Revista Matemática Iberoamericana

Clifford and Harmonic Analysis on Cylinders and Tori

Rolf Sören Krausshar and John Ryan

Full-text: Open access

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 $1\leq k\leq n$. 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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 21, Number 1 (2005), 87-110.

Dates
First available in Project Euclid: 22 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1114176228

Mathematical Reviews number (MathSciNet)
MR2155015

Zentralblatt MATH identifier
1079.30067

Subjects
Primary: 30G35: Functions of hypercomplex variables and generalized variables 42B30: $H^p$-spaces 53C27: Spin and Spin$^c$ geometry 58J32: Boundary value problems on manifolds

Keywords
Dirac operator Clifford analysis cotangent functions

Citation

Krausshar, Rolf Sören; Ryan, John. Clifford and Harmonic Analysis on Cylinders and Tori. Rev. Mat. Iberoamericana 21 (2005), no. 1, 87--110. https://projecteuclid.org/euclid.rmi/1114176228


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