On the domain of Dirac and Laplace type operators on stratified spaces

Hartmann, Luiz; Lesch, Matthias; Vertman, Boris

doi:10.4171/jst/228

Cited by 12 publications

(16 citation statements)

References 17 publications

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“…Let K ⊂ Γ be a subgroup and ρ ∈ K. Then E ρ := x∈M K E xρ is a smooth vector bundle over M K , the set of fixed points of M with respect to K. Similarly, (E ⊗ ρ) K → M K is a smooth vector bundle (over M K ). Moreover, we have an isomorphism (41) End…”

Section: Applications and Extensionsmentioning

confidence: 99%

“…Let K ∈ Stab Γ0 Γ (M ) (so Γ 0 ⊂ K), ρ ∈ K, and ξ ∈ T * x M {0} with Γ ξ = K. We have that (σ m (P ) ⊗ id ρ )| (E⊗ρ) K is invertible at ξ ∈ (T * M ) K if, and only if, the restriction of σ m (P )(ξ) to E xρ is invertible, since they correspond to each other under the isomorphism of Equation (41). The relation (2) thus means that the restriction of the principal symbol σ m (P ) is invertible on a subset of X α M,Γ , so (1) implies (2) right away.…”

Section: Applications and Extensionsmentioning

confidence: 99%

“…This means that there exists g ∈ Γ such that ρ ′ := g • ρ and α are Γ 0 associated (see Equation (7) and Definition 2.2; alternatively, this is also recalled in Remark 4.3). For this to make sense, it is implicit that (2) for the group K gives that π gξ,ρ ′ (σ m (P )) is invertible, since the irreducible representation ρ ′ of Γ gξ is Γ 0 -associated to α (we have used here again the isomorphism (41)). Furthermore, g : E ξ,ρ → E gξ,ρ ′ is an isomorphism.…”

Section: Applications and Extensionsmentioning

confidence: 99%

“…Some of the most recent papers using related ideas include [7,24,25,27,60,61,82], to which we refer for further references. Besides C * -algebras, pseudodifferential operators were also used to obtain Fredholm conditions, see [32,51,41] and the references therein.…”

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: Applications and Extensionsmentioning

confidence: 99%

Section: Applications and Extensionsmentioning

confidence: 99%

Section: Applications and Extensionsmentioning

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

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“…This paper is motivated by recent works [1,3,27,28,37,38,44] that contribute towards developing elliptic theory for operators on incomplete stratified manifolds. The long-term goal is a robust elliptic theory of ideal boundary conditions associated with the singular strata, and the model operator level is essential for this.…”

Section: Introductionmentioning

confidence: 99%

Extensions of symmetric operators that are invariant under scaling and applications to indicial operators

Krainer¹

2021

Preprint

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Indicial operators are model operators associated to an elliptic differential operator near a corner singularity on a stratified manifold. These model operators are defined on generalized tangent cone configurations and exhibit a natural scaling invariance property with respect to dilations of the radial variable. In this paper we discuss extensions of symmetric indicial operators from a functional analytic point of view. In the first, purely abstract part of this paper, we consider a general unbounded symmetric operator that exhibits invariance with respect to an abstract scaling action on a Hilbert space, and we describe its extensions in terms of generalized eigenspaces of the infinitesimal generator of this action. Among others, we obtain a Green formula for the adjoint pairing, an algebraic formula for the signature, and in the semibounded case explicit descriptions of the Friedrichs and Krein extensions. In the second part we consider differential operators of Fuchs type on the half axis with unbounded operator coefficients that are invariant under dilation, and show that under suitable ellipticity assumptions on the indicial family these operators fit into the abstract framework of the first part, which in this case furnishes a description of extensions in terms of polyhomogeneous asymptotic expansions. We also obtain an analytic formula for the signature of the adjoint pairing in terms of the spectral flow of the indicial family for such operators.

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The fractional porous medium equation on manifolds with conical singularities II

Roidos

Shao

2023

Mathematische Nachrichten

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This is the second of a series of two papers that studies the fractional porous medium equation, 𝜕 𝑡 𝑢 + (−Δ) 𝜎 (|𝑢| 𝑚−1 𝑢) = 0 with 𝑚 > 0 and 𝜎 ∈ (0, 1], posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin-Sobolev spaces including the Rellich-Kondrachov theorem and Sobolev-Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for 𝐿 ∞ −initial data and all 𝑚 > 0. We further investigate a number of properties of the solutions, including comparison principle, 𝐿 𝑝 −contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.

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On the domain of Dirac and Laplace type operators on stratified spaces

Cited by 12 publications

References 17 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

Extensions of symmetric operators that are invariant under scaling and applications to indicial operators

The fractional porous medium equation on manifolds with conical singularities II

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