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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 = {D. Calvo and B. W. Schulze }, 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 AU - D. Calvo & B. W. Schulze 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 and B. W. Schulze . (2006). Operators on Corner Manifolds with Exit to Infinity. Journal of Partial Differential Equations. 19 (2). 147-192. doi:
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