Almost everywhere convergence of Bochner-Riesz means for the Hermite type Laguerre expansions

2025/06/20 by Wei, Longben, Duan, Zhiwen
Mathematics · #35P10 #42B15 #42B25 #42C10 #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Inequalities and Applications

paper · pdf · doi:10.48550/arxiv.2506.16958

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the space ℝ₊^d=(0,∞)^d equipped with Euclidean distance and the Lebesgue measure. For every α=(α₁,...,α_d)∈[-1/2,∞)^d, we consider the Hermite-Laguerre operator L^α=-Δ+\arrowvert x\arrowvert²+∑_i=1^d(α_j²-(1)/(4))(1)/(x_i²). In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with L^α which is defined as S_R^λ(L^α)f(x)=∑_n=0^∞(1-(4n+2\arrowvertα\arrowvert₁+2d)/(R²))₊^λP_nf(x). Here P_nf(x) is the n-th Laguerre spectral projection operator and \arrowvertα\arrowvert₁ denotes ∑_i=1^dα_i. For 2≤ p<∞, we prove that lim_{R → ∞} S_R^λ(L^α)f = f a.e. for all f∈ L^p(ℝ₊^d) provided that λ>λ(p)/2 and λ(p)=max\d(1/2-1/p)-1/2,0\. Conversely, we show that the convergence generally fails if λ<λ(p)/2 in the sense that there exists f∈ L^p(ℝ₊^d) for 2d/(d-1)< p such that the convergence fails.

Citations

• Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian
• Endpoint eigenfunction bounds for the Hermite operator
• Bochner-Riesz means for the Hermite and special Hermite expansions
• Almost everywhere convergence of Bochner-Riesz means for the Hermite operators
• New estimates of maximal Bochner-Riesz operator in the plane
• Bounds on the maximal Bochner-Riesz means for elliptic operators
• Sharp estimates for oscillatory integral operators via polynomial\n partitioning
• Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means
• Bounds on oscillatory integral operators based on multilinear estimates
• Restriction and spectral multiplier theorems on asymptotically conic\n manifolds
• Sobolev spaces associated to the harmonic oscillator
• Recent progress on the restriction conjecture
• Riesz transform, Gaussian bounds and the method of wave equation
• A bilinear approach to the restriction and Kakeya conjectures
• Summability of Hermite expansions. I
• Summability of Hermite expansions. II

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• On almost everywhere convergence of planar Bochner-Riesz means