Slice Lebesgue Measure of Quaternions

• Published in: Advances in applied Clifford algebras Vol. 26; no. 1; pp. 399 - 416
• Main Authors: Ren, Guangbin, Xu, Zhenghua
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2016
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Mathematical Methods in Physics; Physics; Physics and Astronomy; Theoretical; 32A26; Lipschitz space; 30G35; Slice regular function; Dual space; Bloch space
• ISSN: 0188-7009; 1661-4909
• DOI: 10.1007/s00006-015-0578-1

The slice Lebesgue measure is introduced in this article in the theory of slice regular functions, which replaces the role of Lebesgue measure in the theory of holomorphic functions. It is naturally arisen from the predual of the slice regular Lipschitz space. The slice regular Lipschitz space can be described in terms of the so called σ -distance globally, or alternatively in terms of derivative as a Bloch type space. Its predual turns out to be a new kind of Bergman spaces of slice regular functions in the unit ball of quaternions associated with the slice Lebesgue measure. Associated with the slice Lebesgue measure as well as the related surface measure, the Forelli-Rudin type estimate can be established so that the theory of holomorphic function spaces could be generalized in parallel to the slice regular setting.