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

TL;DR

This paper investigates weight conditions for the boundedness of certain fractional integral operators with rough kernels, extending previous results by characterizing weights for both weak and strong type estimates.

Contribution

It introduces the class of weights _{A,p,q} for these operators and characterizes weights for strong-type estimates under specific kernel conditions.

Findings

Defined the _{A,p,q} weight class for fractional operators

Characterized weights for strong-type boundedness of the operators

Provided testing conditions for strong-type estimates

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.