Littérature scientifique sur le sujet « Geometric theory of regular functions of a quaternionic variable »

The theory of regular functions over the quaternions introduced by Gentili and Struppa in 2006, already quite rich, is in continuous development. Despite their diverse peculiarities, regular functions reproduce numerous properties of holomorphic functions of one complex variable. This Thesis is devoted to investigate properties of regular functions defined on the unit ball B of the quaternions H. As it happens in the complex case, this particular subset of H represents a special domain for the class of regular function. It is the simplest example of the most natural set of definition for a regular function, namely of a "symmetric slice domain". Furthermore, on open balls centred at the origin, regular functions are characterized by having a power series expansion, hence they behave very nicely. The first Chapter, starting from the very first definitions, includes all the preliminary results that will be used in the sequel. The second Chapter discusses some properties of the modulus of regular functions, in particular how it is related with the modulus of the "regular conjugate" of a regular function. The main result presented is an analogue of the Borel-Carathéodory Theorem, a tool useful to bound the modulus of a regular function by means of the modulus of its real part. The central part of the Thesis contains geometric theory results. The third Chapter contains the analogue of the Bohr Theorem concerning power series, together with a weaker version, that follows as in the complex case from the Borel-Carathéodory Theorem. In the fourth Chapter we prove a Bloch-Landau type theorem, showing that in some sense the image of a ball under a regular function can not be too much thin. The fifth Chapter is dedicated to Landau-Toeplitz type theorems, that study the possible shapes that the image of a regular function can assume. The last Chapter is devoted to the study of the quaternionic Hardy spaces. We begin by the definition of the spaces H^p(B) and H^{\infty}(B), then we prove some of their basic properties. We introduce in conclusion the Corona Problem in the quaternionic setting, proving a partial statement of the Corona Theorem.

The textbook outlines the basics of the theory of functions of a complex variable. The geometric principles of analytic (regular) functions, on the basis of which the geometric theory of conformal maps is constructed, are also considered. Some applied aspects of the functions of a complex variable are given. Recommended by the Educational and Methodological Council of Higher Educational Institutions of the Russian Federation for Education in the field of applied mathematics and physics as a textbook for students of higher educational institutions studying in the field of Applied Mathematics and Physics and related fields and specialties. Meets the requirements of the federal state educational standards of higher education of the latest generation. For students and teachers of technical and physical-mathematical universities.