Intersecting Legendrians and blow-ups. (English) Zbl 0998.58021

Hassell, Andrew; Vasy, András

Math. Res. Lett. 8, No. 4, 413-428 (2001).

Summary: The purpose of this note is to describe the relationship between two classes of Legendre distributions. These two classes are distributions associated to an intersecting pair of Legendre submanifolds, introduced in [A. Hassell, Geom. Funct. Anal. 10, No. 1, 1–50 (2000; Zbl 0953.35122)] by analogy with intersecting Lagrangian distributions of R. B. Melrose and G. A. Uhlmann [Lagrangian intersection and the Cauchy problem. Commun. Pure. Appl. Math. 32, 483–519 (1979; Zbl 0396.58006)], and Legendre distributions associated to a fibred scattering structure introduced in [A. Hassell and A. Vasy, J. Anal. Math. 79, 241–298 (1999; Zbl 0981.58025)]. We prove that, given appropriate symbolic orders, the first class is a proper subset of the second. We also give an example in two dimensions, which shows explicitly the relation between the two spaces in a simple setting.

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
58J40	Pseudodifferential and Fourier integral operators on manifolds
53D99	Symplectic geometry, contact geometry

Keywords:

Legendre distributions

Citations:

Zbl 0953.35122; Zbl 0396.58006; Zbl 0981.58025