Best constants in some operator smoothness estimates

We provide a short proof of the inequality (cf. Ben-Artzi and Klainerman [Regularity and decay of evolution equations, preprint] and Kato and Yajima [Rev. Math. Phys. 1 (1989), 481–496]) ∝−∞^∞∥(1 + x^2)^(−12)(1 − Δ)^(14)e^(itΔ)u∥^2dt ⩽ C ∥u∥^2 with explicit (essentially exact) values for C.