# Singular integral operators with non-smooth kernels on irregular domains

### Xuan Thinh Duong

Macquarie University, Sydney, Australia### Alan G.R. McIntosh

Australian National University, Canberra, Australia

## 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 $1 < p ≤ 2$; 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 $1 < p ≤2$. iii) We establish sufficient conditions for the maximal truncated operator $T_*$, which is defined by $T_*u(x)$ = sup$_{\epsilon>0} | T_\epsilon u(x) |$, 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.

## Cite this article

Xuan Thinh Duong, Alan G.R. McIntosh, Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15 (1999), no. 2, pp. 233–265

DOI 10.4171/RMI/255