Hardy's Inequality and the L 1 norm of Exponential Sums

In this paper we generalize Hardy's inequality [3] for measures of analytic type and obtain a proof of the Littlewood conjecture [4] for the L' norm of exponential sums as a simple consequence. Let T be the circle group, Z the additive group of integers and M(T) the customary convolution algebra of Borel measures on T; for pe e M(T) and n G Z put M(%)= e-ino dp(O) Denote by H'(T) the classical space of all measures e e M(T) such that j(n) = 0 for all n < 0 and let C denote the complex numbers. We now state Hardy's inequality for measures of analytic type: THEOREM 1. If M C H'(T) then