Singular integrals in the Cesàro sense

Abstract

The existence of the singular integral$int K(x,y) f(y) dy$associated to a Calder´on-Zygmund kernel where the integral is understood inthe principal value sense$Tf(x)=lim_{epsilon o {0^+}}int_{|x-y|>epsilon} K(x,y) f(y) dy$has been well studied.We study inthis paper the existence of the above integral in the Ces`aro-$alpha$ sense.More precisely, we study the existence of$$lim_{epsilon o {0^+}} int_{|x-y|>epsilon} f(y) K(x,y) left(1 -{{epsilon} over{|x-y|}} ight)^{alpha} dy quad ext{ a.e.}$$for $-1