Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP:
$$i\partial_t u(x,t) + \Delta u(x,t)=0$$ $$u(x,0)=u_0$$
Then the following estimates hold:
$a)$ if $D^\gamma$ is the fractional derivative $D^\gamma _x =\int e^{ix\epsilon } |\epsilon|^{\gamma} \hat{f(\epsilon)}d\epsilon$ then $\|D^\gamma _x e^{it\Delta}u_0\|_{L^q_x(L^2_t)}\leq C\|u_0\|_{L^2}$
$b)$ $\|D^\gamma _x e^{it\Delta}u_0\|_{B^*_{1/2}(L^2)}\leq C\|u_0\|_{L^2}$
And this is know as Strichartz Estimate for the Schrödinger equation. I know this was proved by Strichartz and also by Thao for more general family of functions but the way to do this with polar coordinates is writing $D^{\gamma}_xe^{it\Delta}u_0(x)=\int e^{ix\epsilon} e^{it|\epsilon|^2}|\epsilon|^{\gamma} \hat{u_0}(\epsilon)d\epsilon $ and then writing this in polar coordinates. I dont know how to apply polar coordinates in this step $(r,\theta)$ would it be just:
$\int e^{ix\cdot (r\cos(\theta),r\sin(\theta))} e^{it|r|^2}r^{\gamma} \hat{u_0}(r,\theta)rdr d\theta $ but then changing the variable $s=r^2$ would difficult me the sinus and cosine terms... so I think I am doing something wrong.
Then I have to use Plancherel in $t$ and then Minkowski integral inequality but I don't know how to follow from here.
Thanks in advance.
