A nicely behaved singular integral on a purely unrectifiable set

Huovinen, Petri

doi:10.1090/S0002-9939-01-05955-X

A nicely behaved singular integral on a purely unrectifiable set
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Proc. Amer. Math. Soc. 129 (2001), 3345-3351 Request permission

Abstract:

We construct an example of a purely 1-unrectifiable AD-regular set $E$ in the plane such that the limit \[ \lim _{r\downarrow 0} \int \limits _{E\setminus B(x,r)} K(x-y) d \mathcal {H}^1 (y) \] exists and is finite for $\mathcal {H}^1$ almost every $x\in E$ for some class of antisymmetric Calderón-Zygmund kernels. Moreover, the singular integral operators associated with these kernels are bounded in $L^2(F)$, where $F\subset E$ has a positive $\mathcal {H}^1$ measure.

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