Jean-Michel Bony

Evolution Equations and Generalized Fourier Integral Operators

We consider evolution equations ∂u/∂t = ia w (x, D)u where a is the (real valued) Weyl symbol of the operator A = a w . For instance, Schrodinger-like equations. After recalling what are generalized Fourier integral operators in the framework of the Weyl-Hormander calculus, we give conditions on a and on the dynamics of its hamiltonian flow which imply: 1. The operator a w is essentially self-adjoint and the propagators e itA are bounded between (conveniently related) generalized Sobolev spaces. 2. The propagators e itA are generalized Fourier integral operators.

On the differentiability class of the admissible square roots of regular nonnegative functions

We investigate the possibility of writing f = g 2 when f is a C k nonnegative function with k ≥ 6. We prove that, assuming that f vanishes at all its local minima, it is possible to get g ∈ C 2 and three times differentiable at every point, but that one cannot ensure any additional regularity.

Fourier integral operators and Weyl-Hörmander calculus

It is well known that the space of classical pseudo-differential operators is invariant under conjugation by classical Fourier integral operators. However, the Weyl-Hormander calculus [Ho1] [Ho2] provides a much larger framework for the theory of pseudo-differential operators. Any riemannian metric g on the phase space R n x × R n ξ, satisfying the conditions of definition 1.1, defines a graded algebra of pseudo-differential operators. The classical theory corresponds to a particular metric, namely g(dx,dξ) = dx 2 + dξ2/〈ξ〉2.

Weyl quantization and Fourier integral operators

The Weyl-Hormander calculus [Hol], [Ho2] provides a framework for the theory of pseudo-differential operators which is very general and, in spirit, quite simple. The data is just a riemannian metric on the phase space satisfying some conditions, and to this metric is associated a graded algebra of pseudo-differential operators having essentially the properties one can expect.

On the Characterization of Pseudodifferential Operators (Old and New)

In the framework of the Weyl–Hormander calculus, under a condition of “geodesic temperance”, pseudodifferential operators can be characterized by the boundedness of their iterated commutators. As a corollary, functions of pseudodifferential operators are themselves pseudodifferential. Sufficient conditions are given for the geodesic temperance. In particular, it is valid in the Beals–Fefferman calculus.