Positive Curvature Property for Sub-Laplacian on Nilpotent Lie Group of Rank Two
Positive Curvature Property for Sub-Laplacian on Nilpotent Lie Group of Rank Two
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DOI：
10.1007/s11118-013-9332-2
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发表时间：
2013-02
影响因子：
1.1
通讯作者：
B. Qian
中科院分区：
文献类型：
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作者：
B. Qian
本文主要研究秩为2的幂零李群上的次Laplace算子，它是n布朗运动及其Lévy面积过程生成的扩散的无穷小生成元，是Heisenberg群上的次Laplace算子的简单推广.为了研究伴随热核的收缩性质，我们证明了，与Heisenberg群和三布朗运动模型的情形一样，（见定义3.5）满足正Ricci曲率条件（更精确地说，aCD（0，∞）不等式），见定理4.5，而算子本身不满足任何CD（r，∞）不等式。由此，我们可以推导出一些有用的，尖锐的梯度界限相关的热核。可以看出该论文的概括（Qian，Bull Sci Math 135：262-278，2011）。
In this note, we concentrate on the sub-Laplace operator on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated bynBrownian motions and theirLévy area processes, which is the simple extension of the sub-Laplacian on the Heisenberg group ℍ. In order to study contraction properties of the associated heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motions model, the restriction of the sub-Laplace operator acting on radial functions (see Definition 3.5) satisfies a positive Ricci curvature condition (more precisely aCD(0, ∞ ) inequality), see Theorem 4.5, whereas the operator itself does not satisfy anyCD(r, ∞ ) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel. It can be seen a generalization of the paper (Qian, Bull Sci Math 135:262–278, 2011).