Bochner积分与函数值凸组合之差的范数估计

平面中与粗糙凸区域相关的 Bochner–Riesz 算子的改进 $L^p$ 界

Improved $L^p$ bounds for Bochner--Riesz operators associated with rough convex domains in the plane

本文利用 Bose–Chowla 的大 $B_m$ 集与 Bourgain 的大 $\Lambda(p)$ 集构造平面凸区域，改进了广义 Bochner–Riesz 算子的 $L^p$ 估计。

New $L^p$ bounds for Bochner-Riesz multipliers associated with convex planar domains with rough boundary

We consider generalized Bochner-Riesz multipliers of the form $(1-ρ(ξ))_+^λ$ where $ρ(ξ)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sjölin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the "additive energy" of the boundary of a convex domain. Second, we associate a set of direction…

凸域上的双线性Bochner-Riesz平均与Kakeya极大函数

Bilinear Bochner-Riesz means for convex domains and Kakeya Maximal function

该研究引入平面凸域相关的双线性Bochner-Riesz平均，借助双线性Kakeya极大函数建立其$L^p$估计以证明有界性，并揭示该极大函数与Lacey双线性极大函数的联系。

关于Bochner-Riesz平均的Stein加权$L^2$猜想的$C^k$凸变体的研究

On $C^k$ convex variants of Stein's Weighted $L^2$ Conjecture for Bochner-Riesz means

本文证明了Stein加权$L^2$猜想在$C^k$凸边界乘子变体上的幂损失结果，并关联Mizohata-Takeuchi反例，讨论了端点对数爆炸。

A Characterization of Norm Compactness in the Bochner Space $L^p (G ; B)$ For an Arbitrary Locally Compact Group $G$

In this paper, we generalize a result of N. Dinculeanu which characterizes norm compactness in the Bochner space $L^p(G ; B)$ in terms of an approximate identity and translation operators, where $G$ is a locally compact abelian group and $B$ is a Banach space. Our characterization includes the case where $G$ is nonabelian, and we weaken the hypotheses on the approximate identity used, providing new results even for the case $B = \mathbb{C}$ and $G = \mathbb{R}^n.$

Weak type estimates of the maximal quasiradial Bochner-Riesz operator on certain Hardy spaces

Let $\{A_t\}_{t>0}$ be the dilation group in ${\Bbb R}^n$ generated by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let $\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$ be a $A_t$-homogeneous distance function defined on ${\Bbb R}^n$. For $f\in {\frak S}({\Bbb R}^n)$, we define the maximal quasiradial Bochner-Riesz operator ${\frak M}^δ_{\varrho}$ of index $δ>0$ by $${\frak M}^δ_{\varrho} f(x)=\sup_{t>0}|{\Cal F}^{-1}[(1-\varrho/t)_+^δ\hat f ](x)|.$$ If $A_t=t I$ and $\{ξ\in {\Bbb R}^n| \varrho(ξ)=1\}$ is a smooth convex hypersurface of finite type, then we prove in an extre…