The Calderón Projector for Fibred Cusp Operators

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Karsten Fritzsch
  • Daniel Grieser
  • Elmar Schrohe

Organisationseinheiten

Externe Organisationen

  • Carl von Ossietzky Universität Oldenburg
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Details

OriginalspracheEnglisch
Aufsatznummer110127
FachzeitschriftJournal of Functional Analysis
Jahrgang285
Ausgabenummer10
Frühes Online-Datum9 Aug. 2023
PublikationsstatusVeröffentlicht - 15 Nov. 2023

Abstract

A Calder\'on projector for an elliptic operator \(P\) on a manifold with boundary \(X\) is a projection from general boundary data to the set of boundary data of solutions \(u\) of \(Pu=0\). Seeley proved in 1966 that for compact \(X\) and for \(P\) uniformly elliptic up to the boundary there is a Calder\'on projector which is a pseudodifferential operator on \(\partial X\). We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity. This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the \(\phi\)-pseudodifferential calculus introduced by Mazzeo and Melrose. In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.

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The Calderón Projector for Fibred Cusp Operators. / Fritzsch, Karsten; Grieser, Daniel; Schrohe, Elmar.
in: Journal of Functional Analysis, Jahrgang 285, Nr. 10, 110127, 15.11.2023.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Fritzsch K, Grieser D, Schrohe E. The Calderón Projector for Fibred Cusp Operators. Journal of Functional Analysis. 2023 Nov 15;285(10):110127. Epub 2023 Aug 9. doi: 10.48550/arXiv.2006.04645, 10.1016/j.jfa.2023.110127
Fritzsch, Karsten ; Grieser, Daniel ; Schrohe, Elmar. / The Calderón Projector for Fibred Cusp Operators. in: Journal of Functional Analysis. 2023 ; Jahrgang 285, Nr. 10.
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abstract = "A Calder{\'o}n projector for an elliptic operator P on a manifold with boundary X is a projection from general boundary data to the set of boundary data of solutions u of Pu=0. Seeley proved in 1966 that for compact X and for P uniformly elliptic up to the boundary there is a Calder{\'o}n projector which is a pseudodifferential operator on ∂X. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity. This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the ϕ-pseudodifferential calculus introduced by Mazzeo and Melrose. In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.",
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AU - Schrohe, Elmar

N1 - Funding Information: Part of this work was done while DG was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall semester 2019, supported by the National Science Foundation under Grant No. DMV-1440140 . DG and ES were partially supported by DFG Priority Programme 2026 ‘Geometry at Infinity’.

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N2 - A Calderón projector for an elliptic operator P on a manifold with boundary X is a projection from general boundary data to the set of boundary data of solutions u of Pu=0. Seeley proved in 1966 that for compact X and for P uniformly elliptic up to the boundary there is a Calderón projector which is a pseudodifferential operator on ∂X. We generalize this result to the setting of fibred cusp operators, a class of elliptic operators on certain non-compact manifolds having a special fibred structure at infinity. This applies, for example, to the Laplacian on certain locally symmetric spaces or on particular singular spaces, such as a domain with cusp singularity or the complement of two touching smooth strictly convex domains in Euclidean space. Our main technical tool is the ϕ-pseudodifferential calculus introduced by Mazzeo and Melrose. In our presentation we provide a setting that may be useful for doing analogous constructions for other types of singularities.

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