# Commutators of the fractional maximal function on variable exponent Lebesgue spaces

• Volume: 64, Issue: 1, page 183-197
• ISSN: 0011-4642

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## Abstract

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Let ${M}_{\beta }$ be the fractional maximal function. The commutator generated by ${M}_{\beta }$ and a suitable function $b$ is defined by $\left[{M}_{\beta },b\right]f={M}_{\beta }\left(bf\right)-b{M}_{\beta }\left(f\right)$. Denote by $𝒫\left({ℝ}^{n}\right)$ the set of all measurable functions $p\left(·\right):{ℝ}^{n}\to \left[1,\infty \right)$ such that $1<{p}_{-}:=\underset{x\in {ℝ}^{n}}{\mathrm{ess}\mathrm{inf}}p\left(x\right)\phantom{\rule{1.0em}{0ex}}\text{and}\phantom{\rule{1.0em}{0ex}}{p}_{+}:=\underset{x\in {ℝ}^{n}}{\mathrm{ess}\mathrm{sup}}p\left(x\right)<\infty ,$ and by $ℬ\left({ℝ}^{n}\right)$ the set of all $p\left(·\right)\in 𝒫\left({ℝ}^{n}\right)$ such that the Hardy-Littlewood maximal function $M$ is bounded on ${L}^{p\left(·\right)}\left({ℝ}^{n}\right)$. In this paper, the authors give some characterizations of $b$ for which $\left[{M}_{\beta },b\right]$ is bounded from ${L}^{p\left(·\right)}\left({ℝ}^{n}\right)$ into ${L}^{q\left(·\right)}\left({ℝ}^{n}\right)$, when $p\left(·\right)\in 𝒫\left({ℝ}^{n}\right)$, $0<\beta and $1/q\left(·\right)=1/p\left(·\right)-\beta /n$ with $q\left(·\right)\left(n-\beta \right)/n\in ℬ\left({ℝ}^{n}\right)$.

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