On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{\alpha−1}u$, where $\alpha\in\left[1,\frac{d}{d−2}\right]$, in dimension $d\ge3$ with initial data in $M^1_{p,p'}(\mathbb{R}^d)\times M_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an application of the high-low method
described by Bourgain in [1] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces.