Global versions of the Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations

Abstract:

We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.