Harmonic analysis on constant curvature surfaces with point singularities
type: Item
instance of: scholarly article
title: Harmonic analysis on constant curvature surfaces with point singularities
zbMATH Open document ID: 0726.43011
DOI: 10.1016/0022-1236(90)90048-P
author: Robert S. Strichartz
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 4198770
zbMATH Keywords: covering space
zbMATH Keywords: flat Riemannian metric
zbMATH Keywords: harmonic analysis
zbMATH Keywords: spectral decompositions
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: differential operators
zbMATH Keywords: Fourier transform
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Hilbert transform
zbMATH Keywords: Radon transform
zbMATH Keywords: heat equation
reviewed by: Koichi Saka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4198770