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This is the extended version of a lecture course given at the University of Vienna in the spring term 2005. Many thanks to the audience of this course for many keen questions. The main aim of this course was to understand the papers [ 12 ] and [ 13 ].

We shall present here a survey concerning the (microlocal) well-posedness of the Cauchy problem for a second order hyperbolic operator P around a non-effectively hyperbolic double characteristic ρ and its close relations to the behavior of the bicharacteristics of p , the principal symbol of P , near ρ . Assuming that p vanishes of second order on the smooth doubly characteristic manifold Σ and that the rank of the canonical symplectic two form is constant on Σ , the microlocal Cauchy problem is C ^∞ well posed if and only if p admits an elementary decomposition, and moreover p admits an elementary decomposition if and only if there is no bicharacteristic issuing from a simple characteristic point which has a limit point in Σ . Thus the behavior of bicharacteristics is a real object which controls the (microlocal) C ^∞ well-posedness of the Cauchy problem, while the spectral properties of the Hamilton map H _p, defined through Hesse matrix of p , cannot determine the behavior of bicharacteristics completely in general.

We extend the Fefferman-Phong inequality to certain N × N systems of PDEs, and hence generalize Sung’s result in [ S86 ], that was obtained for systems of ODEs. Our proof uses a Fefferman-Phong Calderón-Zygmund decomposition of the phase-space and induction on the size N of the system.

We examine the memorphic continuation of the cut-off resolvent R _χ( z ) = χ( U ( T , 0) ∊ z )^−1χ, χ( x ) ∊ C _0 ^∞ (ℝ^n), where U ( t, s ) is the propagator related to the wave equation with non-trapping time-periodic perturbations (potential V ( t, x ) or a periodically moving obstacle) and T > 0 is the period. Assuming that R _χ( z ) has no poles z with | z | ≥ 1, we establish a local energy decay and we obtain global Strichartz estimates. We discuss the case of trapping moving obstacles and we present some results and conjectures concerning the behavior of R _χ( z ) for | z | > 1.

We present an elementary, L ^2, proof of Fediĭ’s theorem on arbitrary (e.g., infinite order) degeneracy and extensions. In particular, the proof allows and shows C ^∞, Gevrey, and real analytic hypoellipticity, and allows the coefficents to depend on the remaining variable as well.

In these notes we discuss recent results concerning the long time evolution for variable coefficient time dependent Schrödinger evolutions in ℝ^n. Precisely, we use phase space methods to construct global in time outgoing parametrices and to prove Strichartz type estimates. This is done in the context of C ^2 metrics which satisfy a weak asymptotic flatness condition at infinity.

The note describes, in simple analytic and geometric terms, the global Poisson stratification of the characteristic variety Char L of a second-order linear differential operator − L = X _1 ^2 + ... + X _r ^2 , i.e., a sum-of-squares of real-analytic, real vector fields X _i on an analytic manifold Ω . It is conjectured that the leaves in the bicharacteristic foliation of each Poisson stratum of Char L propagate the analytic singularities of the solutions of the equation Lu = f ∊ C ^ω. Closely related conjectures of necessary and sufficient conditions for local, germ and global analytic hypoellipticity, respectively, are stated. It is an open question whether the new conjecture regarding local analytic hypoellipticity is equivalent to that put forward by the author in earlier articles.