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Volume 19, Issue 2
Operators on Corner Manifolds with Exit to Infinity

D. Calvo & B. W. Schulze

J. Part. Diff. Eq., 19 (2006), pp. 147-192.

Published online: 2006-05

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  • Abstract

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ_ ψ(A), σ_^(A), σ_^(A)), where σ_ ψ is the interior symbol and σ^(A)(y, η), (y, η) ∈ T∗Y \0, the (operator-valued) edge symbol of ‘first generation̻, cf. [1]. The novelty here is the edge symbol σ_^ of ‘second generation’, parametrised by (z, ζ) ∈ T∗Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.

  • Keywords

Operators on manifolds with edge and conical exit to infinity Sobolev spaces with double weights on singular cones parameter-dependent ellipticity edge and corner symbols

  • AMS Subject Headings

35J70 35S05 58J40.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-19-147, author = {}, title = {Operators on Corner Manifolds with Exit to Infinity}, journal = {Journal of Partial Differential Equations}, year = {2006}, volume = {19}, number = {2}, pages = {147--192}, abstract = {

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ_ ψ(A), σ_^(A), σ_^(A)), where σ_ ψ is the interior symbol and σ^(A)(y, η), (y, η) ∈ T∗Y \0, the (operator-valued) edge symbol of ‘first generation̻, cf. [1]. The novelty here is the edge symbol σ_^ of ‘second generation’, parametrised by (z, ζ) ∈ T∗Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5325.html} }
TY - JOUR T1 - Operators on Corner Manifolds with Exit to Infinity JO - Journal of Partial Differential Equations VL - 2 SP - 147 EP - 192 PY - 2006 DA - 2006/05 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5325.html KW - Operators on manifolds with edge and conical exit to infinity KW - Sobolev spaces with double weights on singular cones KW - parameter-dependent ellipticity KW - edge and corner symbols AB -

We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ_ ψ(A), σ_^(A), σ_^(A)), where σ_ ψ is the interior symbol and σ^(A)(y, η), (y, η) ∈ T∗Y \0, the (operator-valued) edge symbol of ‘first generation̻, cf. [1]. The novelty here is the edge symbol σ_^ of ‘second generation’, parametrised by (z, ζ) ∈ T∗Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.

D. Calvo & B. W. Schulze . (2019). Operators on Corner Manifolds with Exit to Infinity. Journal of Partial Differential Equations. 19 (2). 147-192. doi:
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