Uniform Estimates for a Class of Evolution Equations

L∞ estimates are derived for the oscillatory integral ∫+0∞e−i(xλ + (1/m) tλm)a(λ) dλ, where 2 ≤ m ∈ R and (x, t) ∈ R × R+. The amplitude a(λ) can be oscillatory, e.g., a(λ) = eitp(λ) with p(λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)k with 0 ≤ k ≤ 12(m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.