[Paper Review] On the optimality of the hypercontractivity of the complex Bohnenblust--Hille inequality

Jamilson R. Campos, Gustavo A. Muñoz-Fernández+2 more|arXiv (Cornell University)|Jan 8, 2013

Advanced Banach Space Theory14 references4 citations

TL;DR

This paper investigates the optimality of hypercontractivity in the complex polynomial Bohnenblust–Hille inequality by establishing a critical link to the real polynomial Bohnenblust–Hille inequality. Through detailed analysis of absolute and asymptotic hypercontractivity constants for real scalars, the authors derive improved lower bounds, showing that the hypercontractive constant for the complex case is optimal up to a factor of approximately 1.5098.

ABSTRACT

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 ($H_{\infty,\mathbb{R}}$). Among other results, our estimates combined with recent results from \cite{CMPS} show that \[ 1.5098

Motivation & Objective

• To determine whether the hypercontractivity of the complex polynomial Bohnenblust–Hille inequality is an optimal result.
• To explore the connection between the optimality of the complex inequality and the best possible constants in the real polynomial Bohnenblust–Hille inequality.
• To analyze two types of hypercontractivity constants for real scalars: absolute ($H_{a,bR}$) and asymptotic ($H_{ﾻR}$).
• To provide improved lower bounds for the hypercontractivity constants in the real case, which in turn inform the optimality of the complex case.
• To establish that the complex hypercontractivity constant cannot be improved beyond a factor of approximately 1.5098.

Proposed method

• The authors analyze the absolute hypercontractivity constant $H_{a,bR}$ and the asymptotic hypercontractivity constant $H_{ﾻR}$ for real scalars.
• They use recent results from [CMPS] to derive quantitative estimates on the growth of the hypercontractivity constants in the real setting.
• By relating the real and complex cases, they show that the optimality of the complex Bohnenblust–Hille inequality hinges on the behavior of the real constants.
• The analysis involves bounding the best possible constants in the real polynomial Bohnenblust–Hille inequality using functional and harmonic analysis techniques.
• The authors derive a lower bound of $1.5098$ for the ratio between the complex and real hypercontractivity constants, indicating the tightness of the complex result.
• The method relies on the interplay between polynomial inequalities and the structure of multilinear forms over real and complex spaces.

Experimental results

Research questions

• RQ1 Is the hypercontractivity of the complex polynomial Bohnenblust–Hille inequality an optimal result?
• RQ2 What is the precise relationship between the hypercontractivity constants in the real and complex polynomial Bohnenblust–Hille inequalities?
• RQ3 Can the absolute and asymptotic hypercontractivity constants for real scalars be bounded from below with sufficient precision to determine optimality in the complex case?
• RQ4 What is the best possible lower bound for the ratio between the complex and real hypercontractivity constants?
• RQ5 Does the optimality of the complex inequality depend on the asymptotic or absolute behavior of the real constants?

Key findings

• The paper establishes that the hypercontractivity of the complex polynomial Bohnenblust–Hille inequality is optimal up to a factor of approximately 1.5098.
• The absolute hypercontractivity constant for real scalars, $H_{a,bR}$, is bounded below by a value that implies the optimality of the complex case.
• The asymptotic hypercontractivity constant $H_{ﾻR}$ for real scalars plays a crucial role in determining the tightness of the complex inequality.
• By combining their estimates with results from [CMPS], the authors derive a lower bound of $1.5098$ for the ratio between the complex and real hypercontractivity constants.
• The analysis confirms that no further improvement is possible in the complex hypercontractivity constant beyond this factor.
• The results demonstrate a deep structural link between the real and complex versions of the Bohnenblust–Hille inequality through their hypercontractivity constants.