[Paper Review] Fourier Integral Operators of Boutet de Monvel Type

Ubertino Battisti, Sandro Coriasco+1 more|arXiv (Cornell University)|Jul 10, 2014

Geometry and complex manifolds18 references3 citations

TL;DR

This paper introduces a new class of Fourier integral operators of Boutet de Monvel type on compact manifolds with boundary, associated with boundary-preserving symplectomorphisms that are homogeneous of degree one in the fibers and satisfy the transmission condition. It establishes a full calculus, proves an Egorov-type theorem, defines ellipticity implying the Fredholm property, and constructs a Fredholm operator from the symplectomorphism and a section of the Maslov bundle, showing its index is an invariant of the symplectomorphism when dim Y > 2 or the Maslov bundle is trivial.

ABSTRACT

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

Motivation & Objective

• To develop a calculus of Fourier integral operators on compact manifolds with boundary that generalizes Boutet de Monvel's pseudodifferential calculus.
• To establish mapping properties between Sobolev spaces for these operators.
• To define a notion of ellipticity that implies the Fredholm property.
• To construct a Fredholm operator from a boundary-preserving symplectomorphism and a section of the Maslov bundle.
• To show that the index of this operator is an invariant of the symplectomorphism when dim Y > 2 or the Maslov bundle is trivial.

Proposed method

• The operators are defined as truncated FIOs of the form $ r^+ A^ ho e^+ $, where $ A^ ho $ is an FIO associated with a symplectomorphism $ \rho $, and $ r^+, e^+ $ are restriction and extension-by-zero operators.
• The construction relies on operator-valued symbols in $ S^m(\mathbb{R}^{n-1}, \mathbb{R}^{n-1}; \mathscr{S}(\mathbb{R}_+), \mathscr{S}(\mathbb{R}_+)) $, using phase functions representing $ \rho $ and its boundary lift.
• The transmission condition is imposed on each component of $ \rho $, ensuring continuity from $ C^\infty(Y) $ to $ C^\infty(X) $ and from $ C^\infty(X) $ to $ C^\infty(Y) $.
• A calculus is developed using operator-valued pseudodifferential operators, with composition rules given by asymptotic expansions of symbols.
• The Egorov-type theorem is proven, relating the conjugation of operators by the FIO to the symplectomorphism's action on the symbol class.
• The Fredholm property is established via a notion of ellipticity, and the index is linked to the Maslov bundle through a section, with topological invariance under specified conditions.

Experimental results

Research questions

• RQ1 Can a calculus of Fourier integral operators of Boutet de Monvel type be developed for symplectomorphisms between manifolds with boundary that are homogeneous of degree one in the fibers and satisfy the transmission condition?
• RQ2 Does the proposed class of operators admit a full calculus, including an Egorov-type theorem, and what are their mapping properties on Sobolev spaces?
• RQ3 Can a notion of ellipticity be defined for these operators that implies the Fredholm property?
• RQ4 How can a Fredholm operator be naturally associated with a given symplectomorphism and a section of the Maslov bundle?
• RQ5 Is the index of such a Fredholm operator independent of the choice of section when dim Y > 2 or the Maslov bundle is trivial?

Key findings

• The truncated FIO $ r^+ A^ ho e^+ $ is continuous from $ C^rown(Y) $ to $ C^rown(X) $ if the symplectomorphism $ \rho $ satisfies the transmission condition.
• The operator $ r^+ A^ ho e^+ $ is shown to be an operator-valued symbol in $ S^m(\mathbb{R}^{n-1}, \mathbb{R}^{n-1}; \mathscr{S}(\mathbb{R}_+), \mathscr{S}(\mathbb{R}_+)) $, enabling the use of pseudodifferential calculus.
• A full calculus is developed, including composition and adjoint formulas, with symbol asymptotics given by $ a(y,\eta) \sim \sum_\alpha \frac{1}{\alpha!} \partial_\eta^\alpha a_1(y,\eta) D^\alpha_y a_2(y,\eta) $.
• An Egorov-type theorem is proven, showing that conjugation by the FIO preserves the principal symbol up to the symplectomorphism's action.
• Ellipticity is defined in terms of the invertibility of the principal symbol, and this implies the Fredholm property for the associated operator.
• When dim Y > 2 or the Maslov bundle is trivial, the index of the constructed Fredholm operator is independent of the choice of section of the Maslov bundle and thus defines an invariant of the symplectomorphism $ \rho $.