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Removable singularities for weighted Bergman spaces

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Abstract

We develop a theory of removable singularities for the weighted Bergman space

, where μ is a Radon measure on ℂ. The set A is weakly removable for

, and strongly removable for

.

The general theory developed is in many ways similar to the theory of removable singularities for Hardy H p spaces, BMO and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable.

In the case when weak and strong removability are the same for all sets, in particular if μ is absolutely continuous with respect to the Lebesgue measure m, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable.

When dμ = wdm and w is a Muckenhoupt A p weight, 1 < p < ∞, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent p′ = p/(p − 1) and the dual weight w′ = w 1/(1 − p).

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References

  1. D. R. Adams and L. I. Hedberg: Function Spaces and Potential Theory. Springer, Berlin-Heidelberg, 1995.

    Google Scholar 

  2. L. V. Ahlfors and A. Beurling: Conformal invariants and function-theoretic null-sets. Acta Math. 83 (1950), 101–129; Also in: Lars Valerian Ahlfors: Collected Papers, vol. 1. Birkhäuser, Boston, 1982, pp. 406–434; and in: Collected Works of Arne Beurling, vol. 1. Birkhäuser, Boston, 1989, pp. 171–199.

    Article  MathSciNet  Google Scholar 

  3. N. Arcozzi and A. Björn: Dominating sets for analytic and harmonic functions and completeness of weighted Bergman spaces. Math. Proc. Roy. Irish Acad. 102A (2002), 175–192.

    Google Scholar 

  4. A. Björn: Removable singularities for Hardy spaces. Complex Variables Theory Appl. 35 (1998), 1–25.

    MATH  MathSciNet  Google Scholar 

  5. A. Björn: Removable singularities on rectifiable curves for Hardy spaces of analytic functions. Math. Scand. 83 (1998), 87–102.

    MATH  MathSciNet  Google Scholar 

  6. A. Björn: Removable singularities for weighted Bergman spaces. Preprint, LiTH-MATR-1999-23, Linköpings universitet, Linköping, 1999.

    Google Scholar 

  7. A. Björn: Removable singularities for H p spaces of analytic functions, 0 < p < 1. Ann. Acad. Sci. Fenn. Math. 26 (2001), 155–174.

    MATH  MathSciNet  Google Scholar 

  8. A. Björn: Properties of removable singularities for Hardy spaces of analytic functions. J. London Math. Soc. 66 (2002), 651–670.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Björn: Removable singularities for analytic functions in BMO and locally Lipschitz spaces. Math. Z. 244 (2003), 805–835.

    Article  MATH  MathSciNet  Google Scholar 

  10. L. Carleson: Selected Problems on Exceptional Sets. Van Nostrand, Princeton, N. J., 1967.

    Google Scholar 

  11. J. J. Carmona and J. J. Donaire: On removable singularities for the analytic Zygmund class. Michigan Math. J. 43 (1996), 51–65.

    Article  MathSciNet  Google Scholar 

  12. E. P. Dolzhenko: On the removal of singularities of analytic functions. Uspekhi Mat. Nauk 18, No. 4 (1963), 135–142 (In Russian.); English transl.: Amer. Math. Soc. Transl. 97 (1970), 33–41.

    MATH  Google Scholar 

  13. J. García-Cuerva and J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam, 1985.

    Google Scholar 

  14. J. B. Garnett: Analytic Capacity and Measure. Lecture Notes in Math. Vol. 297, Springer, Berlin-Heidelberg, 1972.

    Google Scholar 

  15. V. P. Havin and V. G. Maz’ya: On approximation in the mean by analytic functions. Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 23, No. 13 (1968), 62–74 (In Russian.); English transl.: Vestnik Leningrad Univ. Math. 1 (1974), 231–245.

    Google Scholar 

  16. L. I. Hedberg: Approximation in the mean by analytic functions. Trans. Amer. Math. Soc. 163 (1972), 157–171.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. I. Hedberg: Removable singularities and condenser capacities. Ark. Mat. 12 (1974), 181–201.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Heinonen, T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, Oxford, 1993.

    Google Scholar 

  19. L. Hörmander: The Analysis of Linear Partial Differential Operators I. 2nd ed., Springer, Berlin-Heidelberg, 1990.

    Google Scholar 

  20. R. Kaufman: Hausdorff measure, BMO and analytic functions. Pacific J. Math. 102 (1982), 369–371.

    MATH  MathSciNet  Google Scholar 

  21. S. Ya. Khavinson: Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains. Mat. Sb. 54 (1961), 3–50 (In Russian.); English transl.: Amer. Math. Soc. Transl. 43 (1964), 215–266.

    MATH  MathSciNet  Google Scholar 

  22. S. Ya. Khavinson: Removable singularities of analytic functions of the V. I. Smirnov class. Problems in Modern Function Theory, Proceedings of a Conference (P. P. Belinskii, ed.). Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1976, pp. 160–166. (In Russian.)

    Google Scholar 

  23. S. V. Khrushchëv: A simple proof of a removable singularity theorem for a class of Lipschitz functions. Investigations on Linear Operators and the Theory of Functions XI. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) Vol. 113, Nauka, Leningrad, 1981, pp. 199–203, 267. (In Russian.)

    Google Scholar 

  24. T. Kilpeläinen: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 95–113.

    MATH  MathSciNet  Google Scholar 

  25. P. Koskela: Removable singularities for analytic functions. Michigan Math. J. 40 (1993), 459–466.

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Král: Singularités non essentielles des solutions des équations aux dérivées partielles. Séminaire de Théorie du Potentiel (Paris, 1972–1974). Lecture Notes in Math. Vol. 518, Springer, Berlin-Heidelberg, 1976, pp. 95–106.

    Google Scholar 

  27. X. U. Nguyen: Removable sets of analytic functions satisfying a Lipschitz condition. Ark. Mat. 17 (1979), 19–27.

    Article  MathSciNet  Google Scholar 

  28. W. Rudin: Analytic functions of class H p . Trans. Amer. Math. Soc. 78 (1955), 46–66.

    Article  MATH  MathSciNet  Google Scholar 

  29. W. Rudin: Functional Analysis. 2nd ed., McGraw-Hill, New York, 1991.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Väisälä: Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Math. vol. 229, Springer, Berlin-Heidelberg, 1971.

    Google Scholar 

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  1. Department of Mathematics, Linköping University, SE-581 83, Linköping, Sweden

    Anders Björn

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  1. Anders Björn
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Björn, A. Removable singularities for weighted Bergman spaces. Czech Math J 56, 179–227 (2006). https://doi.org/10.1007/s10587-006-0012-x

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