拟纤维化边界伪微分算子
Quasi-fibered boundary pseudodifferential operators
AI总结：
本文为拟纤维化边界度量发展伪微分演算，引入完全椭圆QFB算子并证明其Fredholm性质，进而构造Hodge-deRham算子的拟基本解，证明L²调和形式空间有限维且具有更快的无穷远衰减。
摘要：
我们为与拟纤维化边界度量（QFB度量）相关的微分算子发展了一套伪微分演算，这类度量包括Degeratu-Mazzeo的拟渐近锥形度量（QAC度量）和Joyce的拟渐近局部欧几里得度量（QALE度量）。通过引入各种主符号，我们引入了完全椭圆QFB算子的概念，并证明这些算子在作用于QFB Sobolev空间时是Fredholm算子。对于QAC度量，我们还为共形相关的Qb度量类发展了一套伪微分演算。我们利用这些演算为某些QFB度量的Hodge-deRham算子构造了一个拟基本解（parametrix），从而证明它在适当的Sobolev空间上是Fredholm算子，并且$L^2$调和形式空间是有限维的。我们的拟基本解是通过在无穷远处反转某些模型算子得到的，这些反转部分是通过对Hodge-deRham算子预解式的低能极限的精细理解来实现的。我们的拟基本解还表明，$L^2$调和形式在无穷远处的衰减比任意$L^2$形式更快，额外的衰减量以距离函数的小负幂来量化。$L^2$调和形式的这种衰减在一篇配套论文中被用于研究某些$QFB$度量的$L^2$上同调。
英文摘要：
We develop a pseudodifferential calculus for differential operators associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics including the quasi-asymptotically conical metrics (QAC metrics) of Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE metrics) of Joyce. Introducing various principal symbols, we introduce the notion of fully elliptic QFB operators and show that those are Fredholm when acting on QFB Sobolev spaces. For QAC metrics, we also develop a pseudodifferential calculus for the conformally related class of Qb metrics. We use these calculi to construct a parametrix for the Hodge-deRham operator of certain QFB metrics, allowing us to show that it is Fredholm on suitable Sobolev spaces and that the space of $L^2$ harmonic forms is finite dimensional. Our parametrix is obtained by inverting certain model operators at infinity, inversions that we achieve in part through a fine understanding of the low energy limit of the resolvent of the Hodge-deRham operator. Our parametrix also implies that $L^2$ harmonic forms decay faster at infinity than an arbitrary $L^2$ form, the extra decay being quantified in terms of a small negative power of the distance function. This decay of $L^2$ harmonic forms is used in a companion paper to study the $L^2$ cohomology of some $QFB$ metrics.
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