TY - JOUR

T1 - Fourier integral operators and the index of symplectomorphisms on manifolds with boundary

AU - Battisti, Ubertino

AU - Coriasco, Sandro

AU - Schrohe, Elmar

N1 - Funding Information: Thanks are due to L. Fatibene, A. Fino and R. Melrose for fruitful discussions, and to C. Epstein for explaining part of his work to us. We also want to express our special gratitude to R. Nest, with whom we worked on the proof of the Fredholm property. The first author has been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by the DAAD . Publisher Copyright: © 2015 Elsevier Inc. Copyright: Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - Given two compact manifolds with boundary X, Y, and a boundary preserving symplectomorphism χ:T*Y{set minus}0→T*X{set minus}0, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with χ. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how-in the spirit of a classical construction by A. Weinstein-a Fredholm operator of this type can be associated with χ and a section of the Maslov bundle. If dim Y>2 or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.

AB - Given two compact manifolds with boundary X, Y, and a boundary preserving symplectomorphism χ:T*Y{set minus}0→T*X{set minus}0, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with χ. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how-in the spirit of a classical construction by A. Weinstein-a Fredholm operator of this type can be associated with χ and a section of the Maslov bundle. If dim Y>2 or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.

KW - Boutet de Monvel algebra

KW - Canonical transformation

KW - Fourier integral operator

KW - Manifold with boundary

UR - http://www.scopus.com/inward/record.url?scp=84945437106&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2015.06.001

DO - 10.1016/j.jfa.2015.06.001

M3 - Article

AN - SCOPUS:84945437106

VL - 269

SP - 3528

EP - 3574

JO - Journal of functional analysis

JF - Journal of functional analysis

SN - 0022-1236

IS - 11

ER -