Details
| Original language | English |
|---|---|
| Article number | 110127 |
| Journal | Journal of Functional Analysis |
| Volume | 285 |
| Issue number | 10 |
| Early online date | 9 Aug 2023 |
| Publication status | Published - 15 Nov 2023 |
Abstract
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.
Keywords
- math.AP, math.DG, 58J40 (Primary) 35J75, 58J32, 35J58 (Secondary), Boundary value problems, Pseudodifferential operators, Cusp singularities, Calderón method
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
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In: Journal of Functional Analysis, Vol. 285, No. 10, 110127, 15.11.2023.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Calderón Projector for Fibred Cusp Operators
AU - Fritzsch, Karsten
AU - Grieser, Daniel
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’.
PY - 2023/11/15
Y1 - 2023/11/15
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.
AB - 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.
KW - math.AP
KW - math.DG
KW - 58J40 (Primary) 35J75, 58J32, 35J58 (Secondary)
KW - Boundary value problems
KW - Pseudodifferential operators
KW - Cusp singularities
KW - Calderón method
UR - http://www.scopus.com/inward/record.url?scp=85168487899&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2006.04645
DO - 10.48550/arXiv.2006.04645
M3 - Article
VL - 285
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 10
M1 - 110127
ER -