Singular integral operators with non-smooth kernels on irregular domains

Abstract

Let $\chi$ be a space of homogeneous type. The aims of this paper are as follows:

i) Assuming that $T$ is a bounded linear operator on $L_2(\chi )$ we give a sufficient condition on the kernel of $T$ so that $T$ is of weak type $(1,1)$, hence bounded on $L_p(\chi )$ for $1<p\le 2$; our condition is weaker than the usual Hörmander integral condition.

ii) Assuming that $T$ is a bounded linear operator on $L_2(\Omega )$ where $\Omega$ is a measurable subset of $\chi$, we give a sufficient condition on the kernel of $T$ so that $T$ is of weak type $(1,1)$, hence bounded on $L_p(\Omega )$ for $1<p\le 2$.

iii) We establish sufficient conditions for the maximal truncated operator $T_{\ast }$, which is defined by $T_{\ast }u(x)$ = sup${}_{ϵ>0}|T_{ϵ}u(x)|$, to be $L_p$ bounded, $1<p<\infty$. Applications include weak $(1,1)$ estimates of certain Riesz transforms and $L_p$ boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.