## 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.## References

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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
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1845012