Revista Matemática Iberoamericana


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Volume 15, Issue 2, 1999, pp. 233–265
DOI: 10.4171/RMI/255

Published online: 1999-08-31

Singular integral operators with non-smooth kernels on irregular domains

Xuan Thinh Duong[1] and Alan G.R. McIntosh[2]

(1) Macquarie University, Sydney, Australia
(2) Australian National University, Canberra, Australia

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 defi ned 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.

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Duong Xuan Thinh, McIntosh Alan: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15 (1999), 233-265. doi: 10.4171/RMI/255