Abstract random polynomial inequalities in Banach spaces

Abstract random polynomial inequalities in Banach spaces

We will discuss some recent work with Andreas Defant, concerning  variants of the celebrated Kahane--Salem--Zygmund random polynomial inequalities. We will present estimates for exponential Orlicz norms of random variables \,$\sup_{1\leq j\leq N}\big|\sum_{i=1}^K a_i(j) \gamma_i\big|$, where $(a_i(j))_{j=1}^N \in \ell_\infty^N, \, 1 \leq i \leq K$  and $(\gamma_i)$ forms a~sequence of real or complex subgaussian random variables. Lifting these inequalities to finite dimensional Banach spaces, we get new Kahane--Salem--Zygmund type inequalities -- in particular, for spaces of subgaussian random polynomials on finite dimensional Banach spaces, and also for subgaussian random Dirichlet polynomials. We use interpolation methods to widen our approach considerably.

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