Lagrangian Distributions and Fourier Integral Operators

Summary

In Section 18.2 we introduced the space of conormal distributions associated with a submanifold Y of a manifold X. This is a natural extension of the classical notion of multiple layer on Y. All such distributions have their wave front sets in the normal bundle of Y which is a conic Lagrangian manifold. In Section 25.1 we generalize the notion of conormal distribution by defining the space of Lagrangian distributions associated with an arbitrary conic Lagrangian Λ ⊂ T*(X)\0. This is the space of distributions u such that there is a fixed bound for the order of P₁, ... P _N u for any sequence of first order pseudo-differential operators P₁,...,P_N with principal symbols vanishing on Λ. This implies that WF(u) ⊂ Λ. Symbols can be defined for Lagrangian distributions in much the same way as for conormal distributions. The only essential difference is that the symbols obtained are half densities on the Lagrangian tensored with the Maslov bundle of Section 21.6.

In Section 25.2 we introduce the notion of Fourier integral operator; this is the class of operators having Lagrangian distribution kernels. As in the discussion of wave front sets in Section 8.2 (see also Section 21.2) it is preferable to associate a Fourier integral operator with the canonical relation ⊂(T*(X)\0)×(T*(Y)\0) obtained by twisting the Lagrangian with reflection in the zero section of T*(Y). We prove that the adjoint of a Fourier integral operator associated with the canonical relation C is associated with the inverse of C, and that the composition of operators associated with C₁ and C₂ is associated with the composition C₁ ο C₂ when the compositions are defined. Precise results on continuity in the H_(s) spaces are proved in Section 25.3 when the canonical relation is the graph of a canonical transformation. We also study in some detail the case where the canonical relation projects into T*(X) and T*(Y) with only fold type of singularities.

The real valued C∞ functions vanishing on a Lagrangian ⊂T*(X)\0 form an ideal with dim X generators which is closed under Poisson brackets. We define general Lagrangian ideals by taking complex valued functions instead. With suitable local coordinates in X they always have a local system of generators of the form

$$ x_j - \partial H\left( \xi \right)/\partial \xi _j ,\quad j = 1, \ldots ,n, $$

, just as in the real case. The ideal is called positive if Im H≦0. This condition is crucial in the analysis and turns out to have an invariant meaning. Distributions associated with positive Lagrangian ideals are studied in Section 25.4. The corresponding Fourier integral operators are discussed in Section 25.5. The results are completely parallel to those of Sections 25.1, 25.2 and 25.3 apart from the fact that for the sake of brevity we do not extend the notion of principal symbol.