This seminar aims to present recent progress around harmonic analysis and its applications. The topics will cover the theory of harmonic analysis（also called Fourier analysis）and related fields in a large sense, for example partial differential equations, analytic number theory and geometric measure theory and so on.
The seminar is held on Thursday in a hybrid form; online access will be provided for speakers and audiences who cannot come to Harbin in person.
Contact: Xudong Lai (xudonglai AT hit.edu.cn)
Upcoming talks
Winter break: January - February 2023
Past talks
December 21, 2022. 16:00 (Beijing time)
Speaker: Danqing He（Fudan University）
Title: Counter example for multilinear Hörmander multipliers
Abstract
The study of multilinear multipliers goes back to the work of Coifman and Meyer in 1970s, who extended Mihlin's result to the bilinear setting. Their result is known as the Coifman-Meyer theorem, which is useful in harmonic analysis and PDE.
The multilinear H\ormander multiplier became am important topic in last decade. Many positive and negative results were obtained. In this talk, we survey the development of multilinear H\ormander multipliers, and present some related counterexamples.
November 17, 2022. 16:00(Beijing time)
Speaker: Mingming Cao（CSIC, ICMAT）
Title: The Dirichlet problem for elliptic operators on rough domains
Abstract
In capacity density open sets, we give a geometric characterization of well-posedness of the Dirichlet problem for elliptic operators with boundary data in Holder spaces. By means of a weak connectivity assumption, we establish the well-posedness of the Dirichlet problem when the solution satisfies a Carleson measure estimate and the boundary data belongs to Holder or Morrey-Campanato spaces. We also present a characterization of the Holder spaces in terms of the boundary traces of solutions.Finally, in 1-sided chord-arc domains, we establish an equivalence between Holder and Carleson norms.
November 12, 2022. 09:30 (Beijing time)
Speaker: Qingquan Deng（Central China Normal University）
Title: The existence of wave operators for multi-channel problem
Abstract
We consider the nonlinear Schr\{o}dinger equation $$i\psi_{t}=-\Delta\psi+F(|\psi|^{2})\psi$$ in dimension 3. we prove the existence of wave operators for multi-channel problem with large final state. That is, for given final state $(E^{+},\gamma^{+},\varphi^{+})$, we can find initial data $\varphi_{0}$ such that its corresponding solution $\psi(t)$ is asymptotically give by $$\psi(t)\approx e^{it\Delta}\varphi^{+}+e^{it E^{+}+i\gamma^{+}}\phi(E^{+}).$$
November 3, 2022. 15:00(Beijing time)
Speaker: Qingying Xue (Beijing Normal University)
Title: 粗糙核算子的弱极限行为问题和二个刻画
Abstract
主要介绍粗糙核弱极限行为的研究背景和进展，包括奇异积分算子，极大函数，平方函数，Calder\'{o}n交换子，在此基础上用小波分析的方法得到了弱极限行为的二个刻画。
Octobor 27, 2022. 16:00(Beijing time)
Speaker: Xiaohua Yao (Central China Normal University)
Title: Uniform Sobolev estimates on Euclidian space and compact manifolds with some applications
In this talk, I will review some results of uniform Sobolev estimates for the Laplace operator on Euclidian space and compact manifolds, which also were extended to the cases with potentials and higher order operator. Meanwhile, I also mention some applications related to the inequalities.Abstract
Octobor 25, 2022. 15:00(Beijing time)
Speaker: Zhuan Ye (Jiangsu Normal University )
Title: Global regularity of 2D MHD equations with almost Laplacian magnetic diffusion
Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion are globally well-posed is a difficult problem and remains completely open. In this talk, we establish the global regularity of smooth solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regularity issue on the 2D MHD equations with only standard Laplacian magnetic diffusion.Abstract
