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.

中文翻译：

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Gagliardo-Nirenberg-Sobolev 不等式的全球版本及其在波方程和 Klein-Gordon 方程中的应用

我们证明了 Gagliardo-Nirenberg 插值不等式的全局或时空加权版本，具有 $L^p$ ($p < \infty$) 端点，适用于双曲面叶理。具有 $L^\infty$ 端点的相应版本由 Klainerman 首次引入，是经典矢量场方法的基础，现在已成为研究非线性演化方程长期行为的标准技术之一。我们的追求受到将向量场方法应用于具有不断增长的更高阶能量的能量层次的设置的激励。在这些设置中，使用 Sobolev 不等式的 $L^p$ 端点版本可以允许人们在估计中获得本质上的一个导数，然后将给出相应的衰减率增益。论文最后分析了一个这样的模型问题，