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

Abstract

Let $0\leq \alpha<n$, $m\in \mathbb{N}$ and let consider $T_{\alpha,m}$ be a of integral operator, given by kernel of the form $$K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),$$ where $A_i$ are invertible matrices and each $k_i$ satisfies a fractional size and generalized fractional H\"ormander condition. In [Iba\~nez-Firnkorn, G. H., and Riveros, M. S. (2018). Certain fractional type operators with H\"ormander conditions. To appear in Ann. Acad. Sci. Fenn. Math.] it was proved that $T_{\alpha,m}$ is controlled in $L^p(w)$-norms, $w\in A_{\infty}$, by the sum of maximal operators $M_{A_i^{-1},\alpha}$. In this paper we present the class of weights $\mathcal{A}_{A,p,q}$, where $A$ is an invertible matrix. This class are the good weights for the weak-type estimate of $M_{A^{-1},\alpha}$. For certain kernels $k_i$ we can characterize the weights for the strong-type estimate of $T_{\alpha,m}$. Also, we give a the strong-type estimate using testing conditions.