数学学科2025系列学术报告之二十七

　　报告题目： Heat kernel-based p-energy norms on metric measure spaces

　　报告人：高晋

　　报告时间：11月20日（星期四）14:30-15:30

　　报告地点：理学院1-301

　　英文摘要：We investigate heat kernel-based and other p-energy norms (1<p<∞)on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blowups. With the weak-monotonicity properties for these semi-norms, we generalize the celebrated Bourgain-Brezis-Mironescu (BBM) type characterization for p≥2. When the underlying space admits a heat kernel satisfying the sub-Gaussian estimates, we establish the equivalence of various p-energy semi-norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when p=2 (that is the case of Dirichlet form). Our paper's key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p-energy norms hold on nested fractals and their blowups, including the BBM type characterization and Gagliardo-Nirenberg inequality.This paper is a joint work with Zhenyu Yu and Junda Zhang.

　　中文摘要：在有界和无界度量测度空间上，特别是嵌套分形及其膨胀集上，我们研究了基于热核的p-能量范数等，其中1<p<∞。利用这些半范数的弱单调性质，我们推广了著名的Bourgain-Brezis-Mironescu（BBM）型刻画至p≥2的情形。当基础空间存在满足次高斯估计的热核时，我们建立了各类p-能量半范数的等价性与弱单调性质，并证明这些弱单调性质在p=2（即狄利克雷形式情形）时成立。本文的核心工作是在分形结构上验证各类弱单调性质的等价性，由此使得许多关于p-能量范数的经典结论在嵌套分形及其膨胀集上依然成立，包括BBM型刻画和Gagliardo-Nirenberg不等式。上述工作是与余振宇和张骏达合作完成。

　　报告人简介：高晋，杭州师范大学讲师，博士毕业于清华大学数学系，研究方向是分形分析和热核估计。在Potential Anal., J Fractal Geom., Acta Math. Sci., J Math. Anal. Appl.等杂志发表论文。