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Removable singularities for analytic functions in the little Zygmund space. (English) Zbl 1380.30003
The Zygmund space \(\Lambda_\ast({\mathbb C})\) is defined as the set of complex-valued functions \(f\) in \({\mathbb C}\) which are bounded and such that \[ \|f\|_\ast = \sup_{z,h\in {\mathbb C}} \frac{|f(z+h) + f(z-h) - 2 f(z)|}{2 |h|} < \infty. \] The little Zygmund space \(\lambda_\ast({\mathbb C})\) is the closure in \(\Lambda_\ast({\mathbb C})\) of the set of bounded \({\mathcal C}^{\infty}\) functions.
The plane compact set \(\mathbf{K}\) is called removable if it has the property that all functions, analytic outside \(\mathbf{K}\), which belong to some space of functions \(X\) can be extended analytically to the entire plane.
The author provides a sharp sufficient condition for the \(\lambda_\ast({\mathbb C})\)-removability in terms of a lower Hausdorff content.
30B40 Analytic continuation of functions of one complex variable
30H35 BMO-spaces
60G46 Martingales and classical analysis
Full Text: DOI
[1] Carmona, JJ; Donaire, JJ, On removable singularities for the analytic Zygmund class, Michigan Math. J., 43, 51-65, (1996) · Zbl 0862.30035
[2] Chow, Y.S., Teicher, H.: Probability Theory. Springer-Verlag, New York (1978) · Zbl 0399.60001
[3] Dolzenko, EP, On the removable singularities of analytic functions, Amer. Math. Soc. Transl., 97, 33-41, (1970) · Zbl 0187.02504
[4] Donaire, JJ, Porosity of sets and the Zygmund class, Bull. London Math. Soc., 34, 659-666, (2002) · Zbl 1026.30005
[5] Feller, W.: An introduction to Probability Theory and its applications I. Wiley, New York (1968) · Zbl 0155.23101
[6] Garnett, J.: Analytic capacity and measure. Lecture Notes in Math, vol. 297. Springer-Verlag, Berlin and New York (1972) · Zbl 0253.30014
[7] Kahane, JP, Trois notes sur LES ensembles parfaits linéaires, Enseign. Math., 15, 185-192, (1969) · Zbl 0175.33902
[8] Kaufman, R, Hausdorff measure, BMO, and analytic functions, Pacific J. Math., 102, 369-371, (1982) · Zbl 0511.30001
[9] Kaufman, R, Smooth functions and porous sets, Proc. Roy. Irish Acad. Sect, A 93, 195-204, (1993) · Zbl 0864.30022
[10] Makarov, NG, Smooth measures and the law of the iterated logarithm, Math. USSR-Izv., 34, 455-469, (1990) · Zbl 0691.30026
[11] Makarov, NG, Probability methods in the theory of conformal mappings, Leningrad Math. J., 1, 1-56, (1990) · Zbl 0736.30006
[12] Makarov, NG, On a class of exceptional sets in the theory of conformal mappings, Math. USSR-Sb., 68, 19-30, (1991) · Zbl 0708.30011
[13] Mattila, P.: Geometry of sets and measures in Euclidean spaces. Cambridge Univ. Press, Cambridge (1995) · Zbl 0819.28004
[14] Nicolau, A; Orobitg, J, Joint approximation in BMO, J. Funct. Anal., 173, 21-48, (2000) · Zbl 0996.46012
[15] O’Farrell, AG, Estimates for capacities, and approximation in Lipschitz norms, J. reine angew. Math., 311, 101-115, (1979) · Zbl 0409.30031
[16] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton (1970) · Zbl 0207.13501
[17] Uy, NX, Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Mat., 17, 19-27, (1979) · Zbl 0442.30033
[18] Uy, NX, A characterization on Cauchy transforms of measures, Complex Variables, 4, 267-275, (1985) · Zbl 0575.30037
[19] Uy, NX, A non-removable set for analytic functions satisfying a Zygmund condition, Illinois J. Math., 30, 1-8, (1986) · Zbl 0575.30036
[20] Verdera, J, BMO rational approximation and one-dimensional Hausdorff content, Trans. Amer. Math. Soc., 297, 283-304, (1986) · Zbl 0642.30029
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