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Singular integral operators with non-smooth kernels on irregular domains. (English) Zbl 0980.42007

This important paper shows how to establish weak type $\left(1,1\right)$ estimates for singular integral operators $K$ (and related operators) when the kernel $k$ is not smooth enough to satisfy a Hörmander condition:

${\int }_{d\left(x,y\right)\ge \delta d\left(x,{y}_{1}\right)}|k\left(x,y\right)-k\left(x,{y}_{1}\right)|dx\le c$

for some $\delta$ in $\left(1,\infty \right)$. Roughly speaking, this estimate is replaced by

${\int }_{d\left(x,y\right)\ge ct}|k\left(x,y\right)-{k}_{t}\left(x,y\right)|dx\le c,$

where ${k}_{t}$ is the kernel of $K{A}_{t}$, ${A}_{t}$ being a suitable approximate identity, with kernel ${a}_{t}$ dominated by ${|B\left(x,t\right)|}^{-1}s\left(d\left(x,y\right)/t\right)$, where $s$ is a bounded positive decreasing function vanishing at a prescribed polynomial rate at infinity. A Calderón-Zygmund decomposition is used, and a tricky argument applies to estimate the “bad part”. It turns out to be possible to prove the second estimate in a range of contexts. The applications include estimates for singular integrals on domains with irregular boundaries, maximal functions associated to singular integrals, Riesz transforms and functions of operators which generate semigroups with kernels which satisfy estimates like ${a}_{t}$ above.

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 43A85 Analysis on homogeneous spaces 35R05 PDEs with discontinuous coefficients or data 42B25 Maximal functions, Littlewood-Paley theory