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Abstract
Let H=−Δ+x^{2} be the Hermite operator in R^{n}. In this paper we study almost everywhere convergence of the BochnerRiesz means associated with H which is defined by S_{R}^{λ}(H)f(x)=∑k=0∞(1−[Formula presented])_{+}^{λ}P_{k}f(x). Here P_{k}f is the kth Hermite spectral projection operator. For 2≤p<∞, we prove that limR→∞S_{R}^{λ}(H)f=fa.e. for all f∈L^{p}(R^{n}) 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}(R^{n}) for 2n/(n−1)≤p such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical BochnerRiesz 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 BochnerRiesz 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.
Original language  English 

Article number  108042 
Pages (fromto)  142 
Number of pages  42 
Journal  Advances in Mathematics 
Volume  392 
DOIs  
Publication status  Published  3 Dec 2021 
Keywords
 Almost everywhere convergence
 BochnerRiesz means
 Hermite operator
 Trace lemma
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 1 Finished

Harmonic analysis: function spaces and partial differential equations
Duong, X., Hofmann, S., Ouhabaz, E. M. & Wick, B.
11/02/19 → 10/02/22
Project: Other