Bochner-Riesz summability for analytic functions on the m-complex unit sphere and for cylindrically symmetric functions on R^{n-1} times R

Bochner-Riesz means for the Hermite and special Hermite expansions

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of $p$ and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This invalidates the conventional conjecture which was expected to be true.

Bochner-Riesz Means Convergence of Prolate Spheroidal Series and Their Extensions

In this paper, we study the $L^p$-Bochner-Riesz mean summability problem related to the spectrum of some particular Sturm-Liouville operators in the weighted $L^p([a,b],ω).$ Our purpose is to establish suitable conditions under which the Bochner-Riesz expansion of a function $f \in L^p([a,b],ω)$,$1<p<\infty$, in two generalisations of Slepian's basis, converges to $f$ in $L^p([a,b],ω)$.

Almost everywhere convergence of Bochner--Riesz means for the twisted Laplacian

Let $\mathcal L$ denote the twisted Laplacian in $\mathbb C^d$. We study almost everywhere convergence of the Bochner--Riesz mean $S^δ_{t}(\mathcal L) f$ of $f\in L^p(\mathbb C^d)$ as $t\to \infty$, which is an expansion of $f$ in the special Hermite functions. For $2\le p\le \infty$, we obtain the sharp range of the summability indices $δ$ for which the convergence of $S^δ_{t}(\mathcal L) f$ holds for all $f\in L^p(\mathbb C^d)$.

L^p有界性与Bochner-Riesz算子

L^p-boundedness of the Bochner-Riesz operator

本文提出新的Bochner-Riesz求和方法，证明当0<Reδ<1/2时，Bochner-Riesz算子S^δ在L^p(R^n)上有界，其中(n-1)/(2n)≤1/p≤(n+1)/(2n)

The bilinear Bochner-Riesz problem

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|ξ|^2-|η|^2)^δ_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $δ>0$, and we obtain estimates for other pairs of spaces for larger values of $δ$. Our study is broad enough to encompass general bilinear multipliers $m(ξ,η)$ radial in $ξ$ and $η$ with minimal smoothness, measured in Sobolev space norms. Our…

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

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 gen…