On Bloom-type estimates for commutators of the fractional maximal function

Abstract

Let \(0<\alpha <n\) and \(M_{\alpha }\) be the fractional maximal function. For a locally integrable function b, we denote by \(M_{\alpha ,b}\) and \([b,M_{\alpha }]\) the maximal commutator and the commutator of \(M_{\alpha }\) with b. In this paper, we consider Bloom-type estimates for \(M_{\alpha ,b}\) and \([b,M_{\alpha }]\). Some necessary and sufficient conditions to characterize the Bloom-type two-weight norm inequalities are given.