2026京师调和分析论坛(4) (2026.8.8)
会议安排
腾讯会议号： 884 610 952，入会密码：888888
会议由以下项目支持:
北京师范大学教育部&科技部“调和分析及其应用创新引智基地”
国家自然科学基金重点项目“相关于球平均、Chirp函数和区域上算子的函数空间公理化实变理论及其应用”
组委会：戴峰, 杨大春, 袁文, 张阳阳
2026年8月4日
报告信息
报告题目: Embeddings of Potentials Along The Moment Curve
报告人: 付星 (湖北大学)
报告摘要: In this talk, let $\alpha\in\mathbb{R}$ and $P^\gamma_\alpha$ be the potential along the moment curve $\gamma=(t,\ldots,t^n)$ ($t\in(0,\infty)$) defined by setting, for any suitable function $f$, $$P^\gamma_\alpha f(x):=\int_0^\infty f(x-\gamma(t))t^{\alpha-1}\,dt \ \forall\,x\in\mathbb{R}^n. $$ We introduce some equivalent characterizations via capacities of the embedding $P^\gamma_\alpha$ from Lebesgue spaces $L^p(\mathbb{R}^n)$ associated with the $n$-dimensional Lebesgue measure to $L^q(\mathbb{R}^n,\mu)$ endowed with the non-negative Radon measure $\mu$ in the form of $d\mu(x):=w(x)\,dx$ for a weight $w$. The main ingredient lies in the different approach to the estimate of the capacity of the anisotropic cubes $Q^\gamma_R$ along $\gamma$.
报告题目: Empirical Approximation of $L_p$ Norms in High-Dimensional Function Spaces
报告人: Feng Dai (University of Alberta, Edmonton, Alberta, Canada)
报告摘要: In this talk, I will discuss the Marcinkiewicz discretization problem for finite-dimensional function spaces using random sampling. The goal is to obtain two-sided estimates for the integral $L_p$-norm of every function in a given space in terms of a finite sum of its values at randomly selected sample points. Importantly, the sample points are chosen independently of the individual functions.
A central challenge is to determine the nearly optimal number of random sample points needed for the Marcinkiewicz discretization inequalities to hold uniformly over the entire function space with high probability. I will present recent advances on this problem and briefly discuss its close connections with several related topics, including embeddings of finite-dimensional subspaces, approximation of moments of random vectors by empirical moments, construction of random matrices with the restricted isometry property, frame theory, and spectral estimates of random matrices.
报告题目: Oscillation and Difference Characterizations of Inhomogeneous Lorentz-Triebel-Lizorkin Spaces
报告人: 王凡 (河北大学)
报告摘要: We establish real-variable characterizations of the inhomogeneous Lorentz--Triebel--Lizorkin spaces \(F^{s,r}_{p,q}(\mathbb R^n)\). Under suitable assumptions on the parameters and for sufficiently large order \(M\), we prove that the Littlewood--Paley quasi-norm of \(F^{s,r}_{p,q}(\mathbb R^n)\) is equivalent to a quasi-norm defined in terms of local oscillations of order \(M\). We also obtain a characterization by means of \(M\)-th order differences. Moreover, in the range \(1<p<\infty\) and \(1<r<\infty\), these quasi-norms give genuine characterizations of the whole space. Under an additional condition on the auxiliary parameter \(u\), the corresponding discrete quasi-norms are shown to be equivalent to their continuous integral forms.
报告题目: Off-diagonal Muckenhoupt Conditions, Commutators, and Generalized Maximal Operators
报告人: 陶金 (湖北大学)
摘要: In this talk, we consider off-diagonal Muckenhoupt conditions on ball Banach function spaces. First, we use it to characterize the off-diagonal boundedness of the commutator $[b,T]$ from a ball Banach function space $X$ to another ball Banach function space $Y$. This characterization recovers the results of Hytönen [JMPA, (2021)] and Hänninen et al. [JMPA, (2025)] on (weighted) Lebesgue spaces. Next, we use it to characterize the weak boundedness of generalized maximal operator $M^w$ from $X$ to $Y_{weak}$. Moreover, we give a new pointwise sparse domination of $M^w$ and use it to characterize the boundedness of $M^w$ from $X$ to $Y$.
