Maximal Operator in Variable Exponent Lebesgue Spaces on Unbounded Quasimetric Measure Spaces - Adamowicz, Tomasz; Harjulehto, Petteri; Hästö, Peter (MATHEMATICA SCANDINAVICA, 2015)
We study the Hardy-Littlewood maximal operator $M$ on $L^{p({·})}(X)$ when $X$ is an unbounded (quasi)metric measure space, and $p$ may be unbounded. We consider both the doubling and general measure case, and use two versions of the $\log$-Hölder condition. As a special case we obtain the criterion for a boundedness of $M$ on $L^{p({·})}({\mathsf{R}^n},μ)$ for arbitrary, possibly non-doubling, Radon measures.