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) = e^itp(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 ≤ 1 2(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.