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Journal of Function Spaces
Volume 2014, Article ID 404738, 6 pages
Research Article

Sobolev Spaces on Locally Compact Abelian Groups: Compact Embeddings and Local Spaces

1Department of Mathematics and Information Sciences, Warsaw University of Technology, Ul. Koszykowa 75, 00-662 Warsaw, Poland
2Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile

Received 23 May 2013; Accepted 20 December 2013; Published 9 February 2014

Academic Editor: Kehe Zhu

Copyright © 2014 Przemysław Górka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We continue our research on Sobolev spaces on locally compact abelian (LCA) groups motivated by our work on equations with infinitely many derivatives of interest for string theory and cosmology. In this paper, we focus on compact embedding results and we prove an analog for LCA groups of the classical Rellich lemma and of the Rellich-Kondrachov compactness theorem. Furthermore, we introduce Sobolev spaces on subsets of LCA groups and study its main properties, including the existence of compact embeddings into -spaces.