Growth and smooth spectral synthesis in the Fourier algebras of Lie groups

Jean Ludwig¹ ; Lyudmila Turowska²

¹ Department of Mathematics University of Metz F-57045 Metz, France
² Department of Mathematics Chalmers University of Technology SE-412 96 Göteborg, Sweden

Studia Mathematica, Tome 176 (2006) no. 2, pp. 139-158

Résumé

Let $G$ be a Lie group and $A(G)$ the Fourier algebra of $G$. We describe sufficient conditions for complex-valued functions to operate on elements $u\in A(G)$ of certain differentiability classes in terms of the dimension of the group $G$. Furthermore, generalizing a result of Kirsch and Müller [Ark. Mat. 18 (1980), 145–155] we prove that closed subsets $E$ of a smooth $m$-dimensional submanifold of a Lie group $G$ having a certain cone property are sets of smooth spectral synthesis. For such sets we give an estimate of the degree of nilpotency of the quotient algebra $I_A(E)/J_A(E)$, where $I_A(E)$ and $J_A(E)$ are the largest and the smallest closed ideals in $A(G)$ with hull $E$.

Détail

DOI : 10.4064/sm176-2-3

Keywords: lie group fourier algebra describe sufficient conditions complex valued functions operate elements certain differentiability classes terms dimension group furthermore generalizing result kirsch ller ark mat prove closed subsets smooth m dimensional submanifold lie group having certain cone property sets smooth spectral synthesis sets estimate degree nilpotency quotient algebra where largest smallest closed ideals hull