Local smoothing and convergence properties of Schrödinger type equations

The initial value problem for the Schrödinger type equation ∂u ∂t = iP(D)u, u(0, x) = u₀(x), x ∈ Rⁿ, is considered. Here P(ξ) is a principal type polynomial of order m (i.e., |▽P(ξ)| ≥ C(1 + |ξ|)^{m - 1}, |ξ| ≥ R), or a certain type of other real symbol such as P(ξ) = |ξ|^m, m > 1. The following results are proved: (a) If u₀ ∈ L²(Rⁿ), then for any fixed R > 0 write u₀ = u₁ + u₂, where u₁ and u₂ are square integrable and the Fourier transform of u₁, vanishes for |ξ| ≥ R and that of u₂ vanishes for |ξ| ≤ R. If u₁(t, x) = e^itP(D)u₁(x), u₂(t, x) = e^itP(D)u₂(x), then u₁ is real analytic in R × Rⁿ and satisfies for every multi-index α, sup_{t, x} |D^αu₁(t, x)| ≤ C_α ∥u₀∥_L², while (1+|x|²)^{- 1 2}u₂(t,x)∈L²(R_t;H ^{(m-1) 2}(R_xⁿ)). (b) If u₀∈L^2,S(Rⁿ), S>0 then for every t≠0, u(t,·)∈H_loc^(m-1)S(Rⁿ). (c) if u₀∈H^S(Rⁿ), S> 1 2then for a.e. x∈Rⁿ,u(t,x)→u₀(x) as t→0.