Growth and smooth spectral synthesis in the Fourier algebras of Lie groups
Volume 176 / 2006
Studia Mathematica 176 (2006), 139-158
MSC: Primary 46J10; Secondary 46E15, 43A45, 22E30.
DOI: 10.4064/sm176-2-3
Abstract
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 $E$.
