Necessary condition on weights for maximal and integral operators with rough kernels
Volume 263 / 2022
Abstract
Let $0\leq \alpha \lt n$, $m\in \mathbb {N}$ and let $T_{\alpha ,m}$ be an integral operator given by a kernel of the form $$ K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my), $$ where the $A_i$ are invertible matrices and each $k_i$ satisfies a fractional size condition and a generalized fractional Hörmander condition. Ibañez-Firnkorn and Riveros (2018) proved that $T_{\alpha ,m}$ is controlled in $L^p(w)$-norms, $w\in \mathcal {A}_{\infty }$, by the sum of maximal operators $M_{A_i^{-1},\alpha }$. In this paper we present a class $\mathcal {A}_{A,p,q}$ of weights, where $A$ is an invertible matrix. These weights are appropriate for weak-type estimates of $M_{A^{-1},\alpha }$. For certain kernels $k_i$ we can characterize the weights yielding strong-type estimates of $T_{\alpha ,m}$. Also, we give a strong-type estimate using testing conditions.
