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

Abstract

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∞(1−2k+nR2)+λPkf(x). Here Pkf is the k-th Hermite spectral projection operator. For 2≤p<∞, we prove thatlimR→∞⁡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.