Slice Lebesgue Measure of Quaternions

Abstract

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.