Bergman spaces of higher‐dimensional hyperbolic polyhedron‐type domains I

Constales, Denis; Kraußhar, Rolf Sören

doi:10.1002/mma.666

Cited by 11 publications

(4 citation statements)

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“…The literature on Bergman theory is wide and as classical reference books we mention, with no claim of completeness, the books [1], [2] and [21]. The theory has also been developed for hyperholomorphic functions such as the quaternionic regular functions in the sense of Fueter and the theory of monogenic functions, [3], [4], [15], [16], [17], [18], [22], [23], [24] and the literature therein. Recently we have started the study of Bergman theory in the slice hyperholomorphic setting, see [6], [7], [8], [9].…”

Section: Introductionmentioning

confidence: 99%

Further properties of the Bergman spaces of slice regular functions

Colombo

González-Cervantes

Sabadini

2015

Advances in Geometry

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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.

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Section: Introductionmentioning

confidence: 99%

Further properties of the Bergman spaces of slice regular functions

Colombo

González-Cervantes

Sabadini

2015

Advances in Geometry

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“…This can be shown be applying the usual compactification arguments, see Lemma 1 from [5] for details.…”

Section: Cusp Formsmentioning

confidence: 99%

A new class of hypercomplex analytic cusp forms

Constales

Grob

Kraußhar

2012

Trans. Amer. Math. Soc.

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In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator D Delta(k/2) for some even k is an element of Z. They will be called k-holomorphic Cliffordian automorphic forms. k-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. This function class includes the set of k-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing k-hypermonogenic cusp forms for k not equal 0, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms, we turn to the construction of Poincare series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of k-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and of k-holomorphic Cliffordian cusp forms

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“…Fiber bundle theory has been commonly used to interpret some physical and mathematical phenomena, see [2,3,4,23,27]. Bergman's theory has been developed in hypercomplex analysis see [5,6,8,9,10,12,13,14,15,24,25,26]. In particular, works [8,9,10] were the first to introduce the theory of slice regular Bergman spaces.…”

Section: Introductionmentioning

confidence: 99%