Sharpening the gap between L¹ and L² norms

2024/07/05 by Paata Ivanisvili, Ivanisvili, Paata, Yonathan Stone +1
Mathematics · #39B62 #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Numerical methods in inverse problems #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2407.04835

openalex publication_date 2024/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We refine the classical Cauchy--Schwartz inequality ‖X‖₁ ≤ ‖X‖₂ by demonstrating that for any p and q with q>p>2, there exists a constant C=C(p,q) such that ‖X‖₁ ≤ 1 - C (‖X‖_p^p - 1)^(q-2)/(q-p)(‖X‖_q^q - 1)^(2-p)/(q-p) holds true for all Borel measurable random variables X with ‖X‖₂=1 and ‖X‖_p<∞. We illustrate two applications of this result: one for biased Rademacher sums and another for exponential sums.

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