Aspects of the Kahane–Salem–Zygmund inequalities in Banach spaces

Abstract

AbstractThe main aim of this work is to discuss several different approaches to the celebrated Kahane–Salem–Zygmund inequalities. In particular, we prove estimates for exponential Orlicz norms of averages $$\sup _{1\le j\le N}\Big |\sum _{i=1}^K a_i(j) \gamma _i\Big |\,,$$ sup 1 ≤ j ≤ N | ∑ i = 1 K a i ( j ) γ i | , where $$ (a_i(j)) \in \ell _\infty ^N, \, 1 \le i \le K$$ ( a i ( j ) ) ∈ ℓ ∞ N , 1 ≤ i ≤ K and the $$(\gamma _i)$$ ( γ i ) form a sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get some new Kahane–Salem–Zygmund type inequalities—in particular, for spaces of subgaussian random polynomials and multilinear forms on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials.