×

zbMATH — the first resource for mathematics

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.
MSC:
30B40 Analytic continuation of functions of one complex variable
30H35 BMO-spaces
60G46 Martingales and classical analysis
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.