On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

Abstract

Abstract. 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 MaxwellBoltzmann functional equation and hence Gaussian. For u 2 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 u 2 in the classical Sobolev spaces, we show that Gaussians are not extremisers in the latter case for spatial dimensions strictly greater than two.

Some Sharp Bilinear Space–time Estimates for the Wave Equation

“…The question of determining sharp constants and identifying extremisers for weighted linear and bilinear space-time estimates for dispersive and wave-like propagators has been studied in a number of recent papers. In addition to [7] and [15] already mentioned, see [5], [12], [20] and [35] (as well as [15], again) for the Schrödinger equation, [21], [34] and [36] for the Klein-Gordon equation, [6] for the wave equation, and [8], [9] and [33] as well as further references contained in these papers, for more general propagators and spatial weights in the linear setting on L 2 .…”

A sharp trilinear inequality related to Fourier restriction on the circle

“…The literature on sharp Fourier restriction inequalities related to the paraboloid and cone is extensive and we highlight the works [1,4,6,14,19,21,26]. Other interesting works on sharp Strichartz-type estimates and on the existence of extremizers for other Fourier restriction estimates include [2,3,5,12,13,16,18,20,23,25,27,28,29].…”

Some recent progress on sharp Fourier restriction theory

“…The subject is becoming more popular, as shown by the increasing number of works that appeared in the last five years. We have attempted to give a rather complete set of references, which includes several interesting works [3,4,6,7,8,11,29,31,38,41,42,46] that will not be discussed here. Given its young age, there are plenty of open problems in the area.…”