I'm trying to prove global wellposedness for the Klein-Gordon-Equation with **radial** initial data. I'm therefore searching for/trying to prove strichartz estimates of the form: $$ ||e^{it\langle D\rangle}u||\_{L^q\_{t}L^r\_{x}}\lesssim||\langle D\rangle^lu||\_{L^{2}}$$ or $$||P\_{L}((e^{it\langle D\rangle}u\_{M})(e^{it\langle D\rangle}v\_{N}))||\_{L\_{2}}\lesssim L^lM^mN^n||u\_{M}||\_{L^{2}}||v\_{N}||\_{L^{2}}$$ with indices $l,m,n$ as small as possible. Where $P\_{L}$ is the Littlewoodpaley projector, $e^{it\langle D\rangle}$ the Propagator of the (mass 1) Klein-Gordon-Equation and $L,M,N$ frequencies.


I'm aware, that you have more admissible strichartz exponentens $(q,r)$ for the first kind of estimate if you have radial data (the result is outlined here: <http://wiki.math.toronto.edu/DispersiveWiki/index.php/Strichartz_estimates>) but I can't see whether/how that can help me since I will get the same $l$ as with none radial data. Am i missing something here?


Also I don't know how to proof the first inequalitity for non-radial-data. I know it is true for $2\leq r<\infty$, $2/q+n/r=n/2$, $l=1/q-1/r+1/2$ and was told to find the proof in Delort and Fang(2000) but I can't. (I hope the proof will help me understand how I can improve the result for radial functions)



Strichartz Estimates for radial Klein-Gordon equation
You can find a complete set of Strichartz estimates for the nonradial KG equation in Lemma 3 in [this paper](https://projecteuclid.org/euclid.rmi/1049123084) by Machihara, Nakanishi and Ozawa. Let me add that, usually, in proofs of local or global well posedness for the wave or KG equation, the non-endpoint Strichartz estimates are more than enough. Thus the wider range of admissible indices in the radial case is not very helpful in general. It could be useful in some cases in which you expect for some reason a better result for radial data.


