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

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 $K(x,y)=k_1(x-A_{1}y)\mathrm{.}\mathrm{.}\mathrm{.}{k}_m(x-A_{m}y),$ where $k_i(x)=\frac{{\mathrm{\Omega }}_i(x)}{\mathrm{\mid }x{\mathrm{\mid }}^{n\mathrm{/}{q}_i}}$ and ${\mathrm{\Omega }}_i:{\mathbb{R}}^n\to \mathbb{R}$ are homogeneous functions of degree zero, satisfying a size and a Dini condition, $A_i$ are certain invertible matrices, and $\frac{n}{q_1}+\cdots +\frac{n}{q_m}=n-\alpha ,$ $0\le \alpha <n\mathrm{.}$ 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*} $a\mathrm{.}e\mathrm{.}x\in {\mathbb{R}}^n$, $1\le i\le m$.