The initial value problem for one dimensional Shrödinger equation is

$$iu_{t}+u_{xx}=0,$$ $$u(x, 0)= f(x),$$

where $u:\mathbb R \times \mathbb R \rightarrow \mathbb C$ is a complex valued function.

Lemma. For any $f\in S$, where $S$ is space of Schwartz class function on $\mathbb R$, there is a unique solution $u\in C^{\infty} (\mathbb R; S) $ of the above a IVP. The special Fourier transform of the solution is given by $\hat{u}(k, t)= e^{-it |k|^{2}}\hat{f}(k)$ and $$u(x, t)= \frac {1}{(4\pi i t)^{\frac {1} {2}}}\int_{\mathbb R} e^{\frac {-i|x-y|^{2}}{4t}} f(y) dy .$$

My question is: (a) What is Tomas- Stein restriction theorem (inequality ) for paraboloid ? (b) and using this how one does conclude that,

$$\| u \|_{L^{6} (\mathbb R \times \mathbb R)}\leq \| f\| _{L^{2}(\mathbb R)}$$.