Necessary condition on weights for maximal and integral operators with rough kernels

Abstract

Let 0 ≤ α < n, m ∈ N and let Tα,m be an integral operator given by a kernel of the form (Formular Presented) where the Ai are invertible matrices and each ki satisfies a fractional size condition and a generalized fractional Hörmander condition. Ibañez-Firnkorn and Riveros (2018) proved that Tα,m is controlled in Lp(w)-norms, w ∈ A∞, by the sum of maximal operators MA−1,α. In this paper we present a class AA,p,q of weights, where A is an invertible i matrix. These weights are appropriate for weak-type estimates of MA−1,α. For certain kernels ki we can characterize the weights yielding strong-type estimates of Tα,m. Also, we give a strong-type estimate using testing conditions.