  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.
