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

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.

Source: arXiv:2605.05192v1 - http://arxiv.org/abs/2605.05192v1 PDF: https://arxiv.org/pdf/2605.05192v1 Original Link: http://arxiv.org/abs/2605.05192v1