Weak type bounds for a class of rough operators with power weights

Abstract

In this note we show that T Ω,α and M Ω,α , the fractional integral and maximal operators with rough kernel respectively, are bounded operators from L 1 (|x| β(n−α)/n , R n) to L n/(n−α),∞ (|x| β , R n), where 0 < α < n and −1 < β < 0. §1. Introduction Suppose that 0 < α 0 1 r n−α |x−y| < α < n, −1 < β < 0, n/(n − α) ≤ s ≤ ∞ and Ω ∈ L s (S n−1). Then T Ω,α is a bounded operator from L 1 (|x| β(n−α)/n , R n) to L n/(n−α),∞ (|x| β , R n). That is , for any λ > 0 and any f ∈ L 1 (|x| β(n−α)/n , R n), {x:|TΩ,αf (x)|>λ} |x| β dx ≤ C 1 λ R n |f (x)||x| β(n−α)/n dx n/(n−α) , where C is independent of λ and f .