L p boundedness of singular integrals on product domains

Abstract

This paper proves that for Ω ∈ L (log + L ) 2 ( S n -1 × S m -1 ),∫ S n -1 Ω ( x ′, y ′)dσ( x ′)=0( y ′∈ S m -1 ),∫ S m -1 Ω ( x ′, y ′)dσ( y ′)=0( x ′∈ S n -1 ), the singular integral operator T with kernel K ( u,v )= Ω ( u′,v′ )| u| - n |v| - m is bounded on L p ( R n × R m ) for 1 p <∞.