Further properties of the Bergman spaces of slice regular functions

Abstract

In this paper we continue the study of Bergman theory for the class of slice regular functions. In the slice regular setting there are two possibilities to introduce the Bergman spaces, that are called of the first and of the second kind. In this paper we mainly consider the Bergman theory of the second kind, by providing an explicit description of the Bergman kernel in the case of the unit ball and of the half space. In the case of the unit ball, we study the Bergman-Sce transform. We also show that the two Bergman theories can be compared only if suitable weights are taken into account. Finally, we use the Schwarz reflection principle to relate the Bergman kernel with its values on a complex half plane.

Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

“…For various properties of the Bergman spaces of the first and second kind, see, eg, Castillo Villalba et al 11 and Colombo et al 12 The goal of the present paper is to extend the above mentioned results on the density of polynomials in the complex case, to quaternionic weighted Bergman spaces of the first kind and of the second kind. In Section 2, we study the approximation in weighted Bergman spaces of the first kind.…”

“…For various properties of the Bergman spaces of the first and second kind, see, eg, Castillo Villalba et al and Colombo et al…”

“…Proof The proof follows reasoning as in the proof of proposition 3.1 in Colombo et al In fact, it depends just on the analog inequalities in the complex case and the representation formula. More exactly, we take |.| p in the representation formula and then we use the inequalities ( a + b ) p ≤2 p −1 ( a p + b p ), if , and ( a + b ) p ≤ a p + b p , if 0< p <1, for all a , b ≥0.…”

BMO- and VMO-spaces of slice hyperholomorphic functions

“…The slice hyperholomorphic Hardy spaces on the unit ball and on the quaternionic half space , which consists of those quaternions that have positive real part, were introduced in and Hardy spaces with on the unit ball were studied in detail in . Slice hyperholomorphic Bergman spaces were treated in and slice hyperholomorphic Fock spaces in . Furthermore contains a first study on slice hyperholomorphic Bloch, Besov and weighted Bergman spaces and on the Dirichlet space in this setting.…”

On the Zeros of the Slice Bergman Kernels for the Upper Half Space and the Unit Ball on Quaternions and Clifford Algebras

“…Recently many function theoretic problems on the complex analysis have been studied also on quaternion analysis [1,[4][5][6][7][8]. Let be a domain in H which is the algebra of quaternions.…”