On the absence of log terms in the constant curvature case

Brüning, Jochen; Schröder, Herbert

doi:10.3233/asy-1988-1302

Cited by 3 publications

(6 citation statements)

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“…Let us consider now a Γ-equivariant smooth vector bundle E → M . Let us fix x ∈ M and consider as above the tube W x ≃ Γ× Γx U x around x, see Equation (21). We use this diffeomorphism to identify U x to a subset of M , in which case, we can also assume the restriction of E to the slice U x to be trivial.…”

Section: Preliminariesmentioning

confidence: 99%

“…In view of Remark 3.4, we shall concentrate now on the local structure of ker(R M ), that is, on the structure of ker(R O ) for suitable ("small") open, Γ-invariant subsets O ⊂ M . Let us fix then x ∈ M and let W x ≃ Γ × Γx U x be the tube around x, Equation (21). For simplicity, we shall write ( 29)…”

Section: Local Calculationsmentioning

confidence: 99%

“…For these algebras, the role of Γ will be played by Γ x . For the statement of the following lemma, recall the definitions in Subsection (2.4), especially Equation (21).…”

Section: Local Calculationsmentioning

confidence: 99%

“…By contrast, the multiplicity of α ∈ Γ in ind Γ (P ) was much less studied. It did appear, however, implicitly in the work of Brüning [17,21], who initiated the program of studying the "isotypical heat trace" tr(p α e −t∆ ) and its short time asymptotic expansion. See also the recent Preprint [20].…”

Section: Introductionmentioning

confidence: 99%

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Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components

Baldare,

Côme,

Lesch

et al. 2020

Preprint

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Let Γ be a finite group acting on a smooth, compact manifold M , let P ∈ ψ m (M ; E 0 , E 1 ) be a Γ-invariant, classical pseudodifferential operator acting between sections of two equivariant vector bundles E i → M , i = 0, 1, and let α be an irreducible representation of the group Γ. Then P induces a map πα(P ) : H s (M ; E 0 )α → H s−m (M ; E 1 )α between the α-isotypical components of the corresponding Sobolev spaces of sections. We prove that the map πα(P ) is Fredholm if, and only if, P is α-elliptic, an explicit condition that we define in terms of the principal symbol of P and the action of Γ on the vector bundles E i . The result is not true for non-discrete groups. In the process, we also obtain several results on the structure of the algebra of invariant pseudodifferential operators on E 0 ⊕ E 1 , especially in relation to induced representations. We include applications to Hodge theory and to index theory of singular quotient spaces. M 3.2. The restriction morphisms 3.3. Local calculations 3.4. Calculations for the principal orbit bundle 3.5. The non-principal orbit case 4. Applications and extensions 4.1. Reduction to the connected case and α-ellipticity M.L. was partially supported by the Hausdorff Center for Mathematics, Bonn. A.B., R.C., and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR). V.N. was also supported by the NSF grant DMS 1839515.Manuscripts available from http://www.math.psu.edu/nistor/.

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Section: Preliminariesmentioning

confidence: 99%

Section: Local Calculationsmentioning

confidence: 99%

“…For these algebras, the role of Γ will be played by Γ x . For the statement of the following lemma, recall the definitions in Subsection (2.4), especially Equation (21).…”

Section: Local Calculationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components

Baldare,

Côme,

Lesch

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Brüning [9,10] considered the "isotypical heat trace" tr(p α e −t∆ ), which is nothing but the heat trace of π α (∆), and its short time asymptotic expansion. Clearly, carrying out Brüning's programme in full would lead to a heat equation proof of an index theorem for the α isotypical component of Dirac type operators.…”

Section: Introductionmentioning

confidence: 99%

Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

Baldare¹,

Côme²,

Lesch³

et al. 2020

J. Operator Theory

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Let Γ be a finite abelian group acting on a smooth, compact manifold M without boundary and let P∈ψm(M;E0,E1) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles. Let α be an irreducible representation of Γ. We obtain necessary and sufficient conditions for the restriction πα(P):Hs(M;E0)α→Hs−m(M;E1)α of P between the α-isotypical components of Sobolev spaces to be Fredholm.

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Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

Baldare¹,

Côme²,

Lesch³

et al. 2019

Preprint

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We answer the question of when an invariant pseudodifferential operator is Fredholm on a fixed, given isotypical component. More precisely, let Γ be a compact group acting on a smooth, compact, manifold M without boundary and let P ∈ ψ m (M ; E 0 , E 1 ) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles E 0 and E 1 . Let α be an irreducible representation of the group Γ. Then P induces by restriction a map πα(P ) : H s (M ; E 0 )α → H s−m (M ; E 1 )α between the α-isotypical components of the corresponding Sobolev spaces of sections. We study in this paper conditions on the map πα(P ) to be Fredholm. It turns out that the discrete and non-discret cases are quite different. Additionally, the discrete abelian case, which provides some of the most interesting applications, presents some special features and is much easier than the general case. Moreover, some results are true only in the abelian case. We thus concentrate in this paper on the case when Γ is finite abelian. We prove then that the restriction πα(P ) is Fredholm if, and only if, P is "α-elliptic," a condition defined in terms of the principal symbol of P (Definition 1.1). If P is elliptic, then P is also α-elliptic, but the converse is not true in general. However, if Γ acts freely on a dense open subset of M , then P is α-elliptic for the given fixed α if, and only if, it is elliptic. The proofs are based on the study of the structure of the algebra ψ m (M ; E) Γ of classical, Γ-invariant pseudodifferential operators acting on sections of the vector bundle E → M and of the structure of its restrictions to the isotypical components of Γ. These structures are described in terms of the isotropy groups of the action of the group Γ on E → M . M.L. was partially supported by the Hausdorff Center for Mathematics, Bonn. A.B., R.C., and V.N. have been partially supported by ANR-14-CE25-0012-01 (SINGSTAR). Manuscripts available from http://www.math.psu.edu/nistor/. 1 2 A. BALDARE, R. C ÔME, M. LESCH, AND V. NISTOR 2.5. Pseudodifferential operators 13 2.6. Reduction to order-zero operators 13 3. The structure of regularizing operators 14 3.1. Inner actions of Γ: the abstract case 14 3.2. Inner actions of Γ: pseudodifferential operators 15 3.3. The case of free actions 16 4. The principal symbol 17 4.1. The primitive ideal spectrum of the symbol algebra 17 4.2. Factoring out the minimal isotropy 19 5. Applications and extensions 21 5.1. Fredholm conditions 21 5.2. Boundary value problems 22 5.3. The case of non-discrete groups 23 References 23

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On the absence of log terms in the constant curvature case

Cited by 3 publications

References 5 publications

Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components

Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components

Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

Fredholm conditions for invariant operators: finite abelian groups and boundary value problems

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