Slice正则函数理论

Recent development of the theory of Slice regular functions

Slice正则函数理论是单复分析从复数到四元数的自然推广.它由Gentili和Struppa于2006年引入,并迅速地得到广泛的研究,而且在泛函分析、算子理论、Schur分析、四元数量子力学中取得了广泛的应用.与此同时,该理论也被推广到Clifford代数、八元数以及更为一般的交错代数上.本文主要介绍作者在Slice正则函数理论中取得的最新进展.首先,本文证明了Borel-Carath′eodory不等式,并对单位圆盘D上的正规化双全纯函数的Slice正则延拓建立了相应的增长定理与掩盖定理.其次,本文研究了Slice正则函数的边界行为,证明了相应的Julia引理、Julia-Carath′eodory定理以及边界Schwarz引理.特别地,作者发现与单复变不同的是,Slice正则理论中的边界Schwarz引理不能断言四元数空间单位球上的Slice正则自映照在其边界不动点处的导数大于零.最后,本文还研究了正则函数空间理论,建立了相应的Forelli-Rudin估计.

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-Carath′eodory 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-Carath′eodory 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.