关于${\matbb R}^N$中一类系数无界的亚椭圆算子
On a class of hypoelliptic operators with unbounded coefficients in ${\matbb R}^N$
摘要：
考虑退化Ornstein-Uhlenbeck算子在${\mathbb R}^N$中的一类非平凡扰动${\mathscr A}$.事实上，我们扰动算子的扩散和漂移部分（比如$Q$和$B$），使扩散部分在${\mathbb R}^N$中无界。假设矩阵$Q(x)$的核关于$x\in {\mathbb R}^N$是不变的，并且Kalman秩条件在任何$x\in{\mathbb R}^N$处由相同的$m<N$满足，并且发展了Bernstein方法的修正版本，我们证明了我们可以将有界算子的半群$\{T(t)\}$（在有界和连续函数的空间中）与算子${\mathscr A}$相关联。此外，我们还给出了半群$\{T(t)\}$在各向同性和各向异性的（Hölder-）连续函数空间中的空间导数的几个一致估计.最后，我们证明了与算子${\mathscr A}$相关的一些椭圆和抛物问题的Schauder估计.
英文摘要：
We consider a class of non-trivial perturbations ${\mathscr A}$ of the degenerate Ornstein-Uhlenbeck operator in ${\mathbb R}^N$. In fact we perturb both the diffusion and the drift part of the operator (say $Q$ and $B$) allowing the diffusion part to be unbounded in ${\mathbb R}^N$. Assuming that the kernel of the matrix $Q(x)$ is invariant with respect to $x\in {\mathbb R}^N$ and the Kalman rank condition is satisfied at any $x\in{\mathbb R}^N$ by the same $m<N$, and developing a revised version of Bernstein's method we prove that we can associate a semigroup $\{T(t)\}$ of bounded operators (in the space of bounded and continuous functions) with the operator ${\mathscr A}$. Moreover, we provide several uniform estimates for the spatial derivatives of the semigroup $\{T(t)\}$ both in isotropic and anisotropic spaces of (Hölder-) continuous functions. Finally, we prove Schauder estimates for some elliptic and parabolic problems associated with the operator ${\mathscr A}$.
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