[Paper Review] Remarks on the Hardy--Littlewood inequality for $m$-homogeneous polynomials and $m$-linear forms

W. Cavalcante, Daniel Núñez-Alarcón+1 more|arXiv (Cornell University)|Jul 22, 2015

Advanced Harmonic Analysis Research12 references3 citations

TL;DR

This paper establishes an optimal version of the Hardy–Littlewood inequality for $m$-homogeneous polynomials on $\ell_m$ spaces, extending the known range from $p > m$ to $p = m$. It proves that the optimal constant for real 2-homogeneous polynomials on $\ell_2^2$ is exactly 2, and provides a rigorous justification for the non-smooth behavior of optimal exponents in the inequality via the Kahane–Salem–Zygmund inequality.

ABSTRACT

The Hardy--Littlewood inequality for $m$-homogeneous polynomials on $\ell_{p}$ spaces is valid for $p>m.$ In this note, among other results, we present an optimal version of this inequality for the case $p=m.$ We also show that the optimal constant, when restricted to the case of $2$-homogeneous polynomials on $\ell_{2}(\mathbb{R}^{2})$ is precisely $2$. In an Appendix we justify why, curiously, the optimal exponents of the Hardy--Littlewood inequality do not behave smoothly.

Motivation & Objective

• To extend the Hardy–Littlewood inequality to the critical case $p = m$ for $m$-homogeneous polynomials on $\ell_p$ spaces.
• To determine the optimal constant for the case of 2-homogeneous polynomials on $\ell_2^2$ over $\mathbb{R}$, proving it is exactly 2.
• To explain why the optimal exponents in the Hardy–Littlewood inequality do not vary smoothly with $p$, particularly in the bilinear and multilinear settings.
• To unify and refine existing results on the Hardy–Littlewood and Bohnenblust–Hille inequalities using sharpness analysis.

Proposed method

• Adapting a lemma from [3, Proposition 2.2] to relate the $\ell_p$-norm of coefficients of $m$-homogeneous polynomials to the operator norm of their associated symmetric $m$-linear forms.
• Using the Kahane–Salem–Zygmund inequality to test sharpness of exponents in the bilinear case, showing that neither the $\ell^2$ nor the $\ell^\lambda$ exponent can be improved independently.
• Applying the duality between $m$-linear forms and $m$-homogeneous polynomials to derive coefficient estimates in $\ell^{p/(p-m)}$-norms for $m < p < 2m$, and extending this to $p = m$.
• Establishing sharpness of the exponent $\frac{p}{p-m}$ for $m < p < 2m$ and $\frac{2mp}{mp + p - 2m}$ for $p \geq 2m$ via counterexamples and norm comparisons.
• Using interpolation and Hölder-type inequalities for mixed norms to analyze the interplay between the exponents $2$ and $\lambda = \frac{pq}{pq - p - q}$ in the bilinear case.
• Providing a unifying formulation of the Hardy–Littlewood inequality in Theorem 4.3, showing that both exponents are simultaneously optimal and cannot be interchanged when $\frac{1}{p} + \frac{1}{q} > \frac{1}{2}$.

Experimental results

Research questions

• RQ1 What is the optimal constant in the Hardy–Littlewood inequality for 2-homogeneous polynomials on $\ell_2^2(\mathbb{R})$?
• RQ2 Can the Hardy–Littlewood inequality be extended from $p > m$ to $p = m$ for $m$-homogeneous polynomials on $\ell_p$ spaces, and what is the optimal constant in this case?
• RQ3 Why do the optimal exponents in the Hardy–Littlewood inequality fail to behave smoothly as functions of $p$?
• RQ4 Is the exponent $\frac{pq}{pq - p - q}$ in the bilinear case truly optimal, and can it be improved when paired with the $\ell^2$-norm?
• RQ5 Under what conditions can the exponents $2$ and $\lambda = \frac{pq}{pq - p - q}$ in the bilinear inequality be interchanged, and why is this not always possible?

Key findings

• The optimal constant for the Hardy–Littlewood inequality in the case of 2-homogeneous polynomials on $\ell_2^2(\mathbb{R})$ is exactly 2.
• The Hardy–Littlewood inequality for $m$-homogeneous polynomials on $\ell_m$ spaces is valid with the exponent $\frac{p}{p-m}$, and this exponent is sharp for $p = m$.
• The optimal constant $C_{\mathbb{K},m,p}^{\mathrm{pol}}$ for $m$-homogeneous polynomials satisfies $C_{\mathbb{K},m,p}^{\mathrm{pol}} \leq C_{\mathbb{K},m,p}^{\mathrm{mult}} \frac{m^m}{(m!)^{(p-m)/p}}$ when $m < p < 2m$, with equality in the case $m=2$, $p=2$.
• The exponents $2$ and $\lambda = \frac{pq}{pq - p - q}$ in the bilinear Hardy–Littlewood inequality are simultaneously optimal and cannot be improved independently, as shown via the Kahane–Salem–Zygmund inequality.
• For $\frac{1}{2} < \frac{1}{p} + \frac{1}{q} < 1$, the exponents $2$ and $\lambda$ cannot be interchanged due to the lack of symmetry and the impossibility of interpolation, even with advanced inequalities.
• The paper provides a unified and optimal formulation of the Hardy–Littlewood inequality in Theorem 4.3, showing that both exponents are sharp and mutually dependent in the sense of optimality.