Skip to main content
Log in

Uniqueness and free interpolation for logarithmic potentials and the cauchy problem for the laplace equation in 529-1529-1529-1

  • Published:
Geometric & Functional Analysis GAFA Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

References

  • [ATW]H. Aleksander, B.A. Taylor, D.L. Williams, The interpolating sets forA , J. Math. Anal. Appl. 36:1 (1971), 556–568.

    Google Scholar 

  • [Al]A.B. Aleksandrov, On the (A)-integrability of boundary values of harmonic functions, Mat. Zametki 30:1 (1981), 59–72 (Russian).

    Google Scholar 

  • [Ar]Z.A. Arushanyan, A boundary uniqueness theorem for the solutions of the Cauchy problem for the polyharmonic equation, Izvestia AN Arm SSR, Matematica 11:6 (1976), 514–547 (Russian).

    Google Scholar 

  • [B]N.K. Bary, A Treatise on Trigonometric Series, Vol II, Pergamon Press, Oxford, 1964.

    Google Scholar 

  • [BoWo]J. Bourgain, T. Wolff, A remark on gradients of harmonic functions in dimension ≥3, Colloq. Mat. 60–61 (1990), 253–260.

    Google Scholar 

  • [Br]J. Bruna, Les ensembles d'interpolation desA p(D), C.R. Acad. Sci. 290A:1 (1980), 25–27.

    Google Scholar 

  • [C]L. Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345.

    Google Scholar 

  • [D]Ye.P. Dolzhenko, Boundary properties of arbitrary functions, IAN SSSR Ser. Math. 31 (1967), 3–14 (Russian).

    Google Scholar 

  • [DuS]N. Dunford, J.T. Schwartz, Linear Operators, Part I. General Theory, Interscience Publishers, NY, 1967.

    Google Scholar 

  • [Dy1]Ye.M. Dyn'kin, Free interpolation sets for Hölder classes, Mat. Sb. 109 (151):1 (5) (1979), 107–128 (Russian).

    Google Scholar 

  • [Dy2]Ye.M. Dyn'kin, The pseudoanalytic extension, Journ. d'Analyse Math. 60 (1993), 45–70.

    Google Scholar 

  • [DyHr]Ye.M. Dyn'kin, S.V. Hruščev, Interpolation by analytic functions smooth up to the boundary, Zapiski nauchn. sem. LOMI 56 (1976), 59–72.

    Google Scholar 

  • [G]J. Garnett, Bounded Analytic Functions. Academic Press, NY, 1981.

    Google Scholar 

  • [Go]M.G. Goluzina, On multiplication and division of the Cauchy type integrals, Vestnik LGU 19 (1981), 8–15.

    Google Scholar 

  • [HJ]V. Havin, B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Springer Verlag, Berlin-Heidelberg-New York, 1994.

    Google Scholar 

  • [Ho]K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall Inc., N.J. 1962.

    Google Scholar 

  • [Hu]P.D. Humke, A criterion for the nonporosity of a general Cantor set, PAMS 111:2 (Feb. 1991), 365–372.

    Google Scholar 

  • [K]A.M. Kotochigov, Interpolation by analytic functions smooth up to the boundary, Zapiski nauchn. sem. LOMI 30 (1972), 167–169 (Russian).

    Google Scholar 

  • [L]Ye.M. Landis, On some properties of elliptic equations, Doklady AN SSSR 107:5 (1956), 640–643 (Russian).

    Google Scholar 

  • [La]M.M. Lavrentyev, On the Cauchy problem for linear elliptic equations of the second order, Doklady AN SSSR 112:2 (1957), 195–197 (Russian).

    Google Scholar 

  • [MH]V.G. Maz'ya, V.P. Havin, On solutions of the Cauchy problem for the Laplace equation (uniqueness, normality, approximation), Trudy MMO 30 (1974), 61–114.

    Google Scholar 

  • [Me]S.N. Mergelyan, Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation, Uspekhi Mat. Nauk 11:5 (1956), 3–26 (Russian).

    Google Scholar 

  • [P]I.I. Priwalow, Randeigenschaften Analytischer Funktionen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956.

    Google Scholar 

  • [R]V. Rao, A uniqueness theorem for harmonic functions, Mat. Zametki 3:3 (1968), 245–252.

    Google Scholar 

  • [Ru]W. Rudin, Functional Analysis, McGraw-Hill Co. NY, 1973.

    Google Scholar 

  • [Sj]P. Sjögren, WeakL 1-characterization of Poisson Integrals, Green Potentials, andH p-spaces, Trans. Amer. Math. Soc. 233 (1977), 179–196.

    Google Scholar 

  • [Th]B.S. Thomson, Real functions, Lecture Notes in Math. 1170, Springer Verlag, Berlin, NY, 1985.

    Google Scholar 

  • [V]J. Väisälä, Porous sets and quasisymmetric maps, Trans. Amer. Math. Soc. 299:2 (1987), 525–533.

    Google Scholar 

  • [ViH]S.A. Vinogradov, V.P. Havin, Free interpolation inH and in some other function classes, Zapiski Nauchn. Sem. LOMI I 47 (1974), 15–54; II 56 (1976), 12–58.

    Google Scholar 

  • [ViHr]S.A. Vinogradov, S.V. Hruščev, Inner functions and multipliers of Cauchy type integrals, Arkiv för Math. 19:1 (1981), 23–42.

    Google Scholar 

  • [Vy]Yu. Vymenets, On the Cauchy problem for the Laplace equation in dimension two, To appear.

  • [Wo]T. Wolff, Counterexamples with harmonic gradients in ℝ3, Pacific J. Math., To appear.

  • [Z]L. Zajiček, Porosity and σ-porosity, Real Anal. Exchange 13:2 (1987/88), 314–350.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The fourth author is partially supported by the Russian Fund of Fundamental Investigations (RFFI), grant N94-01-01628.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aleksandrov, A., Bourgain, J., Giesecke, M. et al. Uniqueness and free interpolation for logarithmic potentials and the cauchy problem for the laplace equation in 529-1529-1529-1. Geometric and Functional Analysis 5, 529–571 (1995). https://doi.org/10.1007/BF01895831

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01895831

Keywords

Navigation