Pseudodifferential operators on manifolds with fibred corners
Pseudodifferential operators on manifolds with fibred corners
复制标题
具有纤维角的流形上的伪微分算子
DOI：
10.5802/aif.2974
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Frédéric Rochon
中科院分区：
文献类型：
--
作者：
C. Debord;J. Lescure;Frédéric Rochon
对分层伪流形的奇点进行几何编码的一种方法是赋予其内部一个迭代的纤维尖度规。对于这样的度量，我们发展并研究了一种伪微分演算，推广了Mazzeo和Melrose的\ phi演算。我们的出发点是观察，回到梅尔罗斯，分层伪流形可以“分解”成具有纤维角的流形。这允许我们将伪微分算子定义为适当膨胀的双空间上的正态分布。介绍了各种符号映射，引出了全椭圆性的概念。这用于构造精细的参数，并为运算符的映射属性（如Fredholmness或紧凑性）提供标准。我们还引入了微积分的一个半经典版本，并利用它建立了分层伪流形的k -同调与全椭圆算子的k群之间的庞加莱对偶性。
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the \Phi-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Melrose, that a stratified pseudomanifold can be `resolved' into a manifold with fibred corners. This allows us to define pseudodifferential operators as conormal distributions on a suitably blown-up double space. Various symbol maps are introduced, leading to the notion of full ellipticity. This is used to construct refined parametrices and to provide criteria for the mapping properties of operators such as Fredholmness or compactness. We also introduce a semiclassical version of the calculus and use it to establish a Poincar\'e duality between the K-homology of the stratified pseudomanifold and the K-group of fully elliptic operators.