On the optimality of the hypercontractivity of the complex Bohnenblust--Hille inequality

The main motivation of this paper is the following open problem: Is the hypercontractivity of the \emph{complex} polynomial Bohnenblust–Hille inequality an optimal result? We show that the solution to this problem has a close connection with the searching of the optimal constants for the \emph{real} polynomial Bohnenblust–Hille inequality. So we are lead to a detailed study of the hypercontractivity constants for real scalars. In fact we study two notions of constants of hypercontractivity: absolute ($H_{a,\mathbb{R}}$) and asymptotic ($H_{\mathrm{\infty },\mathbb{R}}$). Among other results, our estimates combined with recent results from \cite{CMPS} show that $$1.5098<H_{\mathrm{\infty },\mathbb{R}}<2.829\text{and}1.6561<H_{a,\mathbb{R}}<\mathrm{3.296.}$$