非光滑度量测度空间中的Harnack型不等式和Riesz变换

The Harnack type inequality has drawn more and more attention since its establishment by Feng-Yu Wang on the Riemannian manifold with Ricci curvature bounded below. And there is no doubt that the Riesz transform, which is studied intensively on Riemannian manifolds, is an important topic in analysis. Recently, Ambrosio et al introduced the notion of Riemannian curvature lower bound, stronger than the Ricci curvature lower bound in the sense of Lott-Sturm-Villani, on a class of non-smooth metric measure spaces, which unifies the two approaches of Bakry-Emery and Lott-Sturm-Villani to provide the abstract notions of Ricci curvature lower bounds. The main aim of the present project is to investigate some analytical properties on the non-smooth metric measure space with Riemannian curvature bounded below: establish the Harnack type inequality for the diffusion semigroup, including symmetric and non-symmetric cases, by the semigroup method, apply analytical methods to prove the boundedness of the Riesz transform corresponding to the heat flow, and establish the dimension dependent log-Harnack inequality of the heat flow also by the semigroup method by taking account of the effect of the dimension; furthermore, show some applications of the Harnack type and dimension dependent log-Harnack inequalities, such as the establishment of the transport-cost, log-Sobolev and dimension dependent HWI inequalities, etc.

Harnack型不等式自王凤雨在Ricci曲率有下界的黎曼流形上建立起，已备受关注。Riesz变换更是分析学研究的重点，在黎曼流形上有大量的研究。Ambrosio等最近在非光滑度量测度空间中提出了比Lott-Sturm-Villani意义下的Ricci曲率下界更强的黎曼曲率下界，使得从Bakry-Emery和Lott-Sturm-Villani两种不同角度定义的Ricci曲率下界统一起来。此项目主要考虑在具有黎曼曲率下界的非光滑度量测度空间中，用半群方法建立对称和非对称扩散半群的Harnack型不等式，并用分析方法研究热流的生成元的Riesz变换的有界性；同时，考虑到维数的影响，也用半群方法建立热流的带维数的log-Harnack不等式；进一步地，给出Harnack型和带维数的log-Harnack不等式的一些应用，如建立运费不等式，log-Sobolev不等式和带维数的HWI不等式等。

自从黎曼曲率维数条件提出以来，度量测度空间上的几何与分析得到了丰富的发展. 本项目主要研究了度量测度空间上的分析性质，并取得了以下一些进展. .1.在满足无穷维黎曼曲率维数条件的度量测度空间中建立了Harnack型不等式(又称为与维数无关的Harnack不等式)，并给出了其在Log-Sobolev不等式, 传输不等式, 半群的三种超压缩性方面的应用..2.在满足维数有限的黎曼曲率维数条件的度量测度空间中，给出了热核的高斯型上下界估计, 热核的梯度估计, 研究了热核的长时间渐近行为，并证明了Riesz变换的L^p有界性. 进一步, 在假设体积最大增长的条件下，给出了更精细的热核上下界估计, 并研究了Perelman熵的单调性及其长时间渐近行为..3.在光滑的加权黎曼流形上，分别给出了加权p-Laplacian的非平凡第一Dirichlet和Neumann特征值的新的下界估计; 此估计在Laplacian的情形是一致的..4.在欧氏空间中，考虑了扩散系数一致非退化的随机微分方程, 当漂移系数分别满足Holder连续性和LPS积分型条件时, 建立了Harnack型不等式和Log-Harnack不等式; 对于系数满足Osgood-Sobolev条件的常微分方程, 证明了DiPerna-Lions流的存在唯一性..5.用概率的方法证明了基本谱系猜测.