Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Peng Chen*, Xuan Thinh Duong, Danqing He, Sanghyuk Lee, Lixin Yan

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review


    Let H=−Δ+|x|2 be the Hermite operator in Rn. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with H which is defined by SRλ(H)f(x)=∑k=0∞(1−[Formula presented])+λPkf(x). Here Pkf is the k-th Hermite spectral projection operator. For 2≤p<∞, we prove that limR→∞⁡SRλ(H)f=fa.e. for all f∈Lp(Rn) provided that λ>λ(p)/2 and λ(p)=max⁡{n(1/2−1/p)−1/2,0}. Conversely, we also show the convergence generally fails if λ<λ(p)/2 in the sense that there is an f∈Lp(Rn) for 2n/(n−1)≤p such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For n≥2 and p≥2 our result tells that the critical summability index for a.e. convergence for SRλ(H) is as small as only the half of the critical index for a.e. convergence of the classical Bochner-Riesz means. When n=1, we show a.e. convergence holds for f∈Lp(R) with p≥2 whenever λ>0. Compared with the classical result due to Askey and Wainger who showed the optimal Lp convergence for SRλ(H) on R we only need smaller summability index for a.e. convergence.

    Original languageEnglish
    Article number108042
    Pages (from-to)1-42
    Number of pages42
    JournalAdvances in Mathematics
    Publication statusPublished - 3 Dec 2021


    • Almost everywhere convergence
    • Bochner-Riesz means
    • Hermite operator
    • Trace lemma


    Dive into the research topics of 'Almost everywhere convergence of Bochner-Riesz means for the Hermite operators'. Together they form a unique fingerprint.

    Cite this