Weighted estimates for a class of multilinear fractional type operators

Abstract

In this paper, a class of multiple fractional type weights A ( p → , q ) is defined as sup Q ( 1 | Q | ∫ Q ν ω → q ) 1 q ∏ i = 1 m ( 1 | Q | ∫ Q ω i − p i ′ ) 1 p i ′ < ∞ , where ν ω → = ∏ i = 1 m ω i . And the necessary condition for the characterization of A ( p → , q ) is also obtained. Strong ( L p 1 ( ω 1 p 1 ) × ⋯ × L p m ( ω m p m ) , L q ( ν ω → q ) ) estimates when each p i > 1 and weighted endpoint estimates ( L p 1 ( ω 1 p 1 ) × ⋯ × L p m ( ω m p m ) , L q , ∞ ( ν ω → q ) ) when there exists p i = 1 for some multilinear fractional type operators (e.g. fractional maximal operator, fractional integral, commutators of fractional integral operators) are obtained. As applications of these results, we give some weighted estimates for the above operators with rough homogeneous kernels when suitable conditions were assumed on the kernels. Weighted strong and L ( log L ) type endpoint estimates for commutators of multilinear fractional integral operators are also obtained. Similar results for multilinear Calderón–Zygmund singular integral can be found in A.K. Lerner et al. (in press) [12].