TY - UNPB

T1 - Fourier Integral Operators of Boutet de Monvel Type

AU - Battisti, Ubertino

AU - Coriasco, Sandro

AU - Schrohe, Elmar

PY - 2014

Y1 - 2014

N2 - Given two compact manifolds \(X,Y,\) with boundary and a boundary preserving symplectomorphism \(\chi:T^*Y\setminus0\to T^*X\setminus0\), which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with \(\chi\). 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 \(\chi\) 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 \(X,Y,\) with boundary and a boundary preserving symplectomorphism \(\chi:T^*Y\setminus0\to T^*X\setminus0\), which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with \(\chi\). 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 \(\chi\) 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 - math.FA

KW - math.OA

KW - 35S30, 46F10, 47L80, 58J32, 58J40

M3 - Preprint

BT - Fourier Integral Operators of Boutet de Monvel Type

ER -