Hypoellipticity, Solvability and Construction of Solutions with Prescribed Singularities for Several Classes of Pde Having Symplectic

This paper deals with the hypoellipticity, local solvability and construction of solutions with prescribed singularities for several classes of PDE having double symplectic characteristics. We not only propose a short survey but we investigate several instructive model examples as well. As our results are obtained in the C∞ category, it is interesting to study the same operators in the Gevrey category too. 1. Several definitions and formulation of the main results 1. In the paper under consideration we denote by Lm(X) the set of all classical scalar properly supported pseudodifferential operators of order m and D′(X) stands for the set of all Schwartz distributions on the smooth manifold X . As usual the closed conic in ξ set W F(u), u ∈ D′(X) (wave front set of u) is defined by WF(u) = { ρ ∈ T ∗X\0 : a ∈ L0(X), a(x,D)u ∈C∞(X) ⇒ a0(ρ) = 0 } . We have denoted by a0(ρ) = τ(a) the principal symbol of the operator a(x,D)∈ L0(X). The s-wave front set of u ∈ D′(X), s ∈ R1 is given by WFs(u) = { ρ ∈ T ∗(X)\0 : a ∈ L(X), a(x,D)u ∈ H(X) ⇒ a(ρ) = 0 } . Certainly, ρ = (x,ξ), ξ 6= 0 and WFs(u) is a closed conical in ξ set. Evidently, s′ < s ⇒WFs(u) ⊂WFs′(u). Let V ⊂ T ∗(X)\0 be an open conical in ξ set and N is a closed cone in ξ contained in T ∗(X)\0, N ⊂V . THEOREM 1. [11] Assume that the operator P ∈ Lm(X), s′ < s. Suppose that it does not exist a function u ∈ Hs(X) such that (∗) V ∩WF(Pu) = / 0, V ∩WFs(u) = V ∩WF(u) = N. Then there exists ρ0 ∈ N, pseudodifferential operators ψ, φ, φ′ ∈ L0(X), cone supp φ ⊂ V\N, cone supp φ′ ⊂ V, ψ(ρ) ≡ 1 in a tiny neighborhood of ρ0, C = const > 0, μ ∈ Z+ and such that (1) ‖ψw‖s ≤C [ ‖φ′Pw‖μ +‖φw‖μ +‖w‖s′ ] , ∀w ∈C∞ 0 (X). ∗It is a pleasure to dedicate this paper to Prof. Luigi Rodino on the occasion of his 60th birthday.