Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let be the fractional maximal function. The commutator generated by and a suitable function b is defined by . Denote by P(a"e (n) ) the set of all measurable functions p(center dot): a"e (n) -> [1,a) such that 1 < p- := ess inf x is an element of R-n p(x) and p+ := ess sup x is an element of R-n p(x) < infinity. and by B(a"e (n) ) the set of all p(center dot) a P(a"e (n) ) such that the Hardy-Littlewood maximal function M is bounded on L (p(center dot))(a"e (n) ). In this paper, the authors give some characterizations of b for which is bounded from L (p(center dot))(a"e (n) ) into L (q(center dot))(a"e (n) ), when p(center dot) a P(a"e (n) ), 0 < beta < n/p (+) and 1/q(center dot) = 1/p(center dot) - beta/n with q(center dot)(n - beta)/ b is an element of B (R-n).