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Mat. Zametki, 1998, Volume 63, Issue 6, Pages 923–934 (Mi mz1363)  

This article is cited in 14 scientific papers (total in 14 papers)

Hausdorff measure and capacity associated with Cauchy potentials

V. Ya. Èiderman

Moscow State University of Civil Engineering

Abstract: In the paper the connection between the Hausdorff measure $\Lambda_h(E)$ of sets $E\subset\mathbb C$ and the analytic capacity $\gamma(E)$, and also between $\Lambda_h(E)$ and the capacity $\gamma^+(E)$ generated by Cauchy potentials with nonnegative measures is studied. It is shown that if the integral $\int_0t^{-3}h^2(t)dt$ is divergent and $h$ satisfies the regularity condition, then there exists a plane Cantor set $E$ for which $\Lambda_h(E)>0$, but $\gamma^+(E)=0$. The proof is based on the estimate of $\gamma^+(E_n)$, where $E_n$ is the set appearing at the $n$th step in the construction of a plane Cantor set.

DOI: https://doi.org/10.4213/mzm1363

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English version:
Mathematical Notes, 1998, 63:6, 813–822

Bibliographic databases:

UDC: 517.5
Received: 20.12.1996

Citation: V. Ya. Èiderman, “Hausdorff measure and capacity associated with Cauchy potentials”, Mat. Zametki, 63:6 (1998), 923–934; Math. Notes, 63:6 (1998), 813–822

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. Ya. Havinson, “Golubev sums: a theory of extremal problems like the analytic capacity problem and of related approximation processes”, Russian Math. Surveys, 54:4 (1999), 753–818  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Garnett, J, “Large sets of zero analytic capacity”, Proceedings of the American Mathematical Society, 129:12 (2001), 3543  crossref  mathscinet  zmath  isi
    3. Tolsa, X, “On the analytic capacity gamma(+)”, Indiana University Mathematics Journal, 51:2 (2002), 317  crossref  mathscinet  zmath  isi
    4. Garnett, J, “Analytic capacity, Bilipschitz maps and Cantor sets”, Mathematical Research Letters, 10:4 (2003), 515  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Mateu, J, “The planar Cantor sets of zero analytic capacity and the local T(b)-theorem”, Journal of the American Mathematical Society, 16:1 (2003), 19  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Mateu J., Tolsa X., Verdera J., “On the Semiadditivity of Analytic Capacity and Planar Cantor Sets”, Harmonic Analysis at Mount Holyoke, Contemporary Mathematics Series, 320, eds. Beckner W., Nagel A., Seeger A., Smith H., Amer Mathematical Soc, 2003, 259–278  crossref  mathscinet  zmath  isi
    7. Tolsa X., “Painlevé"S Problem, Analytic Capacity and Curvature of Measures”, European Congress of Mathematics, ed. Laptev A., Eur. Math. Soc., 2005, 459–476  crossref  mathscinet  zmath  isi
    8. Eiderman, VY, “Cartan-type estimates for the Cauchy potential”, Doklady Mathematics, 73:2 (2006), 273  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    9. Tolsa, X, “Painlevé's problem and analytic capacity”, Collectanea Mathematica, 2006, 89  mathscinet  zmath  isi
    10. V. Ya. Èiderman, “Cartan-type estimates for potentials with Cauchy kernels and real-valued kernels”, Sb. Math., 198:8 (2007), 1175–1220  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. Eiderman V., Volberg A., “L-2-Norm and Estimates From Below for Riesz Transforms on Cantor Sets”, Indiana Univ. Math. J., 60:4 (2011), 1077–1112  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Tolsa X., “Caldern-Zygmund Capacities and Wolff Potentials on Cantor Sets”, J. Geom. Anal., 21:1 (2011), 195–223  crossref  mathscinet  zmath  isi  scopus  scopus
    13. B. A. Kats, D. B. Kats, “The Szegö function on a non-rectifiable arc”, Russian Math. (Iz. VUZ), 56:4 (2012), 9–18  mathnet  crossref  mathscinet
    14. A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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