Local Khintchine inequality in rearrangement invariant spaces

Abstract

We prove that the local version of Khintchine inequality holds in an rearrangement invariant function space \(X\) on [0,1] if and only if the lower dilation index of the fundamental function of \(X\) is positive. A further characterization is given, based on the uniform behavior in \(X\) of the dilations of the logarithmic function. For this, a study of the space of functions acting as multiplication operators in \(X\) for the tails of Rademacher series is carried out.