Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Abstract

Let ${M_\beta }$ be the fractional maximal function. The commutator generated by ${M_\beta }$ and a suitable function b is defined by $[{M_\beta },b]f = {M_\beta }(bf) - b{M_\beta }(f)$. Denote by P(ℝⁿ) the set of all measurable functions p(·): ℝⁿ → [1,∞) such that

$1 < p_ - : = \mathop {es\sin fp(x)}\limits_{x \in \mathbb{R}^n } andp_ + : = \mathop {es\operatorname{s} \sup p(x) < \infty }\limits_{x \in \mathbb{R}^n } ,$

and by B(ℝⁿ) the set of all p(·) ∈ P(ℝⁿ) such that the Hardy-Littlewood maximal function M is bounded on L ^p(·)(ℝⁿ). In this paper, the authors give some characterizations of b for which $[{M_\beta },b]$ is bounded from L ^p(·)(ℝⁿ) into L ^q(·)(ℝⁿ), when p(·) ∈ P(ℝⁿ), 0 < β < n/p ₊ and 1/q(·) = 1/p(·) − β/n with q(·)(n − β)/n ∈ B(ℝⁿ).