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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,

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

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1 Answer 1

If $M_0$ is a compact subset of a hypersurface $M$ with novanishing Gauss curvature, then $$ \|\widehat{fd\sigma}\|_{L^p(\mathbb{R}^n)}\leq C(n,p)\|f\|_{L^2(M_0)},\quad p=\frac{2(n+1)}{n-1} $$ for any $f\in C^{\infty}(M_0)$. For example, the truncated paraboloid $\{(x',|x'|^2),x'\in \mathbb{R}^{n-1},|x'|\leq 1\}$.

As an application, let $\mu$ be the measure in $\mathbb{R}\times \mathbb{R}$ satisfies $$ \int_{\mathbb{R}\times \mathbb{R}}\phi(\xi,t)d\mu=\int_{\mathbb{R}\times \mathbb{R}}\phi(\xi,|\xi|^2)d\xi, \quad \phi\in C^0 $$ then $$ \|u\|_{L^6(\mathbb{R}\times \mathbb{R})}=\|\mathcal{F^{-1}}(\hat{f}\mu)\|_{L^6(\mathbb{R}\times \mathbb{R})}\leq C\|\hat{f}\|_{L^2(\mathbb{R})}=C\|f\|_{L^2(\mathbb{R})} $$ by Plancharel.

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