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 $K(x,y)=k_1(x-A_{1}y)...k_m(x-A_{m}y),$ where $k_i(x)=\frac{{\Omega }_i(x)}{|x{|}^{n/q_i}}$ and ${\Omega }_i:{\mathbb{R}}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_i$ are certain invertible matrices, and $\frac{n}{q_1}+\cdots +\frac{n}{q_m}=n-\alpha ,$ $0\le \alpha <n.$ We obtain the $H_{w^p}^p({\mathbb{R}}^n)-L_{w^q}^q({\mathbb{R}}^n)$ boundedness of these operators, for a class of Muckenhoupt weights $w$, satisfying the condition \begin{equation*} w(A_ix)\leq cw(x), \end{equation*} $a.e.x\in {\mathbb{R}}^n$, $1\le i\le m$.