報告題目： Heat kernel-based p-energy norms on metric measure spaces

　　報告人：高晉

　　報告時間：11月20日（星期四）14:30-15:30

　　報告地點：理學院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：在有界和無界度量測度空間上，特別是嵌套分形及其膨脹集上，我們研究了基于熱核的p-能量范數等，其中1<p<∞。利用這些半范數的弱單調性質，我們推廣了著名的Bourgain-Brezis-Mironescu（BBM）型刻畫至p≥2的情形。當基礎空間存在滿足次高斯估計的熱核時，我們建立了各類p-能量半范數的等價性與弱單調性質，并證明這些弱單調性質在p=2（即狄利克雷形式情形）時成立。本文的核心工作是在分形結構上驗證各類弱單調性質的等價性，由此使得許多關于p-能量范數的經典結論在嵌套分形及其膨脹集上依然成立，包括BBM型刻畫和Gagliardo-Nirenberg不等式。上述工作是與余振宇和張駿達合作完成。

　　報告人簡介：高晉，杭州師范大學講師，博士畢業于清華大學數學系，研究方向是分形分析和熱核估計。在Potential Anal., J Fractal Geom., Acta Math. Sci., J Math. Anal. Appl.等雜志發表論文。