Clifford and Harmonic Analysis on Cylinders and Tori

Abstract

Cotangent type functions in ${\mathbb{R}}^n$ are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds ${\mathbb{R}}^n/{\mathbb{Z}}^k$ where $1\le k\le n$. 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.