On the symbol of nonlocal operators in Sobolev spaces
type: Item
instance of: scholarly article
title: On the symbol of nonlocal operators in Sobolev spaces
zbMATH Open document ID: 1236.58034
DOI: 10.1134/S0012266111060176
author: Anton Savin
published in: Differential Equations
publication date: 2011-11-24T00:00:00Z
review text: Let us consider a diffeomorphism \(g: M\to M\) of a smooth closed manifold an denote \(T^k\), \(k\in\mathbb{Z}\), the powers of the operator \(Tu(x)= u(g(x))\). The author studies operators \(D\) in the form of a finite sum, acting on Sobolev spaces: \[ D= \sum_k D_k T^k: H^s(M)\to H^{s-m}(M), \] where \(D_k\) are classical pseudodifferential operators of order \(\leq m\) on the manifold \(M\). Operators of this form were considered, for example, \textit{A. B. Antonevich} [Math. USSR, Izv. 34, No. 1, 1--21 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 1, 3--24 (1989; Zbl 0798.35045)]. They have not pseudo-local character and can be regarded as Fourier integral operators on \(M\). The author defines here the notion of symbol \(\sigma(D)\), as vector-valued function on \(T^*M/\{0\}\). The composition formula \(\sigma(DD')= \sigma(D)\sigma(D')\) is proved and Fredholm properties are deduced. The relevant case when \(M\) is given by the sphere \(\mathbb{S}^n\) is studied in detail and explicit conditions for Fredholmness are obtained.
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 5978722
zbMATH Keywords: pseudodifferential operators on manifolds
zbMATH Keywords: symbolic calculus
zbMATH Keywords: Fredholm property
MaRDI profile type: Publication
cites work: Q3533313
cites work: Elliptic theory and noncommutative geometry. Nonlocal elliptic operators
cites work: On the index of nonlocal elliptic operators for the group of dilations
full work available at URL: s0012266111060176
OpenAlex ID: W2104644515
description: scientific article; zbMATH DE number 5978722