A note on the hypercontractivity of the polynomial Bohnenblust--Hille inequality

TL;DR

This paper investigates the hypercontractivity properties of the polynomial Bohnenblust--Hille inequality, establishing bounds on coefficient norms relative to supremum norms and proving the optimality of the exponent involved.

Contribution

It provides new bounds for the ratio of coefficient norms to supremum norms in polynomial inequalities and proves the optimality of the exponent in these bounds.

Findings

Established a constant C controlling the ratio for all r in [1, 2m/(m+1)]

Derived bounds involving n^{(m/r - (m+1)/2)} for polynomial coefficients

Proved the exponent (m/r - (m+1)/2) is optimal

Abstract

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $m$ a positive integer, we remark that there is a constant $C$ so that, for all $r\in\lbrack1,\frac {2m}{m+1}],$ the supremum of the ratio between the $\ell_{r}$ norm of the coefficients of any nonzero $m$-homogeneous polynomial $P:\ell_{\infty}% ^{n}(\mathbb{K}) \rightarrow\mathbb{K}$ and its supremum norm is dominated by $C^{m}\cdot n^{(\frac{m}{r}-\frac{m+1}{2})}$ and, moreover, we prove that the exponent $\frac{m}{r}-\frac{m+1}{2}$ is optimal.