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On \(C^1\)-approximability of functions by solutions of second order elliptic equations on plane compact sets and C-analytic capacity

  • P. V. Paramonov
  • X. Tolsa
Article

Abstract

Criteria for approximability of functions by solutions of homogeneous second order elliptic equations (with constant complex coefficients) in the norms of the Whitney \(C^1\)-spaces on compact sets in \(\mathbb {R}^2\) are obtained in terms of the respective \(C^1\)-capacities. It is proved that the mentioned \(C^1\)-capacities are comparable to the classic C-analytic capacity, and so have a proper geometric measure characterization.

Keywords

Second order homogeneous elliptic operator \(C^1\)-approximation Localization operator of Vitushkin type L-oscillation \(\mathcal{L}C^1\)-capacity C-analytic capacity Curvature of measure 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. 1.Mechanics and Mathematics Faculty of Moscow State UniversityMoscowRussian Federation
  2. 2.Mathematics and Mechanics Faculty of St-Petersburg State UniversitySaint PetersburgRussian Federation
  3. 3.ICREABarcelonaCatalonia
  4. 4.Departament de Matemàtiques and BGSMathUniversitat Autònoma de BarcelonaBellaterra, BarcelonaCatalonia

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