Skip to main content
SpringerLink
Log in

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

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  Google Scholar 

  2. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. O’Farrell, A.G.: Rational approximation in Lipschitz norms—II. Proc. R. Irish. Acad. 79A(11), 103–114 (1979)

    MathSciNet  MATH  Google Scholar 

  5. Verdera, J.: \({\rm C}^m\) approximation by solutions of elliptic equations, and Calderon–Zygmund operators. Duke Math. J. 55, 157–187 (1987)

    Article  MathSciNet  Google Scholar 

  6. Vitushkin, A.G.: The analytic capacity of sets in problems of approximation theory. Russ. Math. Surv. 22(6), 139–200 (1967)

    Article  MathSciNet  Google Scholar 

  7. Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)

    Article  MathSciNet  Google Scholar 

  8. Tolsa, X.: The semiadditivity of continuous analytic capacity and the inner boundary conjecture. Am. J. Math. 126(3), 523–567 (2004)

    Article  MathSciNet  Google Scholar 

  9. Tolsa, X.: Painleve’s problem and the semiadditivity of analytic capacity. Acta Math. 190(1), 105–149 (2003)

    Article  MathSciNet  Google Scholar 

  10. Melnikov, M.S., Paramonov, P.V., Verdera, J.: \(C^1\)-approximation and extension of subharmonic functions. Sb. Math. 192(4), 515–535 (2001)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  13. Paramonov, P.V.: On harmonic approximation in the \(C^1\)-norm. Math. USSR Sb. 71(1), 183–207 (1992)

    Article  MathSciNet  Google Scholar 

  14. Paramonov, P.V.: Some new criteria for uniform approximability of functions by rational fractions. Sb. Math. 186(9), 1325–1340 (1995)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  16. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  17. Melnikov, M.S.: Analytic capacity: discrete approach and curvature of a measure. Sb. Math. 186(6), 827–846 (1995)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  19. Tolsa, X.: Bilipschitz maps, analytic capacity, and the Cauchy integral. Ann. Math. 162(3), 1241–1302 (2005)

    Article  MathSciNet  Google Scholar 

  20. Mattila, P., Paramonov, P.V.: On geometric properties of harmonic \(Lip_1\)-capacity. Pac. J. Math. 171(2), 469–490 (1995)

    Article  Google Scholar 

  21. Tolsa, X.: Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon–Zygmund Theory. Birkhauser, Basel (2014)

    Book  Google Scholar 

  22. Ruiz de Villa, A., Tolsa, X.: Characterization and semiadditivity of the \(C^1\) harmonic capacity. Trans. Am. Math. Soc. 362, 3641–3675 (2010)

    Article  Google Scholar 

  23. Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36, 63–89 (1934)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanics and Mathematics Faculty of Moscow State University, Moscow, Russian Federation, 119991

    P. V. Paramonov

  2. Mathematics and Mechanics Faculty of St-Petersburg State University, Saint Petersburg, Russian Federation

    P. V. Paramonov

  3. ICREA, Passeig Lluís Companys 23, 08010, Barcelona, Catalonia

    X. Tolsa

  4. Departament de Matemàtiques and BGSMath, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia

    X. Tolsa

Authors
  1. P. V. Paramonov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. Tolsa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. V. Paramonov.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

P.V. Paramonov: Supported by the Russian Science Foundation (Grant No. 17-11-01064).

X. Tolsa: Partially supported by 2017-SGR-0395 (Catalonia) and MTM-2016-77635-P (MICINN, Spain).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paramonov, P.V., Tolsa, X. On \(C^1\)-approximability of functions by solutions of second order elliptic equations on plane compact sets and C-analytic capacity. Anal.Math.Phys. 9, 1133–1161 (2019). https://doi.org/10.1007/s13324-018-0275-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-018-0275-z

Keywords

Search

Navigation