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. I m ~~(1+~~)~ " ~(1-4) " ~e " ~~~/*dt~C~~~~~~-cc with explicit (essentially exact) values for C. Recently Ben-Artzi and Klainerman [l] and Kato and Yajima [S] have focused interest on the estimate I I, 11(1 +x2)) " 2 (1-4)1'4ei'dUJ12dt<C IIU112, (1) a result that implies and extends a number of results in harmonic analysis, e.g. [6, S] (see [ 1 I). Our goal here is to provide an elementary proof with explicit constants. Indeed we prove that if n 3 3, where the constants are best possible. Inequalities equivalent to (2) and (3) by the Kato theory of smooth perturbations [4], but without best constants, have been known for some time, see e.g. Herbst [2, 33.