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We prove global wellposedness of the Klein-Gordon equation with powernonlinearity $|u|^{\alpha-1}u$, where $\alpha\in\left[1,\frac{d}{d-2}\right]$,in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\timesM_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is anapplication of the high-low method described by Bourgain [1] where theKlein-Gordon equation is studied in one dimension with cubic nonlinearity forinitial data in Sobolev spaces.

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