Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form $$T_\alpha f(x) = \int_{\mathbb{R}^n } {\left| {x - A_1 y} \right|^{ - \alpha _1 } \cdots \left| {x - A_m y} \right|^{ - \alpha _m } f(y)dy,}$$ , where A _i are certain invertible matrices, α _i > 0, 1 ≤ i ≤ m, α ₁ + … + α _m = n ? α, 0 ≤ α < n. For $\tfrac{1} {q} = \tfrac{1} {p} - \tfrac{\alpha } {n}$ , we obtain the L ^p (? ⁿ , w ^p ) ? L ^q (? ⁿ ,w ^q ) boundedness for weights w in A(p, q) satisfying that there exists c > 0 such that w(A _i x) ≤ cw(x), a.e. x ∈ ? ⁿ , 1 ≤ i ≤ m. Moreover, we obtain the appropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.