Boundary value problems on manifolds with fibered boundary

Savin, Anton; Sternin, B. Yu.

doi:10.1002/mana.200410308

Cited by 6 publications

(14 citation statements)

References 16 publications

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“…We will show that the homotopy classification enables one to write explicit formula for the obstruction. This formula gives the same result as in the previous computations in [10], [11].…”

Section: Introductionsupporting

confidence: 76%

“…Assume that the base X and the fiber (denoted by Ω) are compact closed manifolds. Consider the algebra generated by pseudodifferential operators of order zero on Y and families of pseudodifferential operators of order zero acting in the fibers, see [11]. Denote this algebra by Ψ(Y, π).…”

Section: Operators With Discontinuous Symbols In Fiber Bundlesmentioning

confidence: 99%

“…Now the Calkin algebra of Ψ(Y, π) can be described (see [11]) as the subalgebra in the direct sum of algebras:…”

Section: Operators With Discontinuous Symbols In Fiber Bundlesmentioning

confidence: 99%

“…Then we formulate the main results of the paper -homotopy classifications for simplest classes of manifolds with singularities: conical points and edges. We would like to mention to simplify the presentation for manifolds with edges we use the algebra of pseudodifferential operators with discontinuous symbols introduced in [11]. The proofs of the classifications are given in the fourth section.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic Operators on Manifolds with Singularities and K-Homology

Savin¹

2005

K-Theory

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Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah-Singer difference construction in the noncommutative case and Poincare isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges. MSC2000: 58J05(Primary) 19K33 35S35 47L15(Secondary)

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“…We will show that the homotopy classification enables one to write explicit formula for the obstruction. This formula gives the same result as in the previous computations in [10], [11].…”

Section: Introductionsupporting

confidence: 76%

Section: Operators With Discontinuous Symbols In Fiber Bundlesmentioning

confidence: 99%

“…Now the Calkin algebra of Ψ(Y, π) can be described (see [11]) as the subalgebra in the direct sum of algebras:…”

Section: Operators With Discontinuous Symbols In Fiber Bundlesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Elliptic Operators on Manifolds with Singularities and K-Homology

Savin¹

2005

K-Theory

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“…The corresponding obstruction was computed in [4] and is a generalization of the Atiyah-Bott obstruction [5] in the theory of classical boundary value problems. For the case in which the obstruction vanishes, we construct a Fredholm realization of the Dirac operator in the class of boundary value problems introduced in [6].…”

Section: Introductionmentioning

confidence: 99%

On the Poincaré isomorphism in K-theory on manifolds with edges

2010

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The aim of this paper is to construct the Poincaré isomorphism in K-theory on manifolds with edges. We show that the Poincaré isomorphism can naturally be constructed in the framework of noncommutative geometry. More precisely, to a manifold with edges we assign a noncommutative algebra and construct an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges viewed as a compact topological space.

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Parameter-dependent pseudodifferential operators of Toeplitz type

Seiler

2013

Annali di Matematica

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Grubb's symbol class of parameter-dependent pseudodifferential symbols of finite regularity on R n is shown to split into so-called weakly and strongly parameter-dependent symbols. For weakly parameter-dependent symbols the regularity is shown to have an interpretation as a polynomial weight in the space of homogeneous components. For a suitable sub-class of weakly parameter-dependent symbols we establish a complete pseudodifferential calculus with ellipticity that implies invertibilty of parameter-dependent operators for large values of the parameter. Combined with strongly parameterdependent symbols we obtain a refined calculus for operators of finite regularity, where ellipticity is also defined in case of vanishing regularity. Applications concern resolvents of pseudodifferential operators and parameter-dependent operators of Toeplitz type. PSEUDODIFFERENTIAL OPERATORS OF FINITE REGULARITY 5 with the modulus |y| outside some compact set. If y = (ξ, µ), we write shortly |ξ, µ| := |(ξ, µ)|, ξ, µ = (ξ, µ) , and [ξ, µ] := [(ξ, µ)].A 0-excision function on R m is a smooth function χ(y) that vanishes in a neighborhood of the origin and such that (1 − χ)(y) has compact support. Functions of the form (1 − χ)(y) are called cut-off functions.

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Boundary value problems on manifolds with fibered boundary

Cited by 6 publications

References 16 publications

Elliptic Operators on Manifolds with Singularities and K-Homology

Elliptic Operators on Manifolds with Singularities and K-Homology

On the Poincaré isomorphism in K-theory on manifolds with edges

Parameter-dependent pseudodifferential operators of Toeplitz type

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