A remark on the almost global existence theorems of Keel, Smith and Sogge

We shall give a new proof of temporally global existence of small solutions for systems of semi-linear wave equations. Our proof uses the Klainerman-Sideris inequality and a space-time L^2-estimate. We shall also discuss whether the scale-invariant version of the space-time L^2-estimates can hold, and obtain some related estimates. Among other things, we prove that the Keel-Smith-Sogge estimate actually holds in all space dimensions