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

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