Optimal constants and maximising functions for Strichartz inequalities

[摘要] We prove sharp weighted bilinear inequalities which are global in time and for general dimensions for the free wave, Schrödinger and Klein-Gordon propagators. This extends work of Ozawa –Rogers for the Klein-Gordon propagator, work of Foschi-Klainerman and Bez-Rogers for the wave propagator, and work of Ozawa-Tsutsumi, Planchon-Vega and Carneiro for the Schrödinger propagator. In each case, we make a connection to estimates involving certain dispersive Sobolev norms. As a consequence of these estimates we obtain, among other things, a new sharp form of a linear Strichartz estimate for the solution of the Klein-Gordon equation in five spatial dimensions for data belonging to H1, and that maximisers do not exist for this estimate. We also obtain a new sharp form of a linear Sobolev- Strichartz estimate for the wave equation in four space dimensions for initial data in H¾ x H-1/4 and characterisation of the maximisers. Finally, we study the variational problems associated to the linear Sobolev-Strichartz estimates for the Schrödinger and wave equations. We establish that Gaussian functions are not maximisers for the Hm to LP inequalities for the Schrödinger propagator, for any m > 0, and make a conjecture about the nature of the maximisers for the H d-1/4xHd-5/4 to L4 inequalities for the wave equation.

[发布机构] University:University of Birmingham;Department:School of Mathematics

[关键词] Q Science;QA Mathematics