On ‘type’ conditions for generic real submanifolds of ℂn
On ‘type’ conditions for generic real submanifolds of ℂn
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关于 ℂn 的泛型实数子流形的“类型”条件
DOI：
10.1007/bf01425740
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发表时间：
1977
影响因子：
3.1
通讯作者：
Ian D. Graham
中科院分区：
文献类型：
--
作者：
T. Bloom;Ian D. Graham
在[13]中，J. J. Kohn研究了r中弱伪凸域的~-Neumann问题的局部边界正则性，他引入了这种域边界上点的类型的概念。该类型被定义为（本质上唯一的）切向全纯向量场及其共轭的交换子的最小阶，以获得“丢失”方向。Kohn得到的次椭圆估计依赖于这个量。见[9]。在[2]中，我们定义了~？是切向全纯向量场和共轭全纯向量场的交换子的最小阶，以获得丢失方向。若P的型是有限的，则P的型可刻划如下（[2，13]）：P是m-1型（根据本文的定义为m型），若在P附近M的任何局部定义函数r具有（0.2）r= 2 Rew + p（z，2 j + f）的形式，以便适当地选择局部全纯坐标。这里p（z，2）是z=（z 1，.，在[1-2]中，我们用复超曲面的最大相切阶给出了P型的一个几何刻画。在偏微分方程中还有很多其他的工作，其中研究的对象是由流形上的向量场形成的算子，使得向量场的许多算子跨越流形的切空间（1-10，14，16，17]）。我们特别提到Rothschild和Stein [17]的工作，其中权的概念和“齐次”情形的使用得到了高度发展。本文研究IE”的余维k< n的一般真实的子流形的类型条件。在这样的子流形M上的点P的类型的适当概念是k元组（ml，...，其中每个m-是> 2或+0 e的整数，并且m1 < m2<. <p>定义（W1.12）是根据切全纯向量场和共轭全纯向量场的分解子的性质给出的。整数λ是有限的，当且仅当通过取这样的λ，可以获得在P处的复化切空间中M的所有缺失方向。主要结果（定理3.2）是，如果k是有限的，那么如果权重m I.....如果将M分配给合适的复法线方向，则存在M在P附近的定义函数，其形式为
In [13] JJ Kohn studied local boundary regularity for the~-Neumann problem for weakly pseudoconvex domains in r He introduced the notion of type of a point on the boundary of such a domain. The type was defined as the minimum order of a commutator of the (essentially unique) tangential holomorphic vector field and its conjugate needed to obtain the'missing'direction. The subelliptic estimates obtained by Kohn depend on this quantity. See also [9]. In [2] we defined the type of a point P on a real hypersurface M in~?" to be the minimal order of a commutator of tangential holomorphic and conjugate holomorphic vector fields needed to obtain the missing direction. If the type of P is finite it can be characterized as follows ([2, 13]): P is of type m-1 (type m according to the definition of the present paper) if any local defining function r for M near P has the form (0.2) r= 2Rew+ p (z, 2j+ f for appropriate choice of local holomorphic coordinates. Here p (z, 2) is a nonzero homogeneous polynomial of degree m in z=(z 1,..., z, _j) without pure terms, and, on assigning weight m to w and 1 to the z variables, f has weight> m at P. In 1-2] we gave a geometric characterization of the type of P in terms of the maximum order of tangency of a complex hypersurface. There has been a good deal of other work in partial differential equations where the object of study is an operator formed from vector fields on a manifold such that finitely many commutators of the vector fields span the tangent space of the manifold (1-10, 14, 16, 17]). We mention particularly the work of Rothschild and Stein [17] in which the idea of weights and the use of the'homogeneous' case are highly developed.In this article we study type conditions for generic real submanifolds of codimension k< n of IE". The appropriate notion of type for a point P on such a submanifold M is a k-tuple (ml,..., ink) where each m~ is an integer> 2 or+ oe and m 1< m2<...< mk. The definition (w 1.12) is in terms of properties of commutators of tangential holomorphic and conjugate-holomorphic vector fields. The integer mk is finite if and only if it is possible to obtain all missing directions in the complexified tangent space to M at P by taking such commutators. The main result (Theorem 3.2) is that if mk is finite, then if weights m I..... mk are assigned to suitable complex normal directions there exist defining functions for M near P of the form