Hardy–Littlewood maximal operator on reflexive variable Lebesgue spaces over spaces of homogeneous type
Volume 254 / 2020
Studia Mathematica 254 (2020), 149-178
MSC: Primary 43A85; Secondary 46E30.
DOI: 10.4064/sm180816-16-9
Published online: 1 April 2020
Abstract
We show that the Hardy–Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot )}$ over a space of homogeneous type $(X,d,\mu )$ if and only if it is bounded on its dual space $L^{p’(\cdot )}$, where $1/p(x)+1/p’(x)=1$ for $x\in X$. This result extends the corresponding result of Lars Diening from the Euclidean setting of $\mathbb {R}^n$ to the setting of spaces $(X,d,\mu )$ of homogeneous type.
