A note on weak type [formula omitted] boundedness of CZOs with rough kernels

Abstract

For f ∈ S ( R 2 ) and Ω ∈ L 1 ( S 1 ) , ∫ S 1 Ω ( x ′ ) d x ′ = 0 , define T Ω ( f ) ( x ) = lim ϵ → 0 + ∫ | x − y | ⩾ ϵ Ω ( y / | y | ) | y | 2 f ( x − y ) d y . In this paper, we shall prove that there is a class of functions in H 1 ( S 1 ) − L ln + L ( S 1 ) such that T Ω is weak type L 1 -bounded.