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Uspekhi Mat. Nauk, 2013, Volume 68, Issue 6(414), Pages 3–58 (Mi umn9556)  

This article is cited in 4 scientific papers (total in 4 papers)

Non-homogeneous harmonic analysis: 16 years of development

A. L. Volberga, V. Ya. Èidermanb

a Michigan State University, East Lansing, MI, USA
b University of Wisconsin-Madison, Madison, WI, USA

Abstract: This survey contains results and methods in the theory of singular integrals, a theory which has been developing dramatically in the last 15–20 years. The central (although not the only) topic of the paper is the connection between the analytic properties of integrals and operators with Calderón–Zygmund kernels and the geometric properties of the measures. The history is traced of the classical Painlevé problem of describing removable singularities of bounded analytic functions, which has provided a strong incentive for the development of this branch of harmonic analysis. The progress of recent decades has largely been based on the creation of an apparatus for dealing with non-homogeneous measures, and much attention is devoted to this apparatus here. Several open questions are stated, first and foremost in the multidimensional case, where the method of curvature of a measure is not available.
Bibliography: 128 titles.

Keywords: analytic capacity, Vitushkin's conjecture, Calderón–Zygmund operators and capacities, $T(1)$- and $T(b)$-theorems, rectifiable sets and measures, singular integrals and operators.

Funding Agency Grant Number
National Science Foundation DMS-0758552


DOI: https://doi.org/10.4213/rm9556

Full text: PDF file (1066 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2013, 68:6, 973–1026

Bibliographic databases:

UDC: 517.53+517.98
MSC: 30C85, 31C15, 42B20
Received: 03.05.2013

Citation: A. L. Volberg, V. Ya. Èiderman, “Non-homogeneous harmonic analysis: 16 years of development”, Uspekhi Mat. Nauk, 68:6(414) (2013), 3–58; Russian Math. Surveys, 68:6 (2013), 973–1026

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    This publication is cited in the following articles:
    1. V. Chousionis, L. Prat, X. Tolsa, “Square functions of fractional homogeneity and Wolff potentials”, Int. Math. Res. Not. IMRN, 2016, no. 8, 2295–2319  crossref  mathscinet  isi  scopus
    2. V. Chousionis, S. Li, “Nonnegative kernels and 1-rectifiability in the Heisenberg group”, Anal. PDE, 10:6 (2017), 1407–1428  crossref  mathscinet  zmath  isi  scopus
    3. V. Eiderman, A. Reznikov, A. Volberg, “Cauchy independent measures and almost-additivity of analytic capacity”, J. Anal. Math., 136:1 (2018), 55–82  crossref  mathscinet  zmath  isi  scopus
    4. Conde-Alonso J.M., Parcet J., “Nondoubling Calderon-Zygmund Theory: a Dyadic Approach”, J. Fourier Anal. Appl., 25:4 (2019), 1267–1292  crossref  isi
  • Успехи математических наук Russian Mathematical Surveys
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