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≥ 2 and any finite sequence (fj)jsubset Lp we have [ Big|sumj fjBig|p ≤ left(supj sumk αjk,cright)1/p' Big(sumj |fj|ppBig)1/p, ] where c=2, 1/p+1/p'=1, and αjk=√frac|fjfk|p/2|fj|p|fk|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 [ Big|sumj=13 fjBig|p ≤ left(1+2Gammac(p)right)1/p' Big(sumj=13 |fj|ppBig)1/p, ] where p ≥ 3, c(p) = frac2ln(2)(p-2)ln(3)+2ln(2) and Gamma=Gamma(f1,f2,f3)∈[0,1] quantifies the degree of orthogonality among f1,f2,f3. 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.