[Paper Review] Hölder's inequality: some recent and unexpected applications

N. Albuquerque, Gustavo Araújo+2 more|arXiv (Cornell University)|Dec 5, 2014

Advanced Banach Space Theory37 references3 citations

TL;DR

This paper demonstrates that a lesser-known variant of Hölder's inequality—Hölder's inequality for mixed $L_p$ spaces—has recently enabled major breakthroughs in Functional Analysis, Complex Analysis, and Quantum Information Theory. By leveraging this inequality as an interpolation tool, the authors achieve optimal subpolynomial bounds for the Bohnenblust–Hille and Hardy–Littlewood inequalities and resolve the asymptotic growth of the Bohr radius at $\sqrt{\frac{\log n}{n}}$, settling a long-standing problem in Dirichlet series theory.

ABSTRACT

Hölder's inequality, since its appearance in 1888, has played a fundamental role in Mathematical Analysis and it is, without any doubt, one of the milestones in Mathematics. It may seem strange that, nowadays, it keeps resurfacing and bringing new insights to the mathematical community. In this expository article we show how a variant of Hölder's inequality (although well-known in PDEs) was essentially overlooked in Functional Analysis and has had a crucial (and in some sense unexpected) influence in very recent and major breakthroughs in Mathematics. Some of these recent advances appeared in 2012-2014 and include the theory of Dirichlet series, the famous Bohr radius problem, certain classical inequalities (such as Bohnenblust--Hille or Hardy--Littlewood), or even Mathematical Physics.

Motivation & Objective

• To demonstrate the overlooked yet powerful role of Hölder's inequality for mixed $L_p$ spaces in recent advances across Functional Analysis, Complex Analysis, and Quantum Information Theory.
• To resolve the asymptotic growth of the $n$-dimensional Bohr radius, a classical problem in Dirichlet series and complex analysis.
• To improve the best-known constants in the Bohnenblust–Hille and Hardy–Littlewood inequalities using this variant of Hölder's inequality.
• To establish tighter bounds for separately summing multilinear operators and related inequalities in multilinear operator theory.

Proposed method

• Application of Hölder's inequality for mixed $\ell_p$ spaces as an interpolation technique, derived from a 1961 result by Benedek and Panzone but only recently recognized in Functional Analysis.
• Use of the generalized Kahane–Salem–Zygmund inequality in conjunction with the mixed-norm Hölder inequality to refine estimates in multilinear operator theory.
• Employment of Wiener's inequality and Harris' inequality to bridge complex analysis tools with the new interpolation framework.
• Combining the mixed-norm Hölder inequality with classical results such as Bayart’s inequality and Minkowski’s inequality to derive improved bounds.
• Utilization of interpolation techniques to generalize $n$-separability to $N$-separability in multilinear operators, improving exponent estimates.
• Derivation of subpolynomial growth estimates for constants in the Bohnenblust–Hille and Hardy–Littlewood inequalities via the new inequality framework.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the $n$-dimensional Bohr radius, and can it be precisely determined using modern functional analytic tools?
• RQ2 Can the constants in the Bohnenblust–Hille and Hardy–Littlewood inequalities be improved beyond the previously known $\left(\sqrt{2}\right)^{m-1}$ bound?
• RQ3 To what extent can Hölder’s inequality for mixed $L_p$ spaces serve as a unifying interpolation tool across disparate fields such as Dirichlet series, quantum information, and multilinear operator theory?
• RQ4 How does the new framework improve the theory of separately summing multilinear operators and their $N$-separability?

Key findings

• The asymptotic growth of the $n$-dimensional Bohr radius is exactly $\sqrt{\frac{\log n}{n}}$, with $\displaystyle\lim_{n\to\infty}\frac{\mathrm{K}_n}{\sqrt{\frac{\log n}{n}}}=1$, resolving a classical problem in complex analysis.
• The Bohnenblust–Hille inequality constants $\mathrm{B}_{\mathbb{K},m}^{\mathrm{mult}}$ exhibit subpolynomial growth, and the new framework improves the upper bound from $\left(\sqrt{2}\right)^{m-1}$ to $\left(\sqrt{2}\right)^{\frac{2m(m-1)}{p}}\left(\mathrm{B}_{\mathbb{R},m}^{\mathrm{mult}}\right)^{\frac{p-2m}{p}}$ for real scalars.
• For complex scalars, the improved bound is $\left(\frac{2}{\sqrt{\pi}}\right)^{\frac{2m(m-1)}{p}}\left(\mathrm{B}_{\mathbb{C},m}^{\mathrm{mult}}\right)^{\frac{p-2m}{p}}$, which yields subpolynomial growth when $p > m^2$.
• The paper establishes that the mixed-norm Hölder inequality enables the derivation of tighter constants in the Hardy–Littlewood inequality, particularly for $p > m^2$, where the constants grow subpolynomially.
• The theory of separately summing operators is generalized: $n$-separability implies $N$-separability for $n < N \leq m$, with improved exponent estimates in special cases.
• The application of the mixed-norm Hölder inequality provides the crucial step in proving the sharp asymptotic behavior of the Bohr radius, resolving uncertainty in earlier estimates of the form $\mathrm{K}_n = b_n\sqrt{\frac{\log n}{n}}$ with $\frac{1}{\sqrt{2}}+o(1) \leq b_n \leq 2$.