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

  • P. V. ParamonovEmail author
  • X. Tolsa


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.


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 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Mazalov, M.Y., Paramonov, P.V., Fedorovskii, K.Y.: Conditions for \(C^m\)-approximability of functions by solutions of elliptic equations. Russ. Math. Surv. 67(6), 1023–1068 (2012)CrossRefGoogle Scholar
  2. 2.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983)zbMATHGoogle Scholar
  3. 3.
    Paramonov, P.V.: Criteria for the individual \(C^m\)-approximability of functions on compact subsets of \(\mathbb{R}^N\) by solutions of second-order homogeneous elliptic equations. Sb. Math. 209(6), 857–870 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    O’Farrell, A.G.: Rational approximation in Lipschitz norms—II. Proc. R. Irish. Acad. 79A(11), 103–114 (1979)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Verdera, J.: \({\rm C}^m\) approximation by solutions of elliptic equations, and Calderon–Zygmund operators. Duke Math. J. 55, 157–187 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Vitushkin, A.G.: The analytic capacity of sets in problems of approximation theory. Russ. Math. Surv. 22(6), 139–200 (1967)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tolsa, X.: The semiadditivity of continuous analytic capacity and the inner boundary conjecture. Am. J. Math. 126(3), 523–567 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tolsa, X.: Painleve’s problem and the semiadditivity of analytic capacity. Acta Math. 190(1), 105–149 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Melnikov, M.S., Paramonov, P.V., Verdera, J.: \(C^1\)-approximation and extension of subharmonic functions. Sb. Math. 192(4), 515–535 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bitsadze, A.V.: Boundary-Value Problems for Second Order Elliptic Equations. North-Holland Series in Applied Mathematics and Mechanics, vol. 5. North-Holland, Amsterdam (1968)Google Scholar
  12. 12.
    Paramonov, P.V., Fedorovskiy, K.Y.: Uniform and \(C^1\)-approximability of functions on compact subsets of \({\mathbb{R}}^2\) by solutions of second-order elliptic equations. Sb. Math. 190(2), 285–307 (1999)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Paramonov, P.V.: On harmonic approximation in the \(C^1\)-norm. Math. USSR Sb. 71(1), 183–207 (1992)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Paramonov, P.V.: Some new criteria for uniform approximability of functions by rational fractions. Sb. Math. 186(9), 1325–1340 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mazalov, M.Y., Paramonov, P.V.: Criteria for \(C^m\)-approximability by bianalytic functions on plane compacts. Sb. Math. 206(2), 77–118 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  17. 17.
    Melnikov, M.S.: Analytic capacity: discrete approach and curvature of a measure. Sb. Math. 186(6), 827–846 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tolsa, X.: \(L^2\) boundedness of the Cauchy transform implies \(L^2\) boundedness of all Calderón–Zygmund operators associated to odd kernels. Publ. Mat. 48, 445–479 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tolsa, X.: Bilipschitz maps, analytic capacity, and the Cauchy integral. Ann. Math. 162(3), 1241–1302 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mattila, P., Paramonov, P.V.: On geometric properties of harmonic \(Lip_1\)-capacity. Pac. J. Math. 171(2), 469–490 (1995)CrossRefGoogle Scholar
  21. 21.
    Tolsa, X.: Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon–Zygmund Theory. Birkhauser, Basel (2014)CrossRefGoogle Scholar
  22. 22.
    Ruiz de Villa, A., Tolsa, X.: Characterization and semiadditivity of the \(C^1\) harmonic capacity. Trans. Am. Math. Soc. 362, 3641–3675 (2010)CrossRefGoogle Scholar
  23. 23.
    Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36, 63–89 (1934)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mergelyan, S.N.: Uniform approximation to functions of a complex variable. Uspehi Mat. Nauk. 7:2, 31–122 (1952). Am. Math. Soc. Transl. 3, 287–293 (1962)Google Scholar
  25. 25.
    Boivin, A., Paramonov, P.V.: Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions. Sb. Math. 189(4), 481–502 (1998)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

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

Personalised recommendations