【04月28日-05月1日】调和分析春季会议

主办单位：中国科学技术大学数学科学学院

组委会：陈杰诚，李嘉禹，麻希南

会议支持：科技部重点研发项目

会议地点：高速徽风皖韵酒店

会议指南：

 1、会议时间：2023年4月28日-5月1日，4月28日报到，5月1日离会

 2、乘车路线：合肥南站/合肥站，新桥机场 出租车到高速徽风皖韵酒店

会议日程

4月28日 全天报到

4月29日

  4月30日

5月1日 全日离会

报告安排

颜立新（中山大学）

Title: Harmonic analysis meets wave equation

Abstract: In this talk I plan to survey some recent progress on  Hardy spaces, Riesz transforms, Bochner-Riesz means and spherical means by using the method of wave equation, and show  interesting connections and interaction of different fields such as harmonic analysis, functional analysis and PDE.

洪桂祥（哈尔滨工业大学）

Title：John-Nirenberg inequalities for noncommutative martingale BMO spaces

Abstract：Recently, we have maken some progresses on John-Nirenberg theorems for BMO/Lipschitz spaces in the noncommutative martingale setting. As conjectured from the classical case, a desired noncommutative ``stopping time argument was discovered to obtain the distribution function inequality form of John-Nirenberg theorem. This not only provides another approach without using duality and interpolation to the results for spaces $\rm{bmo}^c(\mathcal M)$ and ${{\Lambda}^{{c}}_{\beta}}(\mathcal{M})$, but also allows us to find the desired version of John-Nirenberg inequalities for spaces $\mathcal{BMO}^c(\mathcal M)$ and ${\mathcal L^{{c}}_{\beta}}(\mathcal{M})$. And thus we solve two open questions. As an application, we show that Lipschitz space is also the dual space of noncommutative Hardy space defined via symmetric atoms. Finally, our results for ${\mathcal L^{{c}}_{\beta}}(\mathcal{M})$ as well as the approach seem new even going back to the classical setting. Based on joint works with Tao Mei, Chongbian Ma, Yu Wang.

蒋仁进（首都师范大学）

Title: Riesz transform and elliptic equations on exterior domains

Abstract: It has been open for some years what is the behavior of Riesz transform for Dirichlet Laplacian and Neumann Laplacian on exterior domains. In this report, we provide a complete characterization of boundedness of Riesz transform in order to solve the problem. As applications, we provide solutions to well-posedness of elliptic equations for both Dirichlet and Neumann boundary value problem on exterior domains, in the sharp range. The results also confirm some conjectures of Hassell-Sikora IUMJ 2009 and Auscher-Tchamitchian Math. Ann. 2001. This is a joint work with professor F.H. Lin.

李中凯（上海师范大学）

Title：Hardy spaces, area integrals and maximal functions associated with Dunkl operators

Abstract：Associated to the finite reflection group $G$ generated by a root system $R$ on the Euclidean space ${\mathbb{R}}^d$, C. F. Dunkl introduced a class of commuting differential-reflection operators involving a given multiplicity function $\kappa$. In this talk I shall present our recent researches related to the Dunkl operators, mainly about the associated Hardy spaces, Lusin-type area integrals and the non-tangential maximal function on the upper half-space ${\mathbb{R}}_+^{d+1}$, jointly with Dr. Jiaxi Jiu and Dr. Jianquan Liao.

沈益（浙江理工大学）

Title：An Open Problem on Sparse Representations in Unions of Bases

Abstract：We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of sparks are generally combinatorial. For unions of several orthonormal bases, two lower bounds on the spark via the mutual coherence were established in previous work. We constructively prove that both of them are tight. Our main results give positive answers to Gribonval and Nielsen's open problem on sparse representations in unions of orthonormal bases. Constructive proofs rely on a family of mutual unbiased bases which first appears in quantum information theory.  It is joint work with Prof. Song Li, Dr. Yuan Shen and Chenyun Yu.

刘博辰（南方科技大学）

Title: Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures

Abstract: We obtain new mixed-norm estimates and geometric results on projections. In the proof we introduce a new quantity called s-amplitude, and interpolate analytically not only on p,q, but also on dimensions of measures. This mechanism provides new perspectives on operators with measures, thus has its own interest.

张城（清华大学）

报告题目: Eigenfunctions of Schrodinger operators with singular potentials

摘要：We will introduce some recent progress on the eigenfunctions of Schrodinger operators with singular potentials on compact manifolds with/without boundary.

燕敦验（中国科学院大学）

报告题目：Sharp Constant for Generalized Hardy-Littlewood-Sobolev Inequality

摘要：In this talk, we investigate some necessary and sufficient conditions which ensure the validity of the k-Fold Beta integral formula. That is, the k-Fold Beta integral equation is as follows $$\int_{\mathbb{R}^n}\prod\limits^k_{i=1}|x^i-t|^{-d_i}dt=C_{d_1,\cdots,d_k,n} \prod\limits_{1\le i<j\le k}|x^i-x^j|^{-\alpha_{ij}},$$ where $x^{i}\in \mathbb{R}^n$ $d_i$ is nonzero real number, with $i=1,\cdots,k$. Actually, we completely answer the question raised by Grafakos. For some cases, we will prove that the constant number $C_{d_1,\cdots,d_k,n}$ is just the sharp bound of the following generalized Hardy-Littlewood -Sobolev inequality: $$|\Lambda(f_1,\cdots,f_k)|\leg C_{d_1,\cdots,d_k,n}\prod\limits^k_{i=1}\|f\|_{p_i}. $$

陈艳萍（北京科技大学）

Title: Jump and variational inequalities for some operators with rough kernel

Abstract：The jump and variational inequalities have been the subject of many recent articles in probability, ergodic theory and harmonic analysis. We proved a weighted jump/ variational inequalities, weak type boundedness for singular integral operators and average operators with rough kernels. 

陈鹏（中山大学）

Title: Carleson's decomposition of functions in $BMO_L$

Abstract: Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Suppose $ f$ is in the space  $ {\rm BMO}_L(X)$ associated with the operator $L$ and has compact support. We show that there exist $g\in L^\infty(X)$ and a finite Carleson measure $\sigma$ such that $$ f(x)=g(x)+\int_{X\times [0,\infty)} K_{e^{-t^2L}}(x,y)d\sigma(y,t) $$ with $\|g\|_{L^\infty}+\|\sigma\|_{\mathcal C}\leq C\|f\|_{BMO_L}$. This extends the result for the classical John-Nirenberg BMO space by Carleson (1976) (see also Garnett and Jones (1982), Uchiyama (1980) and Wilson (1988)) to the BMO setting associated with operators.

薛庆营（北京师范大学）

Title：A class of multilinear bounded oscillation operators on measure spaces and applications

Abstrat: 近年来，由于二进分析、稀疏控制的广泛应用，多线性算子理论得以迅速发展，特别是加权理论、外推定理和紧性刻画方面。 本报告主要介绍我们最近的一些工作， 涉及在一般测度空间上的一类Banach值多线性有界振荡算子，我们首先发展了一套加权理论，该理论将经典多线性Calderon-Zygmund算子与一些之外的算子都包括在内；其次建立了三种典型的估计：局部指数衰减估计、混合弱型估计和加权范数不等式；另外基于抽象多线性紧算子的外推性质，我们获得了齐次型空间上多线性算子交换子的加权紧性外推定理。

该工作是和M. Cao，G. Iba\~{n}ez-Firnkorn，I.P. Rivera-R\'{i}os，K. Yabuta合作完成的。

宋亮（中山大学）

Title:  Fractional Leibniz rule in Hardy space and its application to energy-critical nonlinear Schrodinger equations

Abstract: We prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrodinger equations assuming data in $L^1\cap H^3$.  The main ingredients are the boundness of the Schr\odinger propagators in Hardy space due to Miyachi  and a fractional Leibniz rule in the Hardy space. We also extend the  fractional chain rule to the Hardy space. This a joint work with Zihua Guo and Chunyan Huang.

李康伟（天津大学）

Title: Singular integrals associated with Zygmund dilations

Abstract: Zygmund dilations are a group of dilations lying in between the standard product theory and the one-parameter setting -- in $\mathbb R^3 = \mathbb R \times \mathbb R \times \mathbb R$ they are the dilations $(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$. In this talk, I will talk about our recent progress on singular integrals associated with Zygmund dilations.

席亚昆（浙江大学）

Title: Square function estimates and Local smoothing for Fourier Integral Operators

Abstract: We discuss some recent progress on the local smoothing conjecture for FIOs. In particular, we prove a variable coefficient version of the square function estimate of Guth--Wang--Zhang, which implies the full range of sharp local smoothing estimates for 2+1 dimensional Fourier integral operators satisfying the cinematic curvature condition. As a consequence, the local smoothing conjecture for wave equations on compact Riemannian surfaces is settled. This is a joint work with Chuanwei Gao, Bochen Liu and Changxing Miao.