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

论文概要

研究领域: 数学
作者: Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, Haozhu Wang
发布时间: 2026-05-06
arXiv: 2605.05192

中文摘要

Carbery提出了以下针对多个函数的强化三角不等式形式：对于任意p>=2和任意有限序列(f_j)j ⊂ L^p，有||sum_j f_j||p <= (sup_j sum_k alpha{jk}^c)^{1/p'} (sum_j ||f_j||p^p)^{1/p}，其中c=2，1/p+1/p'=1，alpha{jk}=sqrt(||f_j f_k||{p/2}/(||f_j||_p ||f_k||p))。在本文第一部分，我们构造了一个反例，证明该不等式对每个p>2都失效。然后我们证明，如果上述形式的估计成立，指数必须满足c<=p'。最后，在临界指数c=p'处，我们对所有整数p>=2建立了该不等式。在本文第二部分，我们得到了一个 sharp 的三函数界：||sum{j=1}^3 f_j||p <= (1+2 Gamma^{c(p)})^{1/p'} (sum{j=1}^3 ||f_j||_p^p)^{1/p}，其中p>=3，c(p)=2ln(2)/((p-2)ln(3)+2ln(2))，Gamma=Gamma(f_1,f_2,f_3)∈[0,1]量化了f_1,f_2,f_3之间的正交程度。指数c(p)是最优的，改进了Carlen、Frank和Lieb先前得到的幂r(p)=6/(5p-4)。本工作中出现的一些中间引理和不等式是在大语言模型Grok的辅助下探索的。

原文摘要

Carbery proposed the following sharpened form of triangle inequality for many functions: for any p >= 2 and any finite sequence (f_j)j subset L^p we have ||sum_j f_j||p <= (sup_j sum_k alpha{jk}^c)^{1/p'} (sum_j ||f_j||p^p)^{1/p}, where c=2, 1/p+1/p'=1, and alpha{jk}=sqrt(||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<=p'. Finally, at the critical exponent c=p', we establish the inequality for all integer values p>=2. In the second part of the paper we obtain a sharp three-function bound ||sum{j=1}^3 f_j||p <= (1+2 Gamma^{c(p)})^{1/p'} (sum{j=1}^3 ||f_j||_p^p)^{1/p}, where p >= 3, c(p) = 2 ln(2)/((p-2)ln(3)+2 ln(2)) and Gamma=Gamma(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) = 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.

自动采集于 2026-05-08

#论文 #arXiv #数学 #小凯