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

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 S_R^λ(H)f(x)=∑k=0∞(1−[Formula presented])₊^λP_kf(x). Here P_kf is the k-th Hermite spectral projection operator. For 2≤p<∞, we prove that limR→∞⁡S_R^λ(H)f=fa.e. for all f∈L^p(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∈L^p(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 S_R^λ(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∈L^p(R) with p≥2 whenever λ>0. Compared with the classical result due to Askey and Wainger who showed the optimal L^p convergence for S_R^λ(H) on R we only need smaller summability index for a.e. convergence.