Harmonic analysis on constant curvature surfaces with point singularities
Let \({\mathbb{R}}^ 2_ n\) denote the n-fold covering space \((0<n\leq \infty)\) of the plane with the origin removed, parametrized by polar coordinates (r,\(\theta\)) with \(\theta\) periodic of period \(2\pi\) n, and equipped with the flat Riemannian metric. The space \({\mathbb{R}}^ 2_ n\) admits a natural harmonic analysis. This harmonic analysis realizes the joint spectral decompositions of the self-adjoint operator i(\(\partial /\partial \theta)\) and the Dirichlet or Friedrichs extension \(\Delta\) of the Laplace-Beltrami operator. This paper gives a detailed development of this harmonic analysis, including the effect of differential operators and multiplication operators on the Fourier transform, the theory of Sobolev spaces. In the process, it is necessary to study certain generalizations of the Hilbert transform. Applications of this harmonic analysis are given to a Radon transform on \({\mathbb{R}}^ 2_ n\), and to heat equation on \({\mathbb{R}}^ 2_ n\). Analogous problems for spaces of non-zero constant curvature are discussed, and in this context extensions to some higher dimensional spaces are given.
scientific article; zbMATH DE number 4198770
scientific article