Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Almost everywhere convergence of Bochner-Riesz means for the Hermite
  operators   Let $H = -\Delta + |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this
paper we study almost everywhere convergence of the Bochner-Riesz means
associated with $H$ which is defined by $S_R^{\lambda}(H)f(x) =
\sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^{\lambda} P_k f(x).$
Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le
p<\infty$, we prove that $$
  \lim\limits_{R\to \infty} S_R^{\lambda}(H) f=f \ \ \ \text{a.e.} $$ for all
$f\in L^p(\mathbb R^n)$ provided that $\lambda> \lambda(p)/2$ and
$\lambda(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely,
we also show the convergence generally fails if $\lambda< \lambda(p)/2$ in the
sense that there is an $f\in L^p(\mathbb R^n)$ for $2n/(n-1)\le p$ such that
the convergence fails. This is in surprising contrast with a.e. convergence of
the classical Bochner-Riesz means for the Laplacian. For $n\geq 2$ and $p\ge 2$
our result tells that the critical summability index for a.e. convergence for
$S_R^{\lambda}(H)$ is as small as only the \emph{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 $f\in L^p({\mathbb R})$ with $ p\geq 2$
whenever $\lambda>0$. Compared with the classical result due to Askey and
Wainger who showed the optimal $L^p$ convergence for $S_R^{\lambda}(H)$ on
${\mathbb R}$ we only need smaller summability index for a.e. convergence.

21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga