Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let M_β be the fractional maximal function. The commutator generated by M_β and a suitable function b is defined by [M_β ,b]f = M_β (bf) - bM_β (f) . Denote by P(ℝ n ) the set of all measurable functions p(·): ℝ n → [1,∞) such that 1 < p_ - : = essin fp(x)x ∈ Rⁿ andp_ + : = ess p(x) < ∞ x ∈ Rⁿ , and by B(ℝ n ) the set of all p(·) ∈ P(ℝ n ) such that the Hardy-Littlewood maximal function M is bounded on L p(·)(ℝ n ). In this paper, the authors give some characterizations of b for which [M_β ,b] is bounded from L p(·)(ℝ n ) into L q(·)(ℝ n ), when p(·) ∈ P(ℝ n ), 0 < β < n/p + and 1/q(·) = 1/p(·) − β/n with q(·)(n − β)/n ∈ B(ℝ n ).