A Calculus of Abstract Edge Pseudodifferential Operators of Type $\rho,\delta$

Krainer, Thomas

doi:10.1007/978-3-319-12547-3_8

“…It is a particular case of the next example, that of a desingularization groupoid, but we nevertheless treat it separately, for the benefit of the reader. See also [22,43,104]. We consider the following framework.…”

Section: 2mentioning

confidence: 99%

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Carvalho

¹

,

Nistor

²

,

Qiao

³

2018

Operator Theory, Operator Algebras, and Matrix Theory

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We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

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“…To summarize the construction of the desingularization, let us denote by φ the natural isomorphism of the following two groupoids: E(S, π, H) U1 U1 (reduction to U 1 ≃ S × (0, 1)) and G U1 U1 = (G 2 ) U1 U1 . Then (25) [ Similar structures arise in other situations; see, for instance, [14,28,19,35,34,43,44,56,53]. See also the discussion at the end of Example 2.4.…”

Section: 1mentioning

confidence: 73%

“…We also introduce an anisotropic version of the desingularized groupoid and determine its Lie algebroid as well. We conclude with an example related to the 'edge'-calculus (see [28] and the references therein).…”

Section: Introductionmentioning

confidence: 96%

Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

Nistor

¹

2019

Communications in Analysis and Geometry

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We introduce and study a "desingularization" of a Lie groupoid G along an "A(G)-tame" submanifold L of the space of units M . An A(G)-tame submanifold L ⊂ M is one that has, by definition, a tubular neighborhood on which A(G) becomes a thick pull-back Lie algebroid. The construction of the desingularization [[G : L]] of G along L is based on a canonical fibered pull-back groupoid structure result for G in a neighborhood of the tame A(G)submanifold L ⊂ M . This local structure result is obtained by integrating a certain groupoid morphism, using results of Moerdijk and Mrcun (Amer. J. Math. 2002). Locally, the desingularization [[G : L]] is defined using a construction of Debord and Skandalis (Advances in Math., 2014). The space of units of the desingularization [[G : L]] is [M : L], the blow up of M along L. The space of units and the desingularization groupoid [[G : L]] are constructed using a gluing construction of Gualtieri and Li (IMRN 2014). We provide an explicit description of the structure of the desingularized groupoid and we identify its Lie algebroid, which is important in analysis applications. We also discuss a variant of our construction that is useful for analysis on asymptotically hyperbolic manifolds. We conclude with an example relating our constructions to the so called "edge pseudodifferential calculus." The paper is written such that it also provides an introduction to Lie groupoids designed for applications to analysis on singular spaces.

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“…In [1], Abels developed a (different) variant of Boutet de Monvel's calculus with nonsmooth symbols in order to construct parametrices to elliptic operators with Hölder regularity. Also, Krainer [25] constructed a calculus with symbols of type ρ, δ taking values in operator ideals in Hilbert spaces. Theorem 1.3 and 1.4 extend [43,Theorem 1.2] in that (i) one can now treat manifolds of bounded geometry instead of bounded domains, (ii) the differentiability assumptions on the coefficients of A are reduced from C ∞ to C τ , τ > 0, for the top order terms and L ∞ for the lower order terms, while T can also be of the form (1.4), and (iii) one obtains the existence of a bounded H ∞ -calculus rather than the existence of a holomorphic semigroup.…”

Section: Relation To Previous Workmentioning

confidence: 99%

Bounded $$H^\infty $$-calculus for a degenerate elliptic boundary value problem

Krietenstein

¹

,

Schrohe

²

2021

Math. Ann.

1

0

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On a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form $$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$ T = φ 0 γ 0 + φ 1 γ 1 . Here $$\gamma _0$$ γ 0 and $$\gamma _1$$ γ 1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $$\varphi _0, \varphi _1\ge 0$$ φ 0 , φ 1 ≥ 0 , and $$\varphi _0+\varphi _1\ge c$$ φ 0 + φ 1 ≥ c , for some $$c>0$$ c > 0 , where either $$\varphi _0,\varphi _1\in C^{\infty }_b(\partial X)$$ φ 0 , φ 1 ∈ C b ∞ ( ∂ X ) or $$\varphi _0=1 $$ φ 0 = 1 and $$\varphi _1=\varphi ^2$$ φ 1 = φ 2 for some $$\varphi \in C^{2+\tau }(\partial X)$$ φ ∈ C 2 + τ ( ∂ X ) , $$\tau >0$$ τ > 0 . We also assume that the highest order coefficients of A belong to $$C^\tau (X)$$ C τ ( X ) and the lower order coefficients are in $$L_\infty (X)$$ L ∞ ( X ) . We show that the $$L_p(X)$$ L p ( X ) -realization of A with respect to the boundary operator T has a bounded $$H^\infty $$ H ∞ -calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.

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A Calculus of Abstract Edge Pseudodifferential Operators of Type $\rho,\delta$

Cited by 6 publications

References 11 publications

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

Bounded $$H^\infty $$-calculus for a degenerate elliptic boundary value problem

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