On the spectral synthesis problem for hypersurfaces of [formula omitted

Abstract

Let F 1( R n ) denote the Fourier algebra on R n , and D ( R n ) the space of test functions on R n . A closed subset E of R n is said to be of spectral synthesis if the only closed ideal J in F 1( R n ) which has E as its hull h(J)={x ϵ R n:f(x)=0 for all f ϵ J} is the ideal k(E)={fϵF 1(R n):f(E)=0} . We consider sufficiently regular compact subsets of smooth submanifolds of R n with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra (k(E)∩D(R n)) − j(E) , where j( E) denotes the smallest closed ideal in F 1( R n ) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k( E)∩ D( R n ) is dense in k( E). Together this solves the synthesis problem for such sets.