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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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


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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Additional Information
  • Petri Huovinen
  • Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35, FIN-40351 Jyväskylä, Finland
  • Email:
  • 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:
  • MathSciNet review: 1845012