Weighted $H^p-L^q$ boundedness of integral operators with rough kernels

Abstract

In this paper, we study integral operators \begin{equation*} T_αf(x)=\int_{\mathbb{R}^{n}}K(x,y) f(y)dy, \end{equation*} with kernels $K(x,y)= k_1( x- A_1y)...k_m( x-A_my),$ where $k_i(x)=\frac{Ω_i(x)}{|x|^{n/q_i}}$ and $Ω_i: \mathbb{R}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_{i}$ are certain invertible matrices, and $\frac n{q_1}+\dots+\frac n{q_m}=n-α,$ $0\leq α<n.$ We obtain the $H^{p}_{w^p}(\mathbb{R}^{n})-L^{q}_{w^q}(\mathbb{R}^{n})$ boundedness of these operators, for a class of Muckenhoupt weights $w$, satisfying the condition \begin{equation*} w(A_ix)\leq cw(x), \end{equation*} $a.e.x\in\mathbb{R} ^n $, $1\leq i\leq m$.