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

Abstract

Let 0 < γ < 1, b be a BMO function and Iγ, b m the commutator of order m for the fractional integral. We prove two type of weighted Lp inequalities for Iγ, b m in the context of the spaces of homogeneous type. The first one establishes that, for A∞ weights, the operator Iγ, b m 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 Iγ, b m is bounded from Lp [Mγ p (M[(m + 1) p] w) (x) d μ (x)] to Lp [w (x) d μ (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 Iγ, b m generalizing the classical results of Muckenhoupt and Wheeden for the fractional integral operator.