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analytic continuation; analytic function; Bergman space; capacity; exceptional set; holomorphic function; Muckenhoupt weight; removable singularity; singular set; Sobolev space; weight
We develop a theory of removable singularities for the weighted Bergman space ${\mathcal A}^p_\mu (\Omega )=\lbrace f \text{analytic} \text{in} \Omega \: \int _\Omega |f|^p \mathrm{d}\mu < \infty \rbrace $, where $\mu $ is a Radon measure on $\mathbb{C}$. The set $A$ is weakly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) \subset \text{Hol}(\Omega )$, and strongly removable for ${\mathcal A}^p_\mu (\Omega \setminus A)$ if ${\mathcal A}^p_\mu (\Omega \setminus A) = {\mathcal A}^p_\mu (\Omega )$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop {\mathrm BMO}$ 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 $\mu $ 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 $\mathrm{d}\mu = w\mathrm{d}m$ and $w$ is a Muckenhoupt $A_p$ weight, $1<p<\infty $, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p^{\prime }=p/(p-1)$ and the dual weight $w^{\prime }=w^{1/(1-p)}$.
[1] D. R. Adams and L. I. Hedberg: Function Spaces and Potential Theory. Springer, Berlin-Heidelberg, 1995. MR 1411441
[2] L. V. Ahlfors and A. Beurling: Conformal invariants and function-theoretic null-sets. Acta Math. 83 (1950), 101–129. DOI 10.1007/BF02392634 | MR 0036841
[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. MR 1961636
[4] A. Björn: Removable singularities for Hardy spaces. Complex Variables Theory Appl. 35 (1998), 1–25. DOI 10.1080/17476939808815069 | MR 1609914
[5] A. Björn: Removable singularities on rectifiable curves for Hardy spaces of analytic functions. Math. Scand. 83 (1998), 87–102. MR 1662084
[6] A. Björn: Removable singularities for weighted Bergman spaces. Preprint, LiTH-MAT-R-1999-23, Linköpings universitet, Linköping, 1999. MR 2207013
[7] A. Björn: Removable singularities for $H^p$ spaces of analytic functions, $0. Ann. Acad. Sci. Fenn. Math. 26 (2001), 155–174. MR 1816565
[8] A. Björn: Properties of removable singularities for Hardy spaces of analytic functions. J. London Math. Soc. 66 (2002), 651–670. DOI 10.1112/S002461070200354X | MR 1934298
[9] A. Björn: Removable singularities for analytic functions in BMO and locally Lipschitz spaces. Math. Z. 244 (2003), 805–835. DOI 10.1007/s00209-003-0524-0 | MR 2000460
[10] L. Carleson: Selected Problems on Exceptional Sets. Van Nostrand, Princeton, N. J., 1967. MR 0225986 | Zbl 0189.10903
[11] J. J. Carmona and J. J. Donaire: On removable singularities for the analytic Zygmund class. Michigan Math. J. 43 (1996), 51–65. DOI 10.1307/mmj/1029005389 | MR 1381599
[12] E. P. Dolzhenko: On the removal of singularities of analytic functions. Uspekhi Mat. Nauk 18, No. 4 (1963), 135–142. (Russian)
[13] J. García-Cuerva and J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam, 1985. MR 0807149
[14] J. B. Garnett: Analytic Capacity and Measure. Lecture Notes in Math. Vol. 297, Springer, Berlin-Heidelberg, 1972. MR 0454006 | Zbl 0253.30014
[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. (Russian) MR 0235131
[16] L. I. Hedberg: Approximation in the mean by analytic functions. Trans. Amer. Math. Soc. 163 (1972), 157–171. DOI 10.1090/S0002-9947-1972-0432886-6 | MR 0432886 | Zbl 0236.31010
[17] L. I. Hedberg: Removable singularities and condenser capacities. Ark. Mat. 12 (1974), 181–201. DOI 10.1007/BF02384755 | MR 0361050 | Zbl 0297.30017
[18] J. Heinonen, T. Kilpeläinen and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Univ. Press, Oxford, 1993. MR 1207810
[19] L. Hörmander: The Analysis of Linear Partial Differential Operators I. 2nd ed., Springer, Berlin-Heidelberg, 1990. MR 1065993
[20] R. Kaufman: Hausdorff measure, BMO and analytic functions. Pacific J. Math. 102 (1982), 369–371. DOI 10.2140/pjm.1982.102.369 | MR 0686557 | Zbl 0511.30001
[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. (Russian) MR 0136720 | Zbl 0147.33203
[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. Belinskiĭ, ed.), Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1976, pp. 160–166. (Russian) MR 0507787
[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. (Russian) MR 0629840
[24] T. Kilpeläinen: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 95–113.
[25] P. Koskela: Removable singularities for analytic functions. Michigan Math. J. 40 (1993), 459–466. DOI 10.1307/mmj/1029004831 | MR 1236172 | Zbl 0805.30001
[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. MR 0509059
[27] X. U. Nguyen: Removable sets of analytic functions satisfying a Lipschitz condition. Ark. Mat. 17 (1979), 19–27. MR 0543500 | Zbl 0442.30033
[28] W. Rudin: Analytic functions of class $H_p$. Trans. Amer. Math. Soc. 78 (1955), 46–66. MR 0067993 | Zbl 0067.30201
[29] W. Rudin: Functional Analysis. 2nd ed., McGraw-Hill, New York, 1991. MR 1157815 | Zbl 0867.46001
[30] X. Tolsa: Painlevé’s problem and the semiadditivity of the analytic capacity. Acta Math. 190 (2003), 105–149. DOI 10.1007/BF02393237 | MR 1982794
[31] J. Väisälä: Lectures on $n$-Dimensional Quasiconformal Mappings. Lecture Notes in Math. vol. 229, Springer, Berlin-Heidelberg, 1971. MR 0454009
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