arXiv 0707.45762007-07-31math.APmath.DG

Grusin算子的热核估计

Heat kernel estimates for the Grusin operator

Martin Paulat

备注：: 32 pages, 8 figures

摘要：

研究了Grusin算子G=Δ x}+ |x| ^{2} \partial_{u}^{2}在\mathbb{R}_{x}^{n} \times \mathbb{R}_{u}上的几何性质，得到了该算子的热核估计.主要工作是在$\mathbb{R}^{n+1}$中寻找连接两个给定点的最短测地线。这给出了与该运算符相关联的卡诺-卡拉西奥托里距离d_{CC}。第二部分的主要结果是给出了热核K_{t}的Carnot-Caratheodory距离的高斯界。特别地，我们得到以下估计：|k_{t}（α，η）|\leq C t^{- \frac{n}{2} -1} \min（1+ \frac{d_{CC}(ζ,η)} {|x+ x|}，1 + \frac{d_{CC}(ζ,η)^{2}}{4t}）^αe^{- \frac{1}{4t} d_{CC}（η，η）^{2}}对于所有$ζ=(x,u_{1}), η=(ξ,u)\in\mathbb{R}^{n+1}$，其中$α= \max{\frac{n}{2}-1,0}$。这里齐次维数是q=n+2，因此$\frac{n}{2}-1=\frac{q-4}{2}$。这表明我们对$n\geq2$的结果与Beals，Gaveau，Greiner在[1]中对Heisenberg群的结果是一致的.

英文摘要：

We study the geometry associated to the Grusin operator G=Δ_{x}+|x|^{2}\partial_{u}^{2} on \mathbb{R}_{x}^{n}\times\mathbb{R}_{u}, to obtain heat kernel estimates for this operator. The main work is to find the shortest geodesics connecting two given points in $\mathbb{R}^{n+1}$. This gives the Carnot-Caratheodory distance d_{CC}, associated to this operator. The main result in the second part is to give Gaussian bounds for the heat kernel K_{t} in terms of the Carnot-Caratheodory distance. In particular we obtain the following estimate |k_{t}(ζ,η)|\leq C t^{-\frac{n}{2}-1}\min(1+\frac{d_{CC}(ζ,η)} {|x+ξ|},1+\frac{d_{CC}(ζ,η)^{2}}{4t})^αe^{-\frac{1}{4t}d_{CC} (ζ,η)^{2}} for all $ζ=(x,u_{1}), η=(ξ,u)\in\mathbb{R}^{n+1}$, where $α= \max{\frac{n}{2}-1,0}$. Here the homogeneous dimension is q=n+2, so that $\frac{n}{2}-1=\frac{q-4}{2}$. This shows that our result for $n\geq2$ corresponds with the result on the Heisenberg group, which was given by Beals, Gaveau, Greiner in [1].

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