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

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 $r\in [1,\frac{2m}{m+1}],$ the supremum of the ratio between the ${\ell }_r$ norm of the coefficients of any nonzero $m$-homogeneous polynomial $P:{\ell }_{\infty }$ 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.

References in corpus (2)

• Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii
• The Sidon constant for homogeneous polynomials