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.

Elements of function theory in the unit ball of quaternions / Giulia Sarfatti. - (2013).

Elements of function theory in the unit ball of quaternions

SARFATTI, GIULIA
2013

Abstract

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.
Graziano Gentili
ITALIA
Giulia Sarfatti
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2158/806320
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