Weighted inequalities for fractional type operators with some homogeneous kernels

Abstract

In this paper, we study integral operators of the form, where Ai are certain invertible matrices, αi > 0, 1 ≤ i ≤ m, α1 + ... + αm = n - α, 0 ≤ α < n. For, we obtain the Lp(ℝn, wp) - Lq(ℝn,wq) boundedness for weights w in A(p, q) satisfying that there exists c > 0 such that w(Aix) ≤ 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.