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SOBOLEV SPACES ON LOCALLY COMPACT ABELIAN GROUPS AND THE BOSONIC STRING EQUATION

Part of: Quantum field theory; related classical field theories Abstract harmonic analysis Miscellaneous topics - Partial differential equations Linear function spaces and their duals

Published online by Cambridge University Press:  14 October 2014

PRZEMYSŁAW GÓRKA  and
ENRIQUE G. REYES
PRZEMYSŁAW GÓRKA
Affiliation:
Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile Department of Mathematics and Information Sciences, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland email P.Gorka@mini.pw.edu.pl
ENRIQUE G. REYES*
Affiliation:
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile email ereyes@fermat.usach.cl, e_g_reyes@yahoo.ca
*
✉ereyes@fermat.usach.cl
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Abstract

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Motivated by a class of nonlinear nonlocal equations of interest for string theory, we introduce Sobolev spaces on arbitrary locally compact abelian groups and we examine some of their properties. Specifically, we focus on analogs of the Sobolev embedding and Rellich–Kondrachov compactness theorems. As an application, we prove the existence of continuous solutions to a generalized bosonic string equation posed on an arbitrary compact abelian group, and we also remark that our approach allows us to solve very general linear equations in a $p$-adic context.

Keywords

abstract harmonic analysisSobolev spacesembedding theoremslocally compact groupsnonlinear nonlocal equationsstring theory

MSC classification

Primary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Secondary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35S: Pseudodifferential operators and other generalizations of partial differential operators 81T30: String and superstring theories; other extended objects (e.g., branes)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 1 , February 2015 , pp. 39 - 53
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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