RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (249 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.5

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
\Bibitem{Eid98} \by V.~Ya.~\Eiderman \paper Hausdorff measure and capacity associated with Cauchy potentials \jour Mat. Zametki \yr 1998 \vol 63 \issue 6 \pages 923--934 \mathnet{http://mi.mathnet.ru/mz1363} \crossref{https://doi.org/10.4213/mzm1363} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1679225} \zmath{https://zbmath.org/?q=an:0919.28004} \transl \jour Math. Notes \yr 1998 \vol 63 \issue 6 \pages 813--822 \crossref{https://doi.org/10.1007/BF02312776} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000076726600035} `

• http://mi.mathnet.ru/eng/mz1363
• https://doi.org/10.4213/mzm1363
• http://mi.mathnet.ru/eng/mz/v63/i6/p923

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Garnett, J, “Large sets of zero analytic capacity”, Proceedings of the American Mathematical Society, 129:12 (2001), 3543
3. Tolsa, X, “On the analytic capacity gamma(+)”, Indiana University Mathematics Journal, 51:2 (2002), 317
4. Garnett, J, “Analytic capacity, Bilipschitz maps and Cantor sets”, Mathematical Research Letters, 10:4 (2003), 515
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
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
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
8. Eiderman, VY, “Cartan-type estimates for the Cauchy potential”, Doklady Mathematics, 73:2 (2006), 273
9. Tolsa, X, “Painlevé's problem and analytic capacity”, Collectanea Mathematica, 2006, 89
10. V. Ya. Èiderman, “Cartan-type estimates for potentials with Cauchy kernels and real-valued kernels”, Sb. Math., 198:8 (2007), 1175–1220
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
12. Tolsa X., “Caldern-Zygmund Capacities and Wolff Potentials on Cantor Sets”, J. Geom. Anal., 21:1 (2011), 195–223
13. B. A. Kats, D. B. Kats, “The Szegö function on a non-rectifiable arc”, Russian Math. (Iz. VUZ), 56:4 (2012), 9–18
14. A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Russian Math. Surveys, 68:6 (2013), 973–1026
•  Number of views: This page: 396 Full text: 145 References: 33 First page: 1