Necessary condition on the weight for maximal and integral operators with rough kernels

Abstract

Let 0 ≤ α < n, m ∈ N and let consider T α,m be a of integral operator, given by kernel of the formwhere A i are invertible matrices and each k i satisfies a fractional size and generalized fractional Hörmander condition. In [7] it was proved that T α,m is controlled in L p (w)-norms, w ∈ A ∞ , by the sum of maximal operators M A −1 i ,α . In this paper we present the class of weights A A,p,q , where A is an invertible matrix. This class are the good weights for the weak-type estimate of M A −1 ,α . For certain kernels k i we can characterize the weights for the strong-type estimate of T α,m . Also, we give a the strong-type estimate using testing conditions.