Complex polynomial Bohnenblust--Hille inequality with polynomial bounds

TL;DR

This paper investigates bounds for complex polynomial Bohnenblust-Hille inequalities, providing new insights into how summability and combinatorial dimensions influence the constants involved, with implications for harmonic analysis and number theory.

Contribution

It introduces a novel inequality linking summability of polynomial restrictions to combinatorial dimensions, refining bounds for the Bohnenblust-Hille constants in complex polynomials.

Findings

Established a new inequality relating summability and combinatorial dimension

Improved understanding of constants in complex polynomial inequalities

Connected summability properties to polynomial index subsets

Abstract

The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for applications in Harmonic Analysis and Number Theory. Up to now, the best known estimates for its constants are dominated by $\kappa\left(1+\varepsilon\right) ^{m}$, where $\varepsilon>0$ is arbitrary and $\kappa>0$ depends on the choice of $\varepsilon$. For the special cases in which the number of variables in each monomial is bounded by some fixed number $M$, it has been shown that the optimal constant is dominated by a constant depending solely on $M$. In this note, based on a deep result of Bayart, we prove an inequality for any subset of the indices, showing how summability of arbitrary restrictions on monomials can be related to the combinatorial dimension associated with them.

Equations

$${\left(\sum _{|\alpha |=m}|c_{\alpha }(P){|}^{\frac{2m}{m+1}}\right)}^{\frac{m+1}{2m}}\le C_m\Vert P\Vert$$

$$C_m\le \kappa {\left(1+\epsilon \right)}^m.$$

$${\left(\sum _{\alpha \in {\Delta }_M}|c_{\alpha }(P){|}^{\frac{2m}{m+1}}\right)}^{\frac{m+1}{2m}}\le 2^{\frac{M}{2}}{m}^{\frac{M+1}{2}}\Vert P\Vert$$

$$w(\alpha )=\text{card }\{j:{\alpha }_j\ne 0\}$$

$${\Delta }_M=\{\alpha :|\alpha |=m,w(\alpha )\le M\}.$$

$$P(x)=\sum _{i=1}^{\infty }{c}_i{x}_{{\sigma }_1\left(i\right)}{x}_{{\sigma }_2\left(i\right)}\cdots x_{{\sigma }_m\left(i\right)},$$

$$P(x)=\sum _{i,j,k=1}^{\infty }{c}_{ijk}{x}_{{\sigma }_1\left(i,j\right)}{x}_{{\sigma }_2\left(j,k\right)}{x}_{{\sigma }_3\left(k,i\right)}.$$

$${\psi }_{\Lambda }\left(n\right):=\max \left\{card\left(\left(A_1\times \cdots \times A_m\right)\cap \Lambda \right);A_i\subset \mathbb{N}\text{, }card\left(A_i\right)\le n\right\}.$$

$$\dim \left(\Lambda \right)$$

$$=\inf \left\{s>0;\exists C>0,{\psi }_{\Lambda }\left(n\right)\le C{n}^s\text{ for all }n\in \mathbb{N}\right\}.$$

$${\left(\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^{\frac{2m}{m+1}}\right)}^{\frac{m+1}{2m}}\le e^{\dim \Lambda }{\left(C_{\Lambda }mm!\right)}^{\frac{\dim \Lambda }{m}}{\left(\frac{2}{\sqrt{\pi }}\right)}^{\frac{\left(m-1\right)\dim \Lambda }{m}}\Vert P\Vert$$

$${\left(\sum _{\left(i_1,\dots ,i_m\right)\in \Lambda }{\left|T\left(e_{i_1},\dots ,e_{i_m}\right)\right|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}\right)}^{\frac{1+\dim \Lambda }{2\dim \Lambda }}$$

$$\le C_{\Lambda }\sum _{k=1}^m\left(\sum _{i_k=1}^{\infty }{\left(\sum _{i_1,...,i_{k-1},i_{k+1},...,i_m=1}^{\infty }{\left|T\left(e_{i_1},\dots ,e_{i_m}\right)\right|}^2\right)}^{\frac{1}{2}}\right)$$

$$\left(\sum _{i_k=1}^{\infty }{\left(\sum _{i_1,...,i_{k-1},i_{k+1},...,i_m=1}^{\infty }{\left|T\left(e_{i_1},\dots ,e_{i_m}\right)\right|}^2\right)}^{\frac{1}{2}}\right)\le {\left(\frac{2}{\sqrt{\pi }}\right)}^{m-1}\Vert T\Vert$$

$${\left(\sum _{\left(i_1,\dots ,i_m\right)\in \Lambda }{\left|T\left(e_{i_1},\dots ,e_{i_m}\right)\right|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}\right)}^{\frac{1+\dim \Lambda }{2\dim \Lambda }}\le m{\left(\frac{2}{\sqrt{\pi }}\right)}^{m-1}{C}_{\Lambda }\Vert T\Vert$$

$$\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}\le \sum _{\left(i_1,\dots ,i_m\right)\in \Lambda }{\left(m!\right)}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}{\left|\hat{P}(e_{i_1},\dots ,e_{i_m})\right|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}.$$

$${\left(\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}\right)}^{\frac{1+\dim \Lambda }{2\dim \Lambda }}\le m!{\left(\sum _{\left(i_1,\dots ,i_m\right)\in \Lambda }|\hat{P}(e_{i_1},\dots ,e_{i_m}){|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}\right)}^{\frac{1+\dim \Lambda }{2\dim \Lambda }}$$

$${\left(\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^{\frac{2\dim \Lambda }{1+\dim \Lambda }}\right)}^{\frac{1+\dim \Lambda }{2\dim \Lambda }}$$

$$\le \left(m!\right)\left(m{\left(\frac{2}{\sqrt{\pi }}\right)}^{m-1}{C}_{\Lambda }\right)e^m\Vert P\Vert .$$

$${\left(\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^2\right)}^{\frac{1}{2}}\le {\left(\sum _{|\alpha |=m}|c_{\alpha }(P){|}^2\right)}^{\frac{1}{2}}\le \Vert P\Vert .$$

$$\frac{1}{\frac{2m}{m+1}}=\frac{\theta }{\frac{2\dim \Lambda }{1+\dim \Lambda }}+\frac{1-\theta }{2}$$

$$\theta =\frac{\dim \Lambda }{m},$$

$${\left(\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^{\frac{2m}{m+1}}\right)}^{\frac{m+1}{2m}}$$

$$\le {\left(\left(m!\right)\left(m{\left(\frac{2}{\sqrt{\pi }}\right)}^{m-1}{C}_{\Lambda }\right)e^m\Vert P\Vert \right)}^{\frac{\dim \Lambda }{m}}\Vert P{\Vert }^{1-\frac{\dim \Lambda }{m}}.$$

$${\left(\sum _{\alpha \in {\Gamma }_{\Lambda }}|c_{\alpha }(P){|}^{\frac{2m}{m+1}}\right)}^{\frac{m+1}{2m}}\le e^{\dim \Lambda }{\left(C_{\Lambda }mm!\right)}^{\frac{\dim \Lambda }{m}}{\left(\frac{2}{\sqrt{\pi }}\right)}^{\frac{\left(m-1\right)\dim \Lambda }{m}}\Vert P\Vert .$$

$${\left(\frac{2C}{\sqrt{\pi }}\right)}^{\dim \Lambda }{\left(m\right)}^{\dim \Lambda }.$$

$$e^{\dim \Lambda }{\left(C_{\Lambda }mm!\right)}^{\frac{\dim \Lambda }{m}}{\left(\frac{2}{\sqrt{\pi }}\right)}^{\frac{\left(m-1\right)\dim \Lambda }{m}}$$

$$\sim {\left(\frac{2Ce}{\sqrt{\pi }}\right)}^{\dim \Lambda }{\left(m!\right)}^{\frac{\dim \Lambda }{m}}$$

$$\sim {\left(\frac{2Ce}{\sqrt{\pi }}\right)}^{\dim \Lambda }{\left(\frac{m^{1+\frac{1}{2m}}}{e}\right)}^{\dim \Lambda }$$

$$\sim {\left(\frac{2C}{\sqrt{\pi }}\right)}^{\dim \Lambda }{\left(m^{1+\frac{1}{2m}}\right)}^{\dim \Lambda }$$

$$\sim {\left(\frac{2C}{\sqrt{\pi }}\right)}^{\dim \Lambda }{\left(m\right)}^{\dim \Lambda }.$$

Full text

1. Introduction

In the investigation of complex $m$-homogeneous polynomials whose monomials have a uniform bound $M$ on the number of variables, Carando, Defant, and Sevilla-Peris have shown (in [5]) that the optimal constants of the Bohnenblust-Hille inequality are dominated by a polynomial on $m$ whenever $M$ is fixed. Maia, Nogueira and Pellegrino ([7]) substantially improved these results, proving that the optimal constants have a universal bound (i.e. they are bounded by a constant dependent solely on $M$) under the same hypotheses. Nevertheless, there has not been any noticeable progress for these bounds as $M$ increases with $m$. In this note, we prove an inequality for any subset of the indices. This is achieved by employing a deep tool recently proved by Bayart [2] which involves the concept of combinatorial dimension. We now turn to the detailed presentation of what has just been outlined.

A mapping $P:c_{0}\rightarrow\mathbb{C}$ is a continuous $m$-homogeneous polynomial if there exists a continuous $m$-linear $\hat{P}:c_{0}\times...\times c_{0}\rightarrow\mathbb{C}$ such that for every $x\in c_{0}$ we have $P(x)=\hat{P}(x,...,x)$. For each sequence $\alpha:\mathbb{N}\rightarrow\mathbb{N}\cup\{0\}$ satisfying $|\alpha|:=\sum_{j}\alpha_{j}=m$, we denote by $c_{\alpha}(P)$ the coefficient of the monomial $x_{i_{1}}^{\alpha_{i_{1}}}\cdots x_{i_{m}}^{\alpha_{i_{m}}}$. Defining the norm of $P$ by $\|P\|:=\sup_{x\in B_{c_{0}}}|P(x)|$, the Bohnenblust–Hille inequality [4] for complex $m$-homogeneous polynomials reads as follows: there is a constant $C_{m}\geq 1$ such that

[TABLE]

for all continuous $m$-homogeneous polynomials $P:c_{0}\rightarrow\mathbb{C}$. The exact control of the growth of the constant $C_{m}$ plays a crucial role in a vast number of applications. In 2011, it was proved in [6] that $C_{m}$ can be chosen with exponential growth, and in 2014 ([3]) the result was improved as follows: for any $\varepsilon>0$ there is a constant $\kappa>0$ such that

[TABLE]

In order to improve the estimate (1) to a polynomial bound the authors of [5] have shown that for integers $m,M$ with $M\leq m$, we have

[TABLE]

for all continuous $m$-homogeneous polynomials $P:c_{0}\rightarrow\mathbb{C}$, where

[TABLE]

and

[TABLE]

In other words, the constant $C_{m}$ can be taken with a polynomial bound provided that the sum is restricted to monomials with uniformly bounded number of variables $M$. In [7] the result of [5] was improved, using techniques of [1], by showing that under the same hypotheses the constant $C_{m}$ can be replaced by a universal constant depending just on $M$.

To illustrate, we provide a pair of examples. First, for $j\in\mathbb{N}$ let $\sigma_{j}:\mathbb{N\rightarrow N}$ be injections such that for each $j,k$ with $j\not=k$ one has $card\left\{i\in\mathbb{N}:\sigma_{j}\left(i\right)\not=\sigma_{k}\left(i\right)\right\}=\aleph_{0}$ (e.g. define $\sigma_{j}(n):=p_{j}^{n}$, where $p_{j}$ is the $j$-th prime number). Notice that the results of [5, 7] are useless for a continuous $m-$homogeneous polynomial

[TABLE]

because in this case $M=m$.

Second, if $\sigma_{1},\sigma_{2},\sigma_{3}$ are injections from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{N}$, consider the following polynomial

[TABLE]

Although it does not make sense at this point, our result will show that the former example behaves somewhat like $M=1$ whereas the latter exhibts fractional quality as though $M=3/2$. This will follow from the fact that, using notation we introduce below, in these examples we have $dim(\Lambda)=1$ and $dim(\Lambda)=3/2$, respectively.

We use notations and notions from combinatorial dimension, as presented in [2]. For $\Lambda\subset\mathbb{N}^{m}$ and $n\geq 0$, define

[TABLE]

The combinatorial dimension of $\Lambda$, denoted by $\dim\left(\Lambda\right)$, is defined as

[TABLE]

Finally, to each monomial $x_{i_{1}}^{\alpha_{i_{1}}}\cdots x_{i_{M}}^{\alpha_{M}}$ of a given $m$-homogeneous polynomial (i.e. $i_{1}<...<i_{M}$, $1\leq M\leq m$, and $\sum_{k=1}^{M}\alpha_{i_{k}}=m$), we associate $(i_{1},\stackrel{{\scriptstyle\alpha_{i_{1}}}}{{...}},i_{1},...,i_{M},\stackrel{{\scriptstyle\alpha_{i_{M}}}}{{...}},i_{M})\in\mathbb{N}^{m}$ and denote by $\Lambda$ the set of all such indices. To keep the sequence notation for indices introduced above, we also define $\Gamma_{\Lambda}$ to be the set of all sequences $\alpha$ in $\mathbb{N}\cup\{0\}$ naturally associated to the indices by defining $\alpha(i_{k}):=\alpha_{i_{k}}$ and zero otherwise.

Theorem 1.1**.**

For all positive integers $m$ and all sets of indices $\Lambda\subset\mathbb{N}^{m},$ there is a constant $C_{\Lambda}$ such that

[TABLE]

for all continuous $m$-homogeneous polynomials $P:c_{0}\rightarrow\mathbb{C}$.

2. The proof of Theorem 1.1

By  [2, Theorem 2.1] we know that there is a constant $C_{\Lambda}>0$ such that

[TABLE]

for all continuous $m$–linear forms $T:c_{0}\times\dots\times c_{0}\rightarrow\mathbb{C}$. It is not difficult to prove that using the Khinchine’s inequality, for Steinhaus variables, we have

[TABLE]

for all $k\in\left\{1,...,m\right\}$ and for all continuous $m$–linear forms $T:c_{0}\times\dots\times c_{0}\rightarrow\mathbb{C}$. In this way, there is a constant $C_{\Lambda}>0$ such that

[TABLE]

for all continuous $m$–linear forms $T:c_{0}\times\dots\times c_{0}\rightarrow\mathbb{C}$.

Let $\hat{P}$ be the symmetric $m$-linear form associated to $P$. Note that

[TABLE]

Therefore

[TABLE]

and

[TABLE]

Since we are dealing only with complex scalars, by the Maximum Modulus Principle, we have

[TABLE]

Since

[TABLE]

with

[TABLE]

by Hölder’s inequality, we have

[TABLE]

Thus

[TABLE]

Remark 2.1**.**

If there is a constant $C$ such that $C_{\Lambda}\leq C^{m}$ for all $m$, then the constant in the above inequality is dominated asymptotically by

[TABLE]

In fact, by Stirling’s Formula we have that

[TABLE]