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

arXiv:2605.05192v1 Announce Type: cross
Abstract: Carbery proposed the following sharpened form of triangle inequality for many functions: for any $pge 2$ and any finite sequence $(f_j)_jsubset L^p$ we have [ Big|sum_j f_jBig|_p le left(sup_j sum_k alpha_jk^,cright)^1/p’ Big(sum_j |f_j|_p^pBig)^1/p, ] where $c=2$, $1/p+1/p’=1$, and $alpha_jk=sqrtfracf_jf_k\$. 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 $cle p’$. Finally, at the critical exponent $c=p’$, we establish the inequality for all integer values $pge 2$.
In the second part of the paper we obtain a sharp three-function bound [ Big|sum_j=1^3 f_jBig|_p le left(1+2Gamma^c(p)right)^1/p’ Big(sum_j=1^3 |f_j|_p^pBig)^1/p, ] where $p geq 3$, $c(p) = frac2ln(2)(p-2)ln(3)+2ln(2)$ 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 $r(p) = frac65p-4$ 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.