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.