The Relative Index for Corner Singularities

Complex Anal. Oper. Theory

Wei

2013

Self Cite

We establish a quantisation of corner-degenerate symbols, here called Mellin-edge quantisation, on a manifold M with second order singularities. The typical ingredients come from the "most singular" stratum of M which is a second order edge where the infinite transversal cone has a base B that is itself a manifold with smooth edge. The resulting operator-valued amplitude functions on the second order edge are formulated purely in terms of Mellin symbols taking values in the edge algebra over B. In this respect our result is formally analogous to a quantisation rule of [9] for the simpler case of edge-degenerate symbols that corresponds to the singularity order 1. However, from the singularity order 2 on there appear new substantial difficulties for the first time, partly caused by the edge singularities of the cone over B that tend to infinity.for any cut-off function ω. Similarly as (1.3) we have K 0,0 (X ∧ ) = r −n/2 L 2 (R + × X). Moreover, for every s, γ, e ∈ R we form the spaces K s,γ;e (X ∧ ) := r −e K s,γ (X ∧ ).For purposes below we introduce a family of isomorphisms κ δ : K s,γ (X ∧ ) → K s,γ (X ∧ ), (κ δ u)(r, x) := δ n+1 2 u(δr, x), where n := dim X. Below we will also employ subspaces K s,γ Θ (X ∧ ) ⊂ K s,γ (X ∧ ) of flatness −ϑ relative to γ for some −∞ ≤ ϑ < 0 and the half-open weight interval Θ = (ϑ, 0]. More precisely, we set K s,γ Θ (X ∧ ) = lim ←− ε>0 K s,γ−ϑ−ε (X ∧ ) in the respective Fréchet topology. In the case Θ = (−∞, 0] the space K s,γ Θ (X ∧ ) = K s,∞ (X ∧ ) is independent of γ.

“…Concerning the proof and more details, cf. [24], [12]. More precisely, we can choose h in such a way that for some ψ ∈ C ∞ 0 (R + ) which is equal to 1 close to 1 we have…”

Section: The Mellin-edge Quantisationmentioning

confidence: 99%

The Mellin-Edge Quantisation for Corner Operators

Complex Anal. Oper. Theory

Wei

2013

Self Cite

“…Another motivation is, of course, to express parametrices of elliptic elements within the calculus which belongs to one of our results; special cases have been treated before, cf., [39], [46]. Other contributions to the higher corner calculus are [45], and the author's joint papers with Maniccia [29], Krainer [27], Calvo and Martin [5], Calvo [6], Harutyunyan [19], [20], [21]. Let B ∈ M 1 ; then a starting point are corner-degenerate families…”

Section: Higher Corner Operatorsmentioning

confidence: 99%

The Iterative Structure of the Corner Calculus

Pseudo-Differential Operators: Analysis, Applications and Computations

2011

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We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference "Elliptic and Hyperbolic Equations on Singular Spaces", October 27 -31, 2008, at the MSRI, University of Berkeley. Mathematics Subject Classification:Primary: 35S35 Secondary: 35J70 Keywords: Categories of stratified spaces, ellipticity of corners operators, principal symbolic hierarchies, boundary value problems, parametrices in algebras of corner operators Boundary symbols are homogeneous in the senseHere κ λ : H s (R + ) → H s−m (R + ) is a strongly continuous group of isomorphisms, defined by (κ λ u)(r) := λ 1/2 u(λr), λ ∈ R + . A similar relation holds for edge symbols, based onIn this presentation we give an idea on how to formulate algebras of (pseudo-differential) operators on int M that contain the (for the nature of singularities typical) differential operators, together with the parametrices of elliptic elements. More details may be found in [45], [46], and in a new monograph in preparation [47], see also the references below. 2 1 The category M k Stratified spaces of different kind occur in numerous fields of mathematics and also in the applied sciences. Here we single out specific categories of such spaces where certain elements of the analysis of PDE can be formulated in an iterative manner. General references on stratified spaces are Fulton and MacPherson [14], or Weinberger [57]. Definition 1.1. A topological space M (under some natural conditions on the topology in general ) is said to be a manifold with singularities of order k ∈ N, k ≥ 1, ifTransition maps X ∆ → X ∆ are induced by restrictions of M k−1 -isomorphisms R×X → R×X to R + × X. This gives rise to corresponding transition maps X ∆ × Ω → X ∆ ×Ω for the respective X ∆ -bundles over Y .Remark 1.2. M k is a category with a natural notion of morphisms and isomorphisms.as a disjoint union of strata Y j ∈ M 0 . We set int M := Y 0 , called the maximal stratum of M, and dim M := dim(int M ). Moreover, M is locally near Y j modelled on an X ∆ j−1 -bundle over Y j , forRemark 1.4. The same topological space M can be stratified in different ways. For instance, we have M = R n ∈ M 0 but also M ∈ M 1 when we setThere are many other interesting properties of the categories M k that we do not discuss in detail here. It would be desirable to develop the connection of our analysis on singular spaces with the work from topological side. For instance, D. Trotman informed me in Berkeley on his works with coauthors, cf.[3] jointly with Bekka, and [23] with King. AbstractWe give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference "

“…In the present case the conditions are to be posed both on edges of first and second generation; this will be done in a future paper. Let us finally give a number of references that have from different point of view connections with this paper, namely, Agranovich and Vishik [1] (parameterdependent calculus), Eskin [6], Rempel and Schulze [13], Grubb [9] (pseudo-differential calculus of boundary value problems), Witt [19] (structure of operator-valued Mellin symbols), Seiler [18] (Green operators in the cone algebra), Schulze [17] (cone calculus when the base has smooth edges), and joint works of the second author with Kapanadze [12] and Harutjunjan [10] (various models of applications, especially, crack theory, and higher corner Mellin symbols).…”

Section: (X ∧ × ω)) Let Diffmentioning

confidence: 99%

Edge symbolic structures of second generation

Calvo

Mathematische Nachrichten

2009

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Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions. Mathematics Subject Classification: 35J70, 35S05, 58J40Keywords: Operators on manifolds with second order singularities, edge quantizations, continuity in Sobolev spaces with double weights, compositions in the edge calculus.