Boundary Julia theory for slice regular functions

TL;DR

This paper develops boundary Julia theory for quaternionic slice regular functions, establishing key theorems and revealing unique quaternionic phenomena like the Lie bracket's role in boundary behavior.

Contribution

It introduces quaternionic versions of classical boundary theorems for slice regular functions, highlighting new phenomena due to quaternion non-commutativity.

Findings

Quaternionic Julia lemma and boundary theorems established.

Classical Hopf lemma may fail; quaternionic variant involves Lie brackets.

New boundary behavior phenomena specific to quaternions discovered.

Abstract

The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of the boundary Julia theory for quaternions. More precisely, we establish the quaternionic versions of the Julia lemma, the Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, the Hopf lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B\subset \mathbb H$ and of the right half-space $\mathbb H_+$. Especially, we find a new phenomenon that the classical Hopf lemma about $f'(\xi)>1$ at the boundary point may fail in general in quaternions, and its quaternionic variant should involve the Lie bracket reflecting the non-commutative feature of quaternions.