Analytic K-theory on manifolds with corners

Melrose, Richard; Piazza, Paolo

doi:10.1016/0001-8708(92)90059-t

Cited by 152 publications

(316 citation statements)

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“…The difference between these two classes is in terms of the behavior of their (full) symbols at fiber-infinity of b T * X; elements of Ψ bc (X) have full symbols that satisfy the usual symbol estimates, while elements of Ψ b (X) have in addition an asymptotic expansion in terms of homogeneous functions, so that Ψ m b (X) ⊂ Ψ m bc (X). Conceptually, these are best defined via the Schwartz kernel of A ∈ Ψ m bc (X) in terms of a certain blow-up X 2 b of X × X; see [20]. The Schwartz kernel is conormal to the lift diag b of the diagonal of X 2 to X 2 b with infinite order vanishing on all boundary faces of X 2 b which are disjoint from diag b .…”

Section: Interaction Of Diff(x) With the B-calculusmentioning

confidence: 99%

“…Indeed, this is true for A ∈ Ψ −∞ b (X) with compact support, as follows from the Schwartz lemma and the explicit description of the Schwartz kernel of A on X 2 b . The standard square root argument then shows the boundedness for A ∈ Ψ 0 bc (X), with norm bounded by a seminorm of A in Ψ 0 bc (X); see [20,Eq. (2.16)].…”

Section: Interaction Of Diff(x) With the B-calculusmentioning

confidence: 99%

“…In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on a manifold with corners M equipped with a smooth Riemannian metric g. We first recall the basic definitions from [12], and refer to [20, §2] as a more accessible reference. Thus, a tied (or t-) manifold with corners X of dimension n is a paracompact Hausdorff topological space with a C ∞ structure with corners.…”

Section: Introductionmentioning

confidence: 99%

“…On manifolds without boundary, this phase space is the standard cotangent bundle. In the presence of boundaries the phase space is the b-cotangent bundle, b T * X, ('b' stands for boundary), which we now briefly describe following [19], which mostly deals with the C ∞ boundary case, and especially [20].…”

Section: Introductionmentioning

confidence: 99%

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Propagation of singularities for the wave equation on manifolds with corners

Vasy

2008

Ann. Math.

219

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In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M × R t , P = D 2 t − ∆ M , and u ∈ H 1 loc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

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Section: Interaction Of Diff(x) With the B-calculusmentioning

confidence: 99%

Section: Interaction Of Diff(x) With the B-calculusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Propagation of singularities for the wave equation on manifolds with corners

Vasy

2008

Ann. Math.

219

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“…For more on these topics, see Melrose [23], Mazzeo [22], or the appendices of Melrose and Piazza [26], and Melrose and Nistor [25]. We now describe the 'small' calculus.…”

Section: Introduction To B-operators On Manifolds With Cornersmentioning

confidence: 99%

Boundary Problems for Dirac-Type Operators on Manifolds with Multi-Cylindrical End Boundaries

Loya

Park

2006

Ann Glob Anal Geom

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Abstract. The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and 'b-admissible' partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderón projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulae, and solve the Bojarski conjecture.

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An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Lauter

1998

Mathematische Nachrichten

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Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A := 9i,c, (X, bnl/e) (cf. 1221, [23]) of dmical, totally characteristic peudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C 'algebra. On the other hand, the symbolic structure of A can be extended continuously to the C 'algebra 8 generated by A 88 a subalgebra of L(pbL2 (XI *n1I2)) by a generalisation of a method of GOHEERG and KRUPNIK. hrthermore, A is densely embedded in a M e t algebra A C B which is a -algebra in the sense of GRAMSCH [9, Definition 5.11, nfiscting .tso smooth properties of the original algebra A.GRAMSCH [9, Definition 5.11 introduced in connection with a perturbation theory for algebras of pseudodifferential operators the notion of Q' -algebras, i. e., symmetric, spectrally invariant, continuously embedded RBchet subalgebras of C 'algebras.Some of the remarkable consequences of the Q' -property and examples will be briefly discussed in Remark 2.10.Nevertheless, there also exist very interesting algebras in the pseudodifferential analysis on singular manifolds where the q* -property fails; for instance, the algebra 1991 Mafhernaticr Subjcct Clortificotion. Primary: 46H35, 58G 15; Secondary: 47D25, 47B47, Keywordr and phmuu. 1y'algebras, totally characterintic pesudodifferential operators. 47C15.Math. Nachr. 196 (1998) *f,cl (x, *Q1l2) of classical, totally characteristic pseudodifferential operators on a compact manifold X with boundary introduced by MELROSE [22] in 1981 carries even no topology such that (X, *all2) is a topological algebra with an open group of invertible elements (Theorem 3.4). On the other hand, it is an algebra with symbolic structure, i. e., it is possible to characterize the F'redholm property by means of the invertibility of symbols; here we think of !Pt,cl(X,bR1/2) as being embedded in C(ehL2(X,bf11/2)) for some weight b E JR.In order to make the Q' -techniques still available also to the small Melrose calculus, one can try to approximate the algebra from above by !P' -algebras in such a way that the approximating !P* -algebras share as many properties of the small calculus as possible. Following a suggestion of GRAMSCH one can consider the intersection of all !P* -algebras in L ( e b L 2 ( X , *all2)) containing @i,cl (X,*R1/2).The resulting algebra is uniquely determined and still a symmetric, spectrally invariant, continuously embedded subalgebra of L(ebL2 (X, which enjoys also many properties of !P' -algebras, although it is in general not a FMchet algebra, but only a sequentially complete, locally convex topological algebra with jointly continuous multiplication and continuous inversion. It is worth pointing out that in these topological algebras the relative inversion within the algebra (cf. (91, [20]) is an appropriate substitute for the parametrix construction for pseudodifferential operators (Theorem 2.11). Th...

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Analytic K-theory on manifolds with corners

Cited by 152 publications

References 11 publications

Propagation of singularities for the wave equation on manifolds with corners

Propagation of singularities for the wave equation on manifolds with corners

Boundary Problems for Dirac-Type Operators on Manifolds with Multi-Cylindrical End Boundaries

An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

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