On the spectral synthesis problem for hypersurfaces of RN

AbstractLet F1(Rn) denote the Fourier algebra on Rn, and D(Rn) the space of test functions on Rn. A closed subset E of Rn is said to be of spectral synthesis if the only closed ideal J in F1(Rn) which has E as its hull h(J)={x ϵ Rn:f(x)=0 for all f ϵ J} is the ideal k(E)={fϵF1(Rn):f(E)=0}. We consider sufficiently regular compact subsets of smooth submanifolds of Rn with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra (k(E)∩D(Rn))−j(E), where j(E) denotes the smallest closed ideal in F1(Rn) 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(Rn) is dense in k(E). Together this solves the synthesis problem for such sets