Bounds on the Hermite spectral projection operator

2022/10/07 by Eun-Hee Jeong, Sanghyuk Lee, Jeong, Eunhee +3
Mathematics · #42B99 (Primary) 42C10 (Secondary) #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2210.03385

openalex publication_date 2022/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study L^p-L^q bounds on the spectral projection operator Π_λ associated to the Hermite operator H=|x|²-Δ in \mathbb R^d. We are mainly concerned with a localized operator χ_EΠ_λχ_E for a subset E⊂\mathbb R^d and undertake the task of characterizing the sharp L^p--L^q bounds. We obtain sharp bounds in extended ranges of p,q. First, we provide a complete characterization of the sharp L^p--L^q bounds when E is away from √λ\mathbb S^d-1. Secondly, we obtain the sharp bounds as the set E gets close to √λ\mathbb S^d-1. Thirdly, we extend the range of p,q for which the operator Π_λ is uniformly bounded from L^p(\mathbb R^d) to L^q(\mathbb R^d).

Related

• Hermite spectral projection operator
• Sharp local L^p estimates for the Hermite eigenfunctions
• Endpoint eigenfunction bounds for the Hermite operator
• The Endpoint Eigenfunction Bound for the Hermite Operator in Two Dimensions
• Sharp L^p-L^q estimates for the spherical harmonic projection