Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 95-120

A SECOND LOOK OF SOBOLEV SPACES ON METRIZABLE GROUPS

Przemyslaw Górka and Tomasz Kostrzewa

Warsaw University of Technology, Department of Mathematics and Information Sciences
Ul. Koszykowa 75, 00-662 Warsaw, Poland; pgorka 'at' mini.pw.edu.pl

Warsaw University of Technology, Department of Mathematics and Information Sciences
Ul. Koszykowa 75, 00-662 Warsaw, Poland; kostrzewat 'at' mini.pw.edu.pl

Abstract. We continue our study of Sobolev spaces on locally compact abelian groups. In this paper we mainly focus on the case of metrizable groups. We show the density of the Bruhat–Schwartz space in Sobolev space. We prove the trace theorem on the cartesian product of topological groups. The comparison of Sobolev and fractional Sobolev spaces are given. In particular, it is proved that in the case of any abelian connected Lie group Sobolev and fractional Sobolev spaces coincide. Most of the theorems are illustrated by p-adic groups.

2010 Mathematics Subject Classification: Primary 32A99, 46E35, 22B99.

Key words: Abstract harmonic analysis, Sobolev spaces, analysis on metric measure spaces, trace theorem, fractional Sobolev spaces, metric group.

Reference to this article: P. Górka and T. Kostrzewa: A second look of Sobolev spaces on metrizable groups. Ann. Acad. Sci. Fenn. Math. 45 (2020), 95-120.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4507