# 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 $χ$ be a space of homogeneous type. The aims of this paper are as follows:

i) Assuming that $T$ is a bounded linear operator on $L_{2}(χ)$ we give a sufficient condition on the kernel of $T$ so that $T$ is of weak type $(1,1)$, hence bounded on $L_{p}(χ)$ for $1<p≤2$; our condition is weaker than the usual Hörmander integral condition.

ii) Assuming that $T$ is a bounded linear operator on $L_{2}(Ω)$ where $Ω$ is a measurable subset of $χ$, we give a sufficient condition on the kernel of $T$ so that $T$ is of weak type $(1,1)$, hence bounded on $L_{p}(Ω)$ for $1<p≤2$.

iii) We establish sufficient conditions for the maximal truncated operator $T_{∗}$, which is defined by $T_{∗}u(x)$ = sup$_{ϵ>0}∣T_{ϵ}u(x)∣$, to be $L_{p}$ bounded, $1<p<∞$. 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