Clifford and Harmonic Analysis on Cylinders and Tori

Abstract

Cotangent type functions in R n are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds R n /Z k where 1 ≤ k ≤ n. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this context. Also the analogues of Calderón-Zygmund type operators are introduced in this context, together with singular Clifford holomorphic, or monogenic, kernels defined on sector domains in the context of cylinders. Fundamental differences in the context of the n-torus arising from a double singularity for the generalized Cauchy kernel on the torus are also discussed.

Some conformally flat spin manifolds, Dirac operators and automorphic forms

“…The cases of n-tori and cylinders which are realized by translation groups equipped with the trivial bundles have already been treated in [16].…”

Aspects of Dirac Operators in Analysis

“…Consequently E 0,k is an automorphic form with respect to the group Λ k with 1 ≤ k ≤ n − 1. From these Eisenstein series we obtain the kernel E 0,k (x − y) and as shown in [17] this kernel projects via the projection p : R n → C k to give the fundamental solution to the Dirac operator defined on the trivial spinor bundle S 0 = C k × Cl n .…”

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

“…In the follow-up paper [7] we developed representation formulas for the Bergman kernel function of wedge-shaped domains which have additional rectangular restrictions in terms of explicit automorphic forms for discrete rotation and translation groups of the Vahlen group. Also for domains with a cylinder symmetry (see References [8,9]) we observed an intrinsic connection to automorphic forms on discrete translation groups. Now, our aim is to develop explicit and closed representation formulas for the reproducing kernel function of the space of square-integrable monogenic functions over hyperbolic polydron type domains of the form…”