Almost-Orthogonality in Lp Spaces: A Case Study with Grok

2026-05-06

AI Summary

This study investigates the validity of Carbery’s proposed sharpened form of the multilinear triangle inequality in $L^p$ spaces for $p > 2$. By constructing explicit counterexamples, the work disproves the universal validity of this inequality in the superquadratic regime and establishes that any such estimate can hold only if the exponent $c$ satisfies $c \leq p'$, where $p'$ denotes the conjugate exponent of $p$. Moreover, at the critical exponent $c = p'$, the inequality is proven to hold for all integer $p \geq 2$. In the three-function case, the authors obtain a sharp bound with an optimal exponent that improves upon the result of Carlen–Frank–Lieb. The analysis combines tools from functional analysis, $L^p$ space theory, and carefully designed counterexamples, with key lemmas aided by large language model–assisted derivations.

Abstract

Carbery proposed the following sharpened form of triangle inequality for many functions: for any $p\ge 2$ and any finite sequence $(f_j)_j\subset L^p$ we have \[ \Big\|\sum_j f_j\Big\|_p \ \le\ \left(\sup_{j} \sum_{k} α_{jk}^{\,c}\right)^{1/p'} \Big(\sum_j \|f_j\|_p^p\Big)^{1/p}, \] where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\sqrt{\frac{\|f_{j}f_{k}\|_{p/2}}{\|f_{j}\|_{p}\|f_{k}\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\ge 2$. In the second part of the paper we obtain a sharp three-function bound \[ \Big\|\sum_{j=1}^{3} f_j\Big\|_p \ \le\ \left(1+2Γ^{c(p)}\right)^{1/p'} \Big(\sum_{j=1}^{3} \|f_j\|_p^p\Big)^{1/p}, \] where $p \geq 3$, $c(p) = \frac{2\ln(2)}{(p-2)\ln(3)+2\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.

Problem

• Almost-Orthogonality
• Lp Spaces
• Triangle Inequality
• Counterexample
• Optimal Exponent

Innovation

• almost-orthogonality
• Lp spaces
• sharp triangle inequality
• critical exponent
• Grok-assisted discovery