Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$

Abstract

In this paper we prove standard and reversed Strichartz estimates for the Klein-Gordon equation in R 3+1 . Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein-Gordon equation. Contents 1. Introduction 1.1. The linear Klein-Gordon equation 1.2. The free equation 1.3. The perturbed equation 1.4. Main results 1.5. The semilinear Klein-Gordon equation 1.6. Discussion of the results 1.7. Notations 1.8. Organization 2. The free Klein-Gordon kernels 2.1. The cosine kernel 2.2. Fractional integration 2.3. Klein-Gordon sine propagator bounds 2.4. Fractional powers 2.5. The free sine propagator 2.6. Pointwise decay estimates 3. The Klein-Gordon equation with a scalar potential 3.1. Standard Strichartz estimates 4. Nonlinear applications Appendix A. Bessel and Hankel functions Appendix B. Sharp Agmon estimates Appendix C. Comparison of Sobolev spaces References