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A nicely behaved singular integral on a purely unrectifiable set


Author: Petri Huovinen
Journal: Proc. Amer. Math. Soc. 129 (2001), 3345-3351
MSC (2000): Primary 28A75, 42B20; Secondary 30E20
DOI: https://doi.org/10.1090/S0002-9939-01-05955-X
Published electronically: April 2, 2001
MathSciNet review: 1845012
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Abstract:

We construct an example of a purely 1-unrectifiable AD-regular set $E$ in the plane such that the limit

\begin{displaymath}\lim_{r\downarrow 0} \int\limits_{E\setminus B(x,r)} K(x-y) \, d \mathcal{H}^1 (y) \end{displaymath}

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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Additional Information

Petri Huovinen
Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
Email: pjh@math.jyu.fi

DOI: https://doi.org/10.1090/S0002-9939-01-05955-X
Keywords: Singular integrals, rectifiability
Received by editor(s): August 31, 1999
Received by editor(s) in revised form: March 22, 2000
Published electronically: April 2, 2001
Additional Notes: The author was supported by EU TMR Grant #ERBFMBICT972410
Communicated by: David Preiss
Article copyright: © Copyright 2001 American Mathematical Society

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