Quantitative behaviour of the norms of an analytic measure

A Littlewood-Paley type inequality for the quotient norms of an analytic measure is obtained; one consequence of this inequality is the classical theorem of F. and M. Riesz. In this paper T is the circle group, Z the additive group of integers and M(T) the customary space of Borel measures on T; for ,u E M(T) and n E Z define Af(n)= e-ino d1(0). If ,u E M(T), put ,u = ,ua + ass where ta, is absolutely continuous with respect to Lebesgue measure on T and i,u is singular with respect to Lebesgue measure on T; let Ma(T) denote the space of all absolutely continuous measures. A measure ,u E M(T) is said to be of analytic type if ,t(n) = 0 for all n 2 for all k and Dk = [n2k, n2k+1) = {m E Z: n2k 2 for all k and Dk = [n2k,fn2k+1)U (-n2k+1,-n2k] C Z. Our quantitative generalization of the F. and M. Riesz Theorem is then THEOREM V. There is a constant C > 0 such that for any sequence (Dn)1 of symmetric dyadic intervals in Z and any ,u E H'(T)