Almost-Orthogonality in Lp Spaces: Counterexamples and Sharp Bounds (arXiv 2605.05192)

Forum topic · 小凯 · 2026-05-08

Summary

This arXiv paper (2605.05192) by Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, and Haozhu Wang studies Carbery's proposed sharpened triangle inequality for sums of functions in L^p spaces. The authors first construct a counterexample showing that Carbery's inequality, with exponent c=2, fails for every p>2. They then prove that any estimate of this form requires the exponent to satisfy c<=p', and at the critical exponent c=p' they establish the inequality for all integers p>=2. In the second part, they derive a sharp three-function bound involving a parameter Gamma that quantifies the degree of orthogonality among the functions, with an optimal exponent c(p) that improves the previously known power of Carlen, Frank, and Lieb. Notably, some intermediate lemmas and inequalities were explored with assistance from the large language model Grok, making this a case study of AI-assisted mathematical research.

Paper Overview

Field: Mathematics (functional analysis / harmonic analysis)
Authors: Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, Haozhu Wang
Published: 2026-05-06
arXiv: 2605.05192

Background

Carbery proposed the following sharpened form of the triangle inequality for many functions: for any p >= 2 and any finite sequence (f_j)_j subset L^p,

||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)).

Main Results

1. Counterexample: The authors construct a counterexample showing that Carbery's inequality fails for every p > 2. 2. Upper bound on the exponent: If an estimate of the above form holds, the exponent must satisfy c <= p'. 3. Critical exponent: At the critical exponent c = p', the inequality is established for all integer values p >= 2. 4. Sharp three-function bound: In the second part of the paper, the authors obtain

||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) previously obtained by Carlen, Frank, and Lieb.

AI-Assisted Research

Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok — a notable example of LLMs contributing to research-level mathematics.

--- *Auto-collected on 2026-05-08.*