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The aim of this paper is to prove a generalization of a well-known con-vexity theorem of M. Riesz [8]. The Riesz theorem was originally deduced by "real-variable" techniques. Later, Thorin [10], Tamarkin and Zygmund [9], and Thorin [ll] introduced convexity properties of analytic functions in their study of Riesz's theorem. These ideas were put in(More)
We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to(More)
The course is devoted to the analysis of differentiable functions of a complex variable. This is a central topic in pure mathematics, as well as a vital computational tool. Syllabus 1. Basics. Geometric description of complex numbers. The complex plane and the Riemann sphere. Conformal mappings. Linear transformations as conformal maps. Representation by(More)
T he subject matter of this essay is Alberto Calderón's pivotal role in the creation of the modern theory of singular integrals. In that great enterprise Calderón had the good fortune of working with Antoni Zygmund, who was at first his teacher and mentor and later his collaborator. For that reason any account of that theory has to be in part the story of(More)