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

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 $c=2$, $1/p+1/p^{\prime }=1$, and $\hat{I}{\pm }_{jk}=\sqrt{\frac{\Vert f_j{f}_k{\Vert }_{p/2}}{\Vert f_j{\Vert }_p\Vert f_k{\Vert }_p}}$. 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 $c\le p^{\prime }$. Finally, at the critical exponent $c=p^{\prime }$, 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 $p\ge 3$, $c(p)=\frac{2\ln (2)}{(p-2)\ln (3)+2\ln (2)}$ and $\hat{I}“=\hat{I}“(f_1,f_2,f_3)\in [0,1]$ 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.