufficient conditions for fractional maximal operators to be bounded on variable Lebesgue spaces over spaces of homogeneous type, extending Euclidean results and simplifying existing proofs.

Findings

Boundedness of fractional maximal operators under new conditions

Weak type inequalities at endpoint cases

Extension of inequalities to fractional integral operators on these spaces

Abstract

Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we also prove the corresponding weak type inequality. As an application we prove norm inequalities for the fractional integral operator {I_{\eta}}. Our proof for the fractional maximal operator uses the theory of dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting it is simpler than existing proofs. For the fractional integral operator we extend a pointwise inequality of Welland to spaces of homogeneous type. Our work generalizes results from the Euclidean case and extends recent work by Adamowicz, et al. on the Hardy-Littlewood maximal operator on spaces of homogeneous type.