On a class of hypoelliptic differential operators with double characteristics (Q1309198)

The author studies hypoellipticity for second-order degenerate elliptic and parabolic operators with real coefficients, giving a unified method with which he can generalize previous results of \textit{Wakabayashi} and \textit{Suzuki} (in print) and \textit{T. Hoshiro} [J. Math. Kyoto Univ. 28, No. 4, 615-632 (1988; Zbl 0692.35034) and ibid. 29, No. 3, 497-513 (1989; Zbl 0725.35036)]. Let \[ P(x,D)= 2 {{\partial^ 2} \over {\partial x_ 1}}+ \sum_{i,j=2}^ n a^{ij}(x) {\partial \over {\partial x_ j}}+ \sum_{i=2}^ n b^ i(x) {\partial \over {\partial x_ i}}+ c(x), \] where the \(a^{ij}\) are the \(C^ \infty\) components of a symmetric contravariant tensor on \(\mathbb{R}^ n\) and \[ \sum_{i,j=2}^ n a^{ij} (x)\xi_ i \xi_ i\geq 0 \quad \text{ on }\quad T^* (\mathbb{R}^ n) \] and \(b^ i,c\in C^ \infty (\mathbb{R}^ n)\). Denote \(\varphi:= (\partial/\partial x_ 1) \otimes (\partial/ \partial x_ 1)+ \sum_{i,j=2}^ n a^{ij} (\partial/ \partial x_ i) \otimes_ s (\partial/\partial x_ j)\), where \(\otimes_ s\) stands for the symmetric tensor product, let \(\Psi: \Gamma( \mathbb{R}^ n, T^* (\mathbb{R}^ n))\to \Gamma(\mathbb{R}^ n, T(\mathbb{R}^ n))\) be the map defined by \(\zeta\to \Phi(\zeta,\cdot)\), \(X_ 1:= \{\Psi (\zeta)\), \(\zeta\in\Gamma (\mathbb{R}^ n, T^* (\mathbb{R}^ n))\), and define the drift vector field \(X_ 0\) by \(X_ 0= \sum_{i=2}^ n b^ i(x) \partial/ \partial x_ i\). Suppose (H): The Lie algebra \({\mathcal L}(X)\) over \(\mathbb{R}\) generated by \(X=X_ 1\cup X_ 0\) has rank \(n\) outside a subset \(S\) of the hypersurface \(x_ 1=0\). The author gives sufficient conditions for hypoellipticity for \(P(x,D)\) under condition (H) (which is Hörmander's celebrated condition). For this, let \(\alpha(x,\xi')= \sum_{i,j=2}^ n a^{ij} (x_ 1, x')\xi_ i \xi_ j\), where \(x_ 1\) is considered as a parameter, and assume: \(\text{A}_ 1)\) \((\exists)\) \(a_ 0>0\) such that \(\sum_{i,j=2}^ n | \partial^ 2\alpha/ \partial x_ i \partial x_ j|^ 2 \leq a_ 0\) for \((x_ 1,x',\xi')\) on \(T^*(\mathbb{R}^{n-1})\), \(\text{A}_ 2)\) If \(\mu(x)= \mu(x_ 1, x')= \min_{|\xi '|=1} \alpha(x_ 1, x', \xi')\) then it is supposed to be Lipschitz in \(x_ 1\) and \(C^ \infty\) in the variables \(x'= (x_ 2,\dots, x_ n)\) and \(\mu(x_ 1, x')>0\) outside \(S\) (condition \(\text{A}_ 2\) implies that \(P(x,D)\) is elliptic outside \(S\) and (H) is satisfied). \(\text{A}_ 3)\) \((\exists)\) \(b_ 0>0\) such that \(\sum_{i=2}^ n | b^ i(x)|\leq b_ 0 \sqrt{\mu(x)}\) on \(\mathbb{R}^ n\). Then, if \[ \lim_{x_ 1\to 0} {{\widetilde{\lambda} (x_ 1,x') \log \mu(x_ 1, x')} \over {\sqrt{\lambda (x_ 1, x')}}} =0 \] uniformly in \(x'\) over compact subsets of \(\mathbb{R}^{n-1}\) which intersect \(S\), \(P(x,D)\) is hypoelliptic in \(\mathbb{R}^ n\). Here \(\lambda:= \sum_{i=2}^ n a^{ii} (x_ 1, x')\) and \(\widetilde{\lambda} (x_ 1, x'):= \int_ 0^{x_ 1} \lambda(t, x')dt\). Analogous conditions allow the author to obtain similar (sufficient) conditions of hypoellipticity for degenerate parabolic operators of second order.