论文标题
Bochner-Riesz几乎到处都是Hermite操作员的意思
Almost everywhere convergence of Bochner-Riesz means for the Hermite operators
论文作者
论文摘要
令$ h =-Δ+ | x |^2 $为$ {\ mathbb r}^n $中的Hermite运算符。在本文中，我们几乎在Bochner-riesz的融合均与$ h $相关的融合的任何地方研究，$ h $由$ s_r^λ（h）f（x）= \ sum \ limits_ {k = 0}^{\ infty} \ big big（1- {2k+n \ forr r^2} \ big big big） $ k $ - th hermite光谱投影操作员。 $ 2 \ le P <\ infty $，我们证明了$$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n \ big（{1/2} - {1/p} \ big） - {1/2}，\，0 \，0 \ big \}。$相反，我们还显示融合通常会失败，如果$λ<λ（p）/2 $在l^p（p（p）中有$ f \ f^p（\ mathbb r^n）$ 2n/e $ $ 2n/（失败。与A.E.形成鲜明对比的是经典的Bochner-Riesz的收敛意思是拉普拉斯人。对于$ n \ geq 2 $和$ p \ ge 2 $，我们的结果告诉A.E.的关键总和指数。 $ s_r^λ（h）$的收敛性仅为A.E.关键索引的\ emph {falf {falf {falcemph {一半}。经典Bochner-Riesz的收敛意思。当$ n = 1 $时，我们显示A.E. Convergence在l^p（{\ Mathbb r}）中的$ f \ contrance保留，每当$λ> 0 $时，$ p \ geq 2 $。与由于Askey和Wainger的经典结果相比，在$ {\ Mathbb r}上显示了$ s_r^λ（h）$的最佳$ l^p $收敛，我们只需要a.e.收敛。
Let $H = -Δ+ |x|^2$ 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 $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $λ< λ(p)/2$ 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 $S_R^λ(H)$ 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 $λ>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^λ(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.
