The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type
Volume 242 / 2018
Abstract
Given a space of homogeneous type $(X,d,\mu )$, we present sufficient conditions on a variable exponent ${p(\cdot )}$ so that the fractional maximal operator $M_\eta $, $0\leq \eta \lt 1$, maps $L^{p(\cdot )}(X)$ to $L^{q(\cdot )}(X)$, where $1/{p(\cdot )}-1/{q(\cdot )}=\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 of Capone et al. (2007) and Cruz-Uribe et al. (2009) from the Euclidean case, and extends recent work by Adamowicz et al. (2015) on the Hardy–Littlewood maximal operator on spaces of homogeneous type.
