On two approaches to the Bergman theory for slice regular functions

In this paper we show that the classical Bergman theory admits two possible settings for the class of slice regular functions. Let Ω be a suitable open subset of the space of quaternions ℍ that intersects the real line and let 𝕊² be the unit sphere of purely imaginary quaternions. Slice regular functions are those functions f: Ω →H whose restriction to the complex planes ℂ(i), for every i ∈ S², are holomorphic maps. One of their crucial properties is that from the knowledge of the values of f on Ω ∩ ℂ(i) for some i ∈ 𝕊², one can reconstruct f on the whole Ω by the so called Representation Formula. We will define the so-called slice regular Bergman theory of the first kind. By the Riesz representation theorem we provide a Bergman kernel which is defined on Ω and is a reproducing kernel. In the slice regular Bergman theory of the second kind we use the Representation Formula to define another Bergman kernel; this time the kernel is still defined on Ω but the integral representation of f requires the calculation of the integral only on Ω ∩ℂ(i) and the integral does not depend on i ∈ 𝕊².