论文标题
全球高斯估算的均匀质量的热核
Global Gaussian estimates for the heat kernel of homogeneous sums of squares
论文作者
论文摘要
令$ \ Mathcal {h} = \ sum_ {j = 1}^{m} x_ {j}^{2}^{2} - \ partial_ {t} $是$ \ mathbb {r}^{n+1} $的热型操作员$ x = \ {x_ {1}，\ ldots，x_ {m} \} $是一种平稳的Hörmander的矢量字段系统的系统，$ \ mathbb {r}^{n} $，每个$ x_ {j {j {j { $ \ mathbb {r}^{n} $，而没有假定基本组结构。在本文中，我们证明了$ \ mathcal {h} $的加热内核$γ（t，x; s，y）$的全球高峰和下高斯的估计，该$ \ nathcal {h} $在carnot-carthéodory的距离方面，$ \ x $ oon $ \ nathbb {r} $ for $}^n} $ suss $任何$γ$的顺序。从高斯边界中，我们得出了凯奇问题的独特解决性，可能是一个可能无限的连续初始基准，可满足无穷大的指数增长。此外，我们研究了H-Dirichlet问题在任意有限域上的解决性。最后，我们为$ \ Mathcal {h} u = 0 $的非负解决方案建立了全局规模不平等不平等。
Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth Hörmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in $\mathbb{R}^{n}$, while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel $Γ(t,x;s,y)$ of $\mathcal{H}$, in terms of the Carnot-Carathéodory distance induced by $X$ on $\mathbb{R}^{n}$, as well as global upper Gaussian estimates for the $t$- or $X$-derivatives of any order of $Γ$. From the Gaussian bounds we derive the unique solvability of the Cauchy problem for a possibly unbounded continuous initial datum satisfying exponential growth at infinity. Also, we study the solvability of the H-Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of $\mathcal{H}u=0$.
