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Proceedings of the American Mathematical Society

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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

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
  • A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 466568, DOI 10.1073/pnas.74.4.1324
  • Michael Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR 1104656
  • Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
  • Guy David, Unrectifiable $1$-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), no. 2, 369–479 (English, with English and French summaries). MR 1654535, DOI 10.4171/RMI/242
  • G. David, P. Mattila, Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoamericana 16 2000, pp. 137–215.
  • Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
  • Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
  • Pertti Mattila, Mark S. Melnikov, and Joan Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), no. 1, 127–136. MR 1405945, DOI 10.2307/2118585
  • F. Nazarov, S. Treil, A. Volberg, Pulling ourselves up by the hair, Preprint.
  • X. Tolsa, Principal values for the Cauchy integral and rectifiability, Proc. Amer. Math. Soc. 128 2000, pp. 2111–2119.
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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