Weighted inequalities for commutators of fractional integrals on spaces of homogeneous type.

Abstract

Let 0 <γ< 1, b be a BMO function and Im γ,b the commutator of order m for the fractional integral. We prove two type of weighted Lp inequalities for Im γ,b in the context of the spaces of homogeneous type. The first one establishes that, for A∞ weights, the operator Im γ,b is bounded in the weighted Lp norm by the maximal operator Mγ (Mm), where Mγ is the fractional maximal operator and Mm is the Hardy– Littlewood maximal operator iterated m times. The second inequality is a consequence of the first one and shows that the operator Im γ,b is bounded from Lp[Mγp(M[(m+1)p] w)(x)dμ(x)] to Lp[w(x)dμ(x)], where [(m + 1)p] is the integer part of (m + 1)p and no condition on the weight w is required. From the first inequality we also obtain weighted Lp–Lq estimates for Im γ,b generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.