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Maximal Operator in Variable Exponent Lebesgue Spaces on Unbounded Quasimetric Measure Spaces

Tomasz Adamowicz, Petteri Harjulehto, Peter Hästö


We study the Hardy-Littlewood maximal operator $M$ on $L^{p({\cdot})}(X)$ 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 $L^{p({\cdot})}({\mathsf{R}^n},\mu)$ for arbitrary, possibly non-doubling, Radon measures.

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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