[Paper Review] Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant

N. Albuquerque, Gustavo Araújo+4 more|arXiv (Cornell University)|Sep 10, 2016

Advanced Harmonic Analysis Research17 references4 citations

TL;DR

This paper establishes that the optimal constants in the Hardy–Littlewood inequalities for $m$-linear forms on $\ell_p$ spaces with $m < p \leq 2m$ are uniformly bounded by $2^{(m-1)(p-m)/p}$, significantly improving the previously known bound of $2^{(m-1)/2}$. The result shows that for $m < p \leq m+1$, the optimal constants are uniformly bounded by 2, resolving a long-standing question about the growth of these constants.

ABSTRACT

The Hardy--Littlewood inequality for $m$-linear forms on $\ell _{p}$ spaces and $m

Motivation & Objective

• To improve the known upper bounds for the optimal constants in the Hardy–Littlewood inequalities for $m$-linear forms on $\ell_p$ spaces when $m < p \leq 2m$.
• To show that the previously used bound $2^{(m-1)/2}$ is not optimal and can be significantly improved.
• To establish that for $m < p \leq m+1$, the optimal constants are uniformly bounded by 2, independent of $m$.
• To provide a refined estimate of the optimal constant as $2^{(m-1)(p-m)/p}$, which interpolates between the known cases at $p = m$ and $p = 2m$.
• To unify and refine previous results on multilinear forms and their $\ell_p$-norm estimates using a recursive, induction-based approach with Khinchin-type inequalities.

Proposed method

• The authors use a recursive induction argument based on the inclusion theorem for multiple summing operators, extending known results from bilinear to $m$-linear forms.
• They apply the Khinchin inequality to control the $\ell_2$-norm of coefficients in the multilinear form, enabling the transition from $\ell_2$-based estimates to $\ell_p$-based ones.
• The proof relies on the structure of the exponent $\frac{1}{1 - \sum_{j=1}^{k} \frac{1}{p_j}}$, which arises naturally in the theory of multiple summing operators.
• The key technical step involves showing that the constant $2^{1 - (1/p_1 + 1/p_2)}$ from the bilinear case can be extended inductively to higher-order forms.
• The authors use the fact that $\|T\|_{\text{mult}} \leq 2^{1 - (1/p_1 + 1/p_2)} \|T\|$ for bilinear forms on $\ell_{p_1} \times \ell_{p_2}$ with $1/p_1 + 1/p_2 \geq 1/2$, and build on this via tensorization and duality.
• The final bound is derived by combining the recursive application of the inclusion theorem with the optimal exponent structure from the Bohnenblust–Hille and Hardy–Littlewood frameworks.

Experimental results

Research questions

• RQ1 Can the optimal constant in the Hardy–Littlewood inequality for $m$-linear forms on $\ell_p$ spaces with $m < p \leq 2m$ be improved beyond $2^{(m-1)/2}$?
• RQ2 Is it possible to achieve a uniform bound on the optimal constants for $m < p \leq m+1$, independent of $m$?
• RQ3 What is the sharp dependence of the optimal constant on $p$ and $m$ in the range $m < p \leq 2m$?
• RQ4 How do the constants behave as $p \to m^+$, given that the constant is 1 at $p = m$?
• RQ5 Can the recursive structure of multiple summing operators be used to derive tighter bounds than previously known?

Key findings

• The optimal constant in the Hardy–Littlewood inequality for $m$-linear forms on $\ell_p$ spaces with $m < p \leq 2m$ is bounded above by $2^{(m-1)(p-m)/p}$, which improves upon the previous bound of $2^{(m-1)/2}$.
• For $m < p \leq m+1$, the optimal constants are uniformly bounded by 2, meaning they do not grow with $m$.
• The bound $2^{(m-1)(p-m)/p}$ interpolates smoothly between the known case $p = m$ (where the constant is 1) and $p = 2m$ (where it matches the previous bound $2^{(m-1)/2}$).
• The result is derived via an inductive application of the inclusion theorem for multiple summing operators, combined with Khinchin-type inequalities.
• The authors show that the constant $2^{1 - (1/p_1 + 1/p_2)}$ for bilinear forms can be extended to $m$-linear forms through a recursive tensorization process.
• The new bound is sharp in the sense that the exponent $\frac{p}{p-m}$ is optimal, as smaller exponents would force the constant to depend on $n$.