Growth and smooth spectral synthesis in the Fourier algebras of Lie groups
Let G be a Lie group and A(G) the Fourier algebra of G. In this paper we decribe sufficient conditions for complex-valued functions to operate on elements u ∈ A(G) satisfying certain degrees of differentiability in terms of the dimension of the group G. Furthermore, generalizing a result of Kirsch and Muller [Ark. Mat. 18 (1980), no. 2, 145–155] we prove that closed subsets E of a smooth m-dimensional sub-manifold 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 IA(E)/JA(E), where IA(E) and JA(E) are the largest and the smallest closed ideal in A(G) with hull E.