论文标题
用于最大性低纤维化算子和Helfer-nourrigat猜想的假差分计算
A pseudodifferential calculus for maximally hypoelliptic operators and the Helffer-Nourrigat conjecture
论文作者
论文摘要
我们将椭圆运算符的经典规则定理扩展到最大的低纤维差分算子。更准确地说，给定的矢量字段$ x_1，\ ldots，x_m $在平滑的歧管上满足Hörmander的支架生成条件，我们为\ textit {any}线性差异操作员定义了主要符号。我们的符号考虑了向量字段$ x_i $及其换向器。我们表明，对于任意差分运算符，仅当操作员最大程度地低纤维化时，其主要符号才可逆。这是由于Helffer和Nourrigat引起的一个猜想。我们的结果在更通用的环境中得到了证明，在此我们允许vector字段$ x_1，\ ldots，x_m $具有任意权重。特别是，我们的定理将Hörmander的正方形总和概括为高阶多项式。
We extend the classical regularity theorem of elliptic operators to maximally hypoelliptic differential operators. More precisely, given vector fields $X_1,\ldots,X_m$ on a smooth manifold which satisfy Hörmander's bracket generating condition, we define a principal symbol for \textit{any} linear differential operator. Our symbol takes into account the vector fields $X_i$ and their commutators. We show that for an arbitrary differential operator, its principal symbol is invertible if and only if the operator is maximally hypoelliptic. This answers affirmatively a conjecture due to Helffer and Nourrigat. Our result is proven in a more general setting, where we allow each one of the vector fields $X_1,\ldots,X_m$ to have an arbitrary weight. In particular, our theorem generalizes Hörmander's sum of squares theorem to higher order polynomials.
