Hardy–Littlewood maximal operator on reflexive variable Lebesgue spaces over spaces of homogeneous type

Abstract

We show that the Hardy–Littlewood maximal operator is bounded on a reflexive variable Lebesgue space Lp(·) over a space of homogeneous type (X, d, µ) if and only if it is bounded on its dual space Lp0(·), where 1/p(x) + 1/p0(x) = 1 for x ∈ X. This result extends the corresponding result of Lars Diening from the Euclidean setting of Rn to the setting of spaces (X, d, µ) of homogeneous type.