Pseudodifferential operators on manifolds with scaled bounded geometry

TL;DR

This paper develops a unified framework for pseudodifferential operators on noncompact manifolds with scaled bounded geometry, enabling advanced microlocal analysis tools for various PDE problems.

Contribution

It introduces the concept of scaled bounded geometry for manifolds, leading to a general algebra of pseudodifferential operators with a precise principal symbol calculus.

Findings

Unified algebra of ps.d.o.s on noncompact manifolds.

Applicable to resolvent bounds, wave analysis, and X-ray transforms.

Captures operators modulo lower order and decay operators.

Abstract

We prove a general black box result which produces algebras of pseudodifferential operators (ps.d.o.s) on noncompact manifolds, together with a precise principal symbol calculus. Our construction (which also applies in parameter-dependent settings, with phase space weights and variable differential and decay orders) recovers most of the ps.d.o. algebras which have been introduced in recent years as tools for the microlocal analysis of non-elliptic partial differential equations. This includes those used for proving resolvent bounds (b- and scattering algebras and resolved or semiclassical versions thereof), studying waves on asymptotically flat spacetimes (3b-, edge-b-, and desc-algebras), inverting geodesic X-ray transforms (semiclassical foliation and 1-cusp algebras), and many others. Our main result rests on the novel notion of manifolds with scaled bounded geometry. A scaling encodes, in each distinguished chart of a manifold with bounded geometry, the amounts in $(0,1]$ by which the components of a uniformly bounded vector field are scaled. This decouples the regularity of the coefficients of elements of the resulting Lie algebra $\mathcal{V}$ of vector fields from the pointwise size of their coefficients. When the scaling tends to $0$ at infinity, the approximate constancy of coefficients of elements of $\mathcal{V}$ on increasingly large cubes, as measured using $\mathcal{V}$, gives rise to a principal symbol which captures $\mathcal{V}$-operators modulo operators of lower differential order and more decay.