Recent development of the theory of Slice regular functions

Abstract

Theory of Slice regular functions is a natural generalization of complex analysis from complex numbers to quaternions. It was introduced initially by Gentili and Struppa in 2006, and has been extensively studied and has found its elegant applications to functional calculus for quaternionic linear operators, operator theory, Schur analysis and quaternionic quantum mechanics. Meanwhile, the notion of Slice regularity was also extended to functions of an octonionic variable and to the setting of Clifford algebras as well as to the setting of alternative real algebras. In this survey, we shall focus mainly on our recent results in the theory of Slice regular functions. Firstly, we establish the Borel-Caratheodory inequality and the sharp growth and distortion theorems for Slice regular extensions of normalized biholomorphic functions on the unit disc in the setting of quaternions. In addition, we study the boundary behavior of Slice regular functions and obtain the Julia lemma, the Julia-Caratheodory theorem as well as the boundary Schwarz lemma. In particular, we find that the boundary Schwarz lemma does not ensure the Slice derivative of a Slice regular self-mappings of the open unit ball B⊂H at its boundary fixed point to be necessarily a real number, in contrast to that in the complex case. Finally, we study theory of function spaces of Slice regular functions and establish the Forelli-Rudin type estimates.