Weighted inequalities for some integral operators with rough kernels

Abstract

AbstractIn this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$$$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, Ai are certain invertible matrices, and n/q1 +…+n/qm = n−α, 0 ≤ α < n. We obtain the appropriate weighted Lp-Lq estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.