K. Jotsaroop, Saurabh Shrivastava, Kalachand Shuin
In this paper we establish weighted estimates for the bilinear Bochner-Riesz operator $\mathcal B^α$ at the critical index $α=n-\frac{1}{2}$ with respect to bilinear weights.
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We prove some new $L^p$ estimates for maximal Bochner-Riesz operator in the plane.
We improve the Bochner-Riesz conjecture in $\mathbb{R}^3$ to $\max\{p,p/(p-1)\}\geq3.25$.
We establish improved and sharp $L^p$ estimates for the maximal bilinear Bochner-Riesz means in all dimensions $n\geq 1$. This work extends the results proved by Jeong and Lee \cite{JL}. We also recover the known results for the bilinear Bochner-Riesz means. The method of proof involves a new decomposition of the bilinear Bochner-Riesz multiplier $(1-|ξ|^2-|η|^2)_+^α$ and delicate analysis in proving $L^p$ estimates for frequency localized square functions.
We prove that the bilinear maximal Bochner-Riesz operator $T_*^λ$ is bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ for appropriate $(p_1,p_2,p)$ when $λ>(4n+3)/5$.
We show that the recent techniques developed to study the Fourier restriction problem apply equally well to the Bochner-Riesz problem. This is achieved via applying a pseudo-conformal transformation and a two-parameter induction-on-scales argument. As a consequence, we improve the Bochner-Riesz problem to the best known range of the Fourier restriction problem in all high dimensions.
We use the very recent approach developed by Lacey in [23] and extended by Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector-valued inequalities are deduced.
$L^{p}$ estimates for multilinear maximal Bochner--Riesz means and square function
本文研究多线性极大Bochner–Riesz平均及其平方函数的$L^{p}$有界性，借鉴Jotsaroop和Shrivastava关于双线性Bochner–Riesz平均极大估计的方法完成证明。
On pointwise convergence of multilinear Bochner-Riesz means
本文改进了多线性Bochner-Riesz平均逐点收敛的指标范围，通过归纳建立k线性极大算子的L^p与加权估计（高维p<2/k时为新结果），并借助Stein平方函数变体及其多线性推广证明了该估计。
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.
The Bochner-Riesz multipliers $ B_{δ}$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< δ< \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ δ$ increases to the critical value, $ δ=\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.
Weighted decoupling estimates and the Bochner-Riesz means
本文证明了新的加权解耦估计，并据此改进了二维和三维空间中Bochner-Riesz均值对L^p函数（1<p<2）几乎处处收敛的充分条件。
We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honzík and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider $L^p\times L^q \to L^r$ estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a conn…
We consider generalized Bochner-Riesz multipliers of the form $(1-ρ(ξ))_+^λ$ where $ρ:\mathbb{R}^2\to\mathbb{R}$ belongs to a class of rough distance functions homogeneous with respect to a nonisotropic dilation group. We prove a critical $L^4$ estimate for the associated square function, which we use to derive multiplier theorems for multipliers of the form $m\circρ$ where $m:\mathbb{R}\to\mathbb{C}$.
A weighted decoupling inequality and its application to the maximal Bochner-Riesz problem
该研究证明了p=2n/(n-1)时的加权L^pℓ^p解耦估计，将其应用于极大Bochner-Riesz问题，得到三维空间中超出实插值指数的结果，并改进了平面情形的相关结论。
We study $L^p\times L^q\to L^r$ bounds for the bilinear Bochner-Riesz operator $\mathcal{B}^α$, $α>0$ in $\mathbb{R}^d,$ $d\ge2$, which is defined by \[ {\mathcal B}^α(f,g)=\iint_{\mathbb{R}^d\times\mathbb{R}^d} e^{2πi x\cdot(ξ+η)} (1-|ξ|^2-|η|^2 )^α_+ ~ \widehat{f}(ξ)\,\widehat{g}(η)\,dξdη.\] We make use of a decomposition which relates the estimates for $\mathcal{B}^α$ to those of the square function estimates for the classical Bochner-Riesz operators. In consequence, we significantly improve the previously known bounds.
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)
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。
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。
On almost everywhere convergence of planar Bochner-Riesz means
本文研究了在$5/3\leq p\leq 2$范围内，$L^p$函数的平面Bochner-Riesz均值几乎处处收敛性，通过建立与Bochner-Riesz乘法算子相关的最大算子的精确$L^{5/3}$估计来实现，该估计依赖于一个新的精细$L^2$估计。
The boundedness of Bochner-Riesz operators on the weighted weak Hardy spaces
设$w$是一个Muckenhoupt权，$WH^p_w(\mathbb R^n)$是加权弱Hardy空间.本文利用$WH^p_w(\mathbb R^n)$的原子分解，证明了当$0<p\le1$和$δ>n/p-{(n+1)}/2$满足一定条件时，极大Bochner-Riesz算子$T^δ_*$是从$WH^p_w(\mathbb R^n)$到$WL^p_w(\mathbb R^n)$的有界算子.此外，我们还证明了Bochner-Riesz算子$T^δ_R$在$WH^p_w(\mathbb R^n)$上对$0<p\le1$和$δ>n/p-{(n+1)}/2$有界.本文的结果即使对于非加权情形也是新的.
Let $G$ be a connected, simply connected, compact semisimple Lie group of dimension $n$. It has been shown by Clerc \cite{Clerc1974} that, for any $f\in L^1(G)$, the Bochner-Riesz mean $S_R^δ(f)$ converges almost everywhere to $f$, provided $δ>(n-1)/2$. In this paper, we show that, at the critical index $δ=(n-1)/2$, there exists an $f\in L^1(G)$ such that $$\limsup_{R\rightarrow\infty} \big|S_{R}^{(n-1)/2}(f)(x)\big|=\infty, \ \text{a.e.}\ x\in G.$$ This is an analogue of a well-known result of Kolmogorov \cite{Kolmogoroff1923} for Fourier series on the circle, and a result of Stein \cite{Stei…
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 $f\in L^p(\mathbb C^d)$.
In this paper we study maximal and square functions associated with bilinear Bochner-Riesz means at the critical index. In particular, we prove that they satisfy weighted estimates from $L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)$ for bilinear weights $(w_1,w_2)\in A_{\vec{P}}$ where $p_1,p_2>1$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Also, we show that both the operators fail to satisfy weak-type estimates at the end-point $(1,1,\frac{1}{2})$.
We consider the square function (known as Stein's square function) estimate associated with the Bochner-Riesz means. The previously known range of sharp estimate is improved. Our results are based on vector valued extensions of Bennett-Carbery-Tao's multilinear (adjoint) restriction estimate and adaption of induction argument due to Bourgain-Guth. Unlike the previous work by Bourgain-Guth on $L^p$ boundedness of Bochner-Riesz means in which oscillatory operators associated to the kernel had been studied, we take more direct approach by working on the Fourier transform side. This enables us to …
In this paper we introduce Stein's square function associated with bilinear Bochner-Riesz means and investigate its $L^p$ boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in~\cite{JS}. Another application concerns the $L^p$ estimates for bilinear fractional Schrödinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik~\cite{GHH} in the context of bili…
Let H be the Hermite operator -Δ+|x|^2 on \mathbb{R}^n. We prove a weighted L^2 estimate of the maximal commutator operator \sup_{R>0}|[b, S_R^λ(H)](f)|, where [b, S_R^λ(H)](f) = bS_R^λ(H) f - S_R^λ(H)(bf) is the commutator of a BMO function b and the Bochner-Riesz means S_R^λ(H) for the Hermite operator H. As an application, we obtain the almost everywhere convergence of [b, S_R^λ(H)](f) for large λand f\in L^p(\mathbb{R}^n).
For $ 0< λ< \frac{1}2$, let $ B_{λ}$ be the Bochner-Riesz multiplier of index $ λ$ on the plane. Associated to this multiplier is the critical index $1 < p_λ= \frac{4} {3+2 λ} < \frac{4}3$. We prove a sparse bound for $ B_{λ}$ with indices $ (p_λ, q)$, where $ p_λ' < q < 4$. This is a further quantification of the endpoint weak $L^{p_λ}$ boundedness of $ B_{λ}$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 \cap RH_{ρ}$, where $ ρ> \frac4 {4 - 3 p_{λ}}$.
Multi-frequency Calderon-Zygmund analysis and connexion to Bochner-Riesz multipliers
在这项工作中，我们描述了在埃斯科里亚尔2012年会议上的演讲中展示的几个结果。我们的目标是继续发展[9]中介绍的多频Calderon-Zygmund分析。定义了广义多频Calderon-Zygmund算子。使用[9]的相应多频率分解获得未加权估计。对于一类新的极大sharp函数，得到了加权估计。
A new estimate for Bochner-Riesz operators at the critical index on the weighted Hardy spaces
设$w$是Muckenhoupt权，$H^p_w(\mathbb R^n)$是加权Hardy空间.本文利用$H^p_w(\mathbb R^n)$的原子分解，证明了当$0<p<1$和$δ=n/p-(n+1)/2$为加权弱Hardy空间时，Bochner-Riesz算子$T^δ_R$从$H^p_w(\mathbb R^n)$到加权弱Hardy空间$WH^p_w(\mathbb R^n)$是有界的.这个结果即使在未加权的情况下也是新的。
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…
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。
The Bochner-Riesz means are shown to have either the unit interval $[0,1]$ or the whole complex plane as their spectra on $L^p, 1\le p<\infty$
The bilinear Bochner-Riesz problem
受傅里叶级数乘积的球面可和性问题的启发，我们研究了双线性Bochner-Riesz乘子$(1-|ξ|^2-|η|^2)^δ_+$的有界性，并在这方面取得了一些进展.我们得到了关于这些平均值从$L^2\times L^2 $到具有最小光滑度的$L^1$的有界性的最佳结果，即，任何$δ>0$，我们得到的估计为其他对空间的较大值的$δ$。我们的研究是广泛的，足以涵盖一般的双线性乘数$m(ξ,η)$径向$ξ$和$η$具有最小的光滑度，测量在索伯列夫空间规范。我们的结果是基于各种技术，包括傅里叶级数扩展，正交性，双线性限制和扩展定理。
Bochner-Riesz mean for the twisted Laplacian in $\mathbb R^2$
本文研究了 $\mathbb R^2$ 上扭曲 Laplacian 的 Bochner-Riesz 问题，通过在 $S_λ^δ(\mathcal L)$ 上获得直至锐利可和性指标的均匀 $L^p$ 界，证明了关于 $L^p$ 收敛条件的猜想。
On approximation methods generated by generalized Bochner-Riesz kernels
得到了与广义Bochner-里斯核生成的算子族和平均收敛性有关的一些尖锐结果。这些方法通过$K$ -功能（或其实现的情况下，空间$L_p$，$0<p<1$）的近似函数的确切顺序推导。
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 $L^p([a,b],ω)$.
We obtain weak-type $(p, p)$ endpoint bounds for Bochner-Riesz means for the Hermite operator $H=-Δ+|x|^2$ in ${\mathbb R}^n, n\geq 2$ and for other related operators for $1\leq p\leq 2n/(n+2)$, extending earlier results of Thangavelu and of Karadzhov.
Bochner-Riesz means at the critical index: Weighted and sparse bounds
本文研究临界指标下Bochner-Riesz平均在加权L^p空间上的稀疏界，推广了Vargas弱型不等式，并在二维获得几乎最优结果，对特定指标证明完全最优稀疏界。
We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.
Bochner-Riesz commutators on Métivier groups: boundedness and compactness
本文针对Métivier群，证明了其与次拉普拉斯算子相关的Bochner-Riesz交换子的有界性与紧性，且该交换子的光滑性参数可由拓扑维数表示。
Bochner-Riesz means on the Heisenberg group
本文证明了 Heisenberg 群上子拉普拉斯算子谱分解的 Bochner-Riesz 平均的 L^p 有界性，范围扩展到 1≤p≤p_n，其中 p_n→2，通过建立 p-敏感的谱乘子定理实现。
A proof of the Bochner-Riesz conjecture
对于$f\in {\frak S}({\Bbb R}^d)$，我们考虑由$$\hat {{\frak R}^δf}(ξ)=(1-|ξ|^2)^δ_+ \hat f (ξ).$$定义的指数为$δ>0$的Bochner-Riesz算子${\frak R}^δ$.然后证明了Bochner-Riesz猜想：如果$δ>\max\{d|1/p-1/2|-1/2,0\}$和$p>1$，则${\frak R}^δ$是从$L^p({\Bbb R}^d)$到$L^p({\Bbb R}^d)$的有界算子;而且，如果$δ(p)=d(1/p-1/2)-1/2$和$1<p<2d/(d+1)$，则${\frak R}^{δ(p)}$是从$L^p({\Bbb R}^d)$到$L^{p,\infty}({\Bbb R}^d)$的有界算子.
Bochner-Riesz summability for analytic functions on the m-complex unit sphere and for cylindrically symmetric functions on R^{n-1} times R
本文证明了Laplace-Beltrami算子在m复单位球面E_{Delta_{S^{2 m-1}（[0，R））上的谱投影作为算子从H^p（S^{2 m-1}）到L^p（S^{2 m-1}）是一致有界的，其中p>（1，无穷大）.我们还证明了Bochner-Riesz猜想在限制于R^{n-1} × R上的柱对称函数时成立。
Fourier-Wigner multipliers and the Bochner-Riesz conjecture for Schatten class operators
本文引入Schatten类算子上的Fourier-Wigner乘子，利用秩一算子卷积和Werner-Young不等式建立相关结果，并证明紧支撑Fourier乘子的等价关系，从而用Schatten类算子重新表述Bochner-Riesz猜想。
Weak type estimates of the maximal quasiradial Bochner-Riesz operator on certain Hardy spaces
令$\{A_t\}_{t>0}$是由无穷小生成器$M$生成的${\Bbb R}^n$中的膨胀群，其中$A_t=\exp(M\log t)$，并且令$\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$是定义在${\Bbb R}^n$上的$A_t$ -齐次距离函数。对于$f\in {\frak S}({\Bbb R}^n)$，我们定义了指数为$δ>0$的极大拟径向Bochner-Riesz算子${\frak M}^δ_{\varrho}$为$${\frak M}^δ_{\varrho} f(x)=\sup_{t>0}|{\Cal F}^{-1}[(1-\varrho/t)_+^δ\hat f ](x)|.$$.如果$A_t=t I$和$\{ξ\in {\Bbb R}^n| \varrho(ξ)=1\}$是有限型光滑凸超曲面，那么当$δ=n(1/p-1/2)-1/2$和$0<p<1$时，我们用一种非常简单的方法证明了${\frak M}^δ_{\varrho}$是定义在$H^p({\Bbb R}^n)$上的，并且是从$H^p({\Bbb R}^n)$到$L^{p,\infty}({\Bbb R}^n)$的有界算子.如果$A_t=t I$和$\varrho\in C^{\infty}({\Bbb R}^n\setminus\{0\})$，我们还…
We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Boc…
A note on almost everywhere convergence of Bochner-Riesz operator
该研究针对Bochner-Riesz算子的几乎处处收敛问题，通过将卷积解释为反常积分给出算子的替代定义，证明了在$2\le p\le \infty$时收敛成立的充要条件为可和性指数$δ\ge δ(d,p)$。
We prove the existence of $L^2$ solutions to the Helmholtz equation $(-Δ- 1)u = f$ in ${\mathbb R}^n$ assuming the given data $f$ belongs to $L^{(2n+2)/(n+5)}({\mathbb R}^n)$ and satisfies the "Fredholm condition" that $\hat{f}$ vanishes on the unit sphere. This problem, and similar results for the perturbed Helmholtz equation $(-Δ-1)u = -Vu + f$, are connected to the Limiting Absorption Principle for Schrödinger operators. The same techniques are then used to prove that a wide range of $L^p \mapsto L^q$ bounds for Bochner-Riesz multipliers are improved if one considers their action on the clo…
Bilinear Bochner-Riesz Means on the Complex Sphere
本文证明复球上双线性Bochner-Riesz平均在特定指数范围及光滑性参数下的$L^p$有界性，并推导了若干具有独立价值的解析估计。
Eigenfunction and Bochner Riesz estimates on manifolds with boundary
本文的目的是给出具有边界$\partial M$的维数为$n\ge 2$的光滑紧致黎曼流形$(M,g)$上的Dirichlet Laplacian的特征函数的$L^\infty$估计的一个简单证明，然后利用这些估计证明了在这种情况下Bochner Riesz平均的新估计.我们的本征函数估计涉及估计的$L^2\to L^\infty$映射属性的运营商$χ_λ$项目到单位频带的频谱的平方根的拉普拉斯算子。这些结果推广了D. Grieser的个别特征函数。我们使用Grieser的一个想法，即使用Seeley，Pham The Lai和Hörmander的早期估计来估计远离边界的情况，然后使用最大值原理证明边界附近的估计。我们对Bochner Riesz估计的证明与C的欧几里德情形的一个论证有关。一个是无边界黎曼情形，一个是作者提出的。这里，与无边界黎曼情形不同，我们不能使用波动方程的参数，而是依赖于一个更经济的论证，只使用波动方程的本征函数估计和有限传播速度。
Square functions and maximal operators associated with radial Fourier multipliers
我们首先概述了Eli Stein介绍的球面和Bochner-Riesz平均的平方函数，并讨论了它们对径向乘数和相关极大函数的影响。然后，我们证明新的端点估计这些平方函数，最大的Bochner-Riesz算子，和更一般的类的径向傅立叶乘数。
We prove that bilinear forms associated to the rough homogeneous singular integrals $T_Ω$ on $\mathbb R^d$, where the angular part $Ω\in L^q (S^{d-1})$ has vanishing average and $1<q\leq \infty$, and to Bochner-Riesz means at the critical index in $\mathbb R^d$ are dominated by sparse forms involving $(1,p)$ averages. This domination is stronger than the weak-$L^1$ estimates for $T_Ω$ and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative $A_p$-weighted estimates for Bochner-Riesz means and for homog…
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…
Weak type estimates for Bochner--Riesz means on Hardy-type spaces associated with ball quasi-Banach function spaces
本文在球拟巴拿赫函数空间关联的哈代型空间框架下，建立了 Bochner–Riesz 平均及其极大算子的弱型有界性，并应用于加权勒贝格、Herz、洛伦兹、变指标勒贝格及 Morrey 空间。
$(L_p, L_q)$ estimates of potentials with oscillating kernel
对振荡势$(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-α}}f(x-y)dy$、$0<α<n$、$n\geq 2$进行了$(L_p, L_q)$估计，其符号在单位球面上具有奇异性.这些势是在傅里叶分析和偏微分方程中著名的Bochner-Riesz算子和Helmholtz势的自然修正。对于$α$的某些值，确定相应的对$(L_p, L_q)$代表一个开放的问题。对于$α$，问题是开放的，其范围与Bochner-Riesz平均值相同。
Estimates at or beyond endpoint in harmonic analysis: Bochner-Riesz means and spherical means
我们引入了一些新的函数空间来研究端点处或端点以外的一些问题。首先证明了Bochner-Riesz平均$B_R^λ$从$L^p_{|x|^α}$的某些子空间到$L^p_{|x|^α}$（对于$ \frac{n-1}{2(n+1)}<λ\leq \frac{n-1}{2}, 0 < p\leq p'_λ=\frac{2n}{n+1+2λ}, n(\frac{p}{p_λ}-1)< α<n(\frac{p}{p'_λ}-1)$和$0<R<\infty,$）是有界的，并且证明了极大Bochner-Riesz平均$B_*^λ$（对于$ \frac{n-1}{2}\leq λ< \infty, 0 < p\leq 1$和$-n< α<n(p-1)$）也是有界的.由此我们得到了$λ,p,$和$α$的$B_R^λ$的$L^p_{|x|^α}$ -范数收敛性质。其次，设$n\geq 3,$，我们证明了对于$0<p\leq \frac{n}{n-1}$和$ -n(1-\frac{p}{2})<α<n(p-1)-n$，从$L^p_{|x|^α}$的某些子空间到$L^p_{|x|^α}$的极大球面平均是有界的。我们还得到了一个$L^p_{|x|^α}$ -范数收敛性质的球形平均，这样的$p$和$α$。最后，证明了一些新的$|x|^α$加权估计在端点处或端点外成立，如Hardy-Littlewood极大算子…
Boundedness of bilinear radial Fourier multipliers
本文研究了双线性径向傅里叶乘法子的有界性条件，并通过光滑性条件证明其在L^2到L^1的有界性，同时给出了多线性Bochner-Riesz算子的另一种证明及广义双线性Bochner-Riesz算子的估计。
Linear dimension-free estimates for the Hermite-Riesz transforms
利用Bellman函数技巧证明了Hermite算子的一个双线性无量纲不等式。贝尔曼技术在这里适用于一个非本地运营商，这在第一次似乎是不可行的。作为我们的双线性不等式的一个结果，我们证明了L^p上的Riesz-Hermite变换的无量纲有界性，其线性增长是关于p的。证明的一个特征是建立L^p（R^n）的定理。给出了一类谱乘子的估计，其界独立于n和p。与海森堡群上的已知结果以及沿抛物线的希尔伯特变换的结果的联系也被给出探讨了我们相信，我们的方法是相当普遍的意义上说，它可以适用于整个范围内的里斯变换所产生的各种微分算子。作为实现这一目标的第一步，我们证明了我们的无量纲双线性嵌入定理相当一般的家庭薛定谔半群。
It was recently conjectured that every system of exceptional orthogonal polynomials is related to classical orthogonal polynomials by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator…
Results analogous to those proved by Rubio de Francia are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results by reducing the dimension restriction from $n\ge 8$ to $n\ge 4$.
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