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Pseudodifferential operators on manifolds with fibred boundaries

Rafe Mazzeo·Richard B. Melrose
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Abstract

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $ϕ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the fibres of $ϕ;$ it is a Lie algebra and $\CI(X)$ module. This Lie algebra is quantized to the `small calculus' of pseudodifferential operators $\PsiF*(X).$ Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an `exact fibred cusp' metric is analyzed as is the wavefront set associated to the calculus.

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Rafe Mazzeo, Richard B. Melrose. 1998-12-18. Pseudodifferential operators on manifolds with fibred boundaries. https://arxiv.org/abs/math/9812120

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