Weighted inequalities for fractional type operators with some homogeneous kernels

Abstract

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(ℝⁿ, w ^p) − L ^q(ℝⁿ,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 ∈ ℝⁿ, 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.