Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under th...Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.展开更多
In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifo...In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).展开更多
In this note, we obtain an asymptotic estimate for the time derivative of the O-entropy in terms of the lower bound of the Bakry-Emery F2 curvature. In the cases of hyperbolic space and the Heisenberg group （more gen...In this note, we obtain an asymptotic estimate for the time derivative of the O-entropy in terms of the lower bound of the Bakry-Emery F2 curvature. In the cases of hyperbolic space and the Heisenberg group （more generally, the nilpotent Lie group of rank two）, we show that the time derivative of the O-entropy is non-increasing and concave in time t, also we get a sharp asymptotic bound for the time derivative of the entropy in these cases.展开更多
基金supported by NSFC (10831008)NKBRPC(2006CB805905)
文摘Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.
基金supported by National Natural Science Foundation of China (Grant Nos. 12271102, 11625102, 11831004 and 11921001)supported by the National Key R&D Program of China (Grant Nos. 2022YFA1006000 and 2020YFA0712900)。
文摘In this paper, we are concerned with the Riesz transform on the direct product manifold H^(n)× M,where H^(n) is the n-dimensional real hyperbolic space, and M is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type(1, 1) property. In addition, we obtain the weak type(1, 1) of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type(1, 1) boundedness of some operators considered in the work of Li et al.(2016).
基金Supported by National Natural Science Foundation of China(Grant No.11201040)
文摘In this note, we obtain an asymptotic estimate for the time derivative of the O-entropy in terms of the lower bound of the Bakry-Emery F2 curvature. In the cases of hyperbolic space and the Heisenberg group （more generally, the nilpotent Lie group of rank two）, we show that the time derivative of the O-entropy is non-increasing and concave in time t, also we get a sharp asymptotic bound for the time derivative of the entropy in these cases.
