The behavior at infinity of the Fourier transform of the random measures that appear in the theory of multiplicative chaos of Mandelbrot, Peyrière, and Kahane is an area quite unexplored. For context and further reference, we first present an overview of this theory and then the result, which is the main objective of this work, generalizing a result previously announced by Kahane. We establish ...

Abstract Let A and B be two unital Banach algebras

(2) r n N I I P Ï 1/p r n N \\} E j ; ^ J >cp ExyTM>*J fr )l lli==1 II ; l l l i = 1 II ) Recalling that in general {E| f ^ } 1 ^ decreases to exp E log \f\ as p decreases to zero, one sees that (1) is a strictly stronger statement than (2); in fact (1) says simply that cp may be taken bounded away from zero in (2). Note that the inequality obtained from (1) by replacing ej with the jth Rademac...