2017 Volume 69 Issue 2 Pages 459-476
We provide a comprehensive analysis of sharp bilinear estimates of Ozawa–Tsutsumi type for solutions u of the free Schrödinger equation, which give sharp control on |u|2 in classical Sobolev spaces. In particular, we generalise their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon–Vega, via entirely different methods, by seeing them all as special cases of a one-parameter family of sharp estimates. The extremal functions are solutions of the Maxwell–Boltzmann functional equation and hence Gaussian. For u2 we argue that the natural analogous results involve certain dispersive Sobolev norms; in particular, despite the validity of the classical Ozawa–Tsutsumi estimates for both |u|2 and u2 in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.
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