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Mat. Sb., 2007, Volume 198, Number 8, Pages 115–160 (Mi msb3742)  

This article is cited in 3 scientific papers (total in 3 papers)

Cartan-type estimates for potentials with Cauchy kernels and real-valued kernels

V. Ya. Èiderman

Moscow State University of Civil Engineering

Abstract: Let $\nu$ be a (complex) Radon measure in $\mathbb C$ with compact support and finite variation and let
$$ \mathscr C_*\nu(z)=\sup_{\varepsilon>0} |\int_{|\zeta-z|>\varepsilon}\frac{d\nu(\zeta)}{\zeta-z}| $$
be the maximal Cauchy integral. Estimates for the Hausdorff $h$-content of the set $\mathscr Z^*(\nu,P)=ż\in\mathbb C:\mathscr C_*\nu(z)>P\}$ are obtained, where $h$ is a measuring function and $P$ is a fixed positive number. These estimates are shown to be sharp up to the values of the absolute constants involved. A similar problem is also considered for potentials with arbitrary real non-increasing kernels of positive measure in $\mathbb R^m$, $m\ge1$. As an application of the so-developed machinery, results on connections between the analytic capacity and the Hausdorff measure are obtained (for instance, an analogue of Frostman's theorem on classical capacities).
Bibliography: 37 titles.

DOI: https://doi.org/10.4213/sm3742

Full text: PDF file (935 kB)
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English version:
Sbornik: Mathematics, 2007, 198:8, 1175–1220

Bibliographic databases:

UDC: 517.535+517.544.5+517.547.73
MSC: Primary 30E20, 30C85; Secondary 30A10
Received: 03.10.2006 and 03.04.2007

Citation: V. Ya. Èiderman, “Cartan-type estimates for potentials with Cauchy kernels and real-valued kernels”, Mat. Sb., 198:8 (2007), 115–160; Sb. Math., 198:8 (2007), 1175–1220

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. 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
    2. 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
    3. V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Ekstremalnye i approksimativnye svoistva naiprosteishikh drobei”, Izv. vuzov. Matem., 2018, no. 12, 9–49  mathnet
  • Математический сборник Sbornik: Mathematics (from 1967)
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