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

## TL;DR

This paper establishes new global weighted Gagliardo-Nirenberg-Sobolev inequalities adapted to hyperboloidal foliation, enabling improved decay estimates in nonlinear wave and Klein-Gordon equations.

## Contribution

It introduces novel space-time weighted inequalities with $L^p$ endpoints, enhancing energy estimates and decay rates in nonlinear evolution equations.

## Key findings

- New $L^p$ endpoint inequalities for hyperboloidal foliation
- Improved decay estimates in wave and Klein-Gordon equations
- Application to a model problem demonstrating effectiveness

## 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.