Microhyperbolic Operators in Gevrey Classes
Microhyperbolic Operators in Gevrey Classes
复制标题
Gevrey 类中的微双曲运算符
DOI：
10.2977/prims/1195173608
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发表时间：
1989
影响因子：
1.2
通讯作者：
S. Wakabayashi
中科院分区：
文献类型：
--
作者：
K. Kajitani;S. Wakabayashi
Kashiwara和Kawai [16]定义了微双曲性，并证明了微双曲拟微分算子的微局部Cauchy问题在微函数的框架下是适定的，这是Bony和Schapira [3]结果的微局部化.在伪微分算子的微局部研究中，微双曲性的概念是非常有用的。从他们的结果可以得到关于微双曲算子解的解析奇点的传播（微解析性的传播）的结果（见[28]）。另一方面，Bronshtein [5]证明了双曲型Cauchy问题在某些Gevrey类中是适定的，这些Gevrey类是介于真实的解析函数空间和C°°之间的中间空间（参见[14]，[15]）.因此，我们至少可以在Gevrey类的框架下推广微双曲性的定义，从而为微双曲性的推广和C°°框架下微双曲算子的微局部研究提供线索。本文考虑Gevrey类中的微双曲算子，证明了微局部Cauchy问题的微局部适定性和微双曲算子的奇性传播定理。我们的目的是展示如何从证明Cauchy问题适定性的方法中获得微局部结果（微局部适定性，因此是Holmgren唯一性定理的微局部版本），并展示奇点传播定理是Holmgren唯一性定理的微局部版本的直接后果，使用
Kashiwara and Kawai [16] defined microhyperbolicity and proved that the microlocal Cauchy problem for microhyperbolic pseudodifferential operators is well-posed in the framework of microfunctions, which is a microlocalization of the results obtained by Bony and Schapira [3]. In the microlocal studies of pseudo-differential operators, the concept of microhyperbolicity is very useful. From their results one can obtain results on propagation of analytic singularities (propagation of micro-analyticities) of solutions for microhyperbolic operators (see [28]). On the other hand, Bronshtein [5] proved that the hyperbolic Cauchy problem is well-posed in some Gevrey classes which are intermediate spaces between the space of real analytic functions and C°° (see, also, [14], [15]). So we can generalize the definition of microhyperbolicity in the framework of some Gevrey classes, to say the least of it. In doing so, we expect to get a clue to a generalization of microhyperbolicity and microlocal studies of microhyperbolic operators in the framework of C°°. In this paper we shall consider microhyperbolic operators in Gevrey classes and prove microlocal well-posedness of the microlocal Cauchy problem and theorems on propagation of singularities for microhyperbolic operators. Our aims are to show how one can obtain microlocal results (microlocal well-posedness and, therefore, a microlocal version of Holmgren's uniqueness theorem) from methods to prove well-posedness of the Cauchy problem and to show that theorems on propagation of singularities are immediate consequences of a microlocal version of Holmgren's uniqueness theorem, using