Commutators of the fractional maximal function on variable exponent Lebesgue spaces |
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Authors: | Pu Zhang Jianglong Wu |
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Institution: | 1. Department of Mathematics, Mudanjiang Normal University, Mudanjiang, 157011, P.R. China
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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(? n ) the set of all measurable functions p(·): ? n → 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(? n ) the set of all p(·) ∈ P(? n ) such that the Hardy-Littlewood maximal function M is bounded on L p(·)(? n ). In this paper, the authors give some characterizations of b for which \({M_\beta },b]\) is bounded from L p(·)(? n ) into L q(·)(? n ), when p(·) ∈ P(? n ), 0 < β < n/p + and 1/q(·) = 1/p(·) ? β/n with q(·)(n ? β)/n ∈ B(? n ). |
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