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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.
MSC:
 30B40 Analytic continuation of functions of one complex variable 30H35 BMO-spaces 60G46 Martingales and classical analysis
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References:
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