Pseudodifferential operator calculus for generalized -rank 1 locally symmetric spaces, I (original) (raw)

A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

Fourier Analysis, 2013

This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, φ-differential operators, that is, they live in the generalised φ-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the φ-calculus. We construct parametrices for them in this paper using a combination of the b-pseudodifferential operator calculus of R. Melrose and the φ-pseudodifferential operator calculus. The construction simplifies and generalizes the construction done by Vaillant in his thesis for the Dirac operator. In addition, we study the mapping properties of these operators and determine the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators.

Recent developments on pseudo-differential operators (II)

Tamkang Journal of Mathematics, 2015

The analysis on manifolds with singularities is a rapidly developing field of research, with new achievements and compelling challenges. We present here elements of an iterative approach to building up pseudo-differential structures. Those participate in operator algebras on singular manifolds and reflect the properties of parametrices of elliptic operators, including boundary value problems.

Pseudodifferential operators on manifolds with a Lie structure at infinity

Annals of Mathematics, 2007

We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a non-compact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ). We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show that Ψ ∞ 1,0,V (M 0 ) is a microlocalization of Diff * V (M 0 ). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).

Elliptic Theory on Manifolds with Corners: I. Dual Manifolds and Pseudodifferential Operators

Trends in Mathematics, 2008

In this first part of the paper, we define a natural dual object for manifolds with corners and show how pseudodifferential calculus on such manifolds can be constructed in terms of the localization principle in C *algebras. In the second part, these results will be applied to the solution of Gelfand's problem on the homotopy classification of elliptic operators for the case of manifolds with corners.

Mapping Properties of Pseudodifferential Operators on Manifolds with Bounded Geometry

Journal of the London Mathematical Society, 1998

We investigate classes of uniform pseudodifferential operators on Riemannian manifolds with bounded geometry. We prove that operators belonging to the classes are bounded in function spaces of Hardy-Sobolev-Besov type defined on the manifold. The classes contain the powers of the Laplace-Beltrami operators so we get the fractional lift property for the function spaces.

A class of metrics on tangent bundles of pseudo-Riemannian manifolds

2011

We provide the tangent bundle TM of pseudo-Riemannian manifold (M, g) with the Sasaki metric gs and the neutral metric gn. First we show that the holonomy group Hs of (TM, gs) contains the one of (M, g). What allows us to show that if (TM, gs) is indecomposable reducible, then the basis manifold (M, g) is also indecomposable-reducible. We determine completely the holonomy group of (TM, gn) according to the one of (M, g). Secondly we found conditions on the base manifold under which (TM, gs) ( respectively (TM, gn) ) is Kählerian, locally symmetric or Einstein manifolds. (TM, gn) is always reducible. We show that it is indecomposable if (M, g) is irreducible.

Algebras of pseudodifferential operators on complete manifolds

2003

In several influential works, Melrose has studied examples of noncompact manifolds M 0 whose large scale geometry is described by a Lie algebra of vector fields V ⊂ Γ(M ; T M) on a compactification of M 0 to a manifold with corners M . The geometry of these manifolds-called "manifolds with a Lie structure at infinity"-was studied from an axiomatic point of view in a previous paper of ours. In this paper, we define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a manifold M 0 with a Lie structure at infinity V ⊂ Γ(M ; T M). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to the algebras that we introduce. In particular, the algebra Ψ ∞ 1,0,V (M 0 ) is a "microlocalization" of the algebra Diff * V (M ) of differential operators with smooth coefficients on M generated by V and C ∞ (M ). This proves a conjecture of Melrose (see his ICM 90 proceedings paper).