Sobre a desigualdade de Kahane-Salem-Zygmund e resultados afins

Abstract

In this work, we explore some classic results that are at the intersection of probability theory and functional analysis, namely: the Kahane-Salem-Zygmund inequality (for simplicity, KSZ), the Gale-Berlekamp unbalanced light game and the Dvoretzky-Rogers Theorem. Our investigation took place, mainly, from the analytical point of view. At first, we present an extended multilinear version of the KSZ, with which we obtain optimal asymptotic estimates for the exponents in cases not covered by previous versions. In particular, we prove that a conjecture proposed by Albuquerque and Rezende is false. Then, inspired by an old result by Bohnenblust and Hille, we investigate how certain matrices of complex scalars can be used to replace the coeficients ±1, to obtain KSZ variants with better properties. In this direction, we propose a continuous version for the famous game of unbalanced lights by Gale-Berlekamp, with good estimates. Finally, using the same class of matrices, we obtained a constructive proof for the Dvoretzky-Rogers Theorem on sequence spaces with complex scalars. More precisely, given p ∈ [1,∞], we provide examples of a series (x(j))∞j=1 unconditionally summable in lp(C) with P∞j=1 kx (j)k2−ε = ∞, for all ε > 0. Still using the "Walsh System", we obtained a similar construction for the case of sequence spaces with real scalars.