A commuting vectorfields approach to Strichartz-type inequalities and applications to quasi-linear wave equations

We give a new, very different, proof of the recent results of Bahouri and Chemin and Tataru concerning improved well posedness results for quasilinear wave equations. At the heart of such results lies a proof of Strichartz type estimate for solutions to linear wave equations with rough coefficients. Instead of relying on parametrices, we develop a different, geometric mehod of deriving such estimates based on generalized energy estimates and commuting vectorfields.