Let $1 \leq p \leq \infty $ and let $p'$ thus $\frac{1}{p} + \frac{1}{p'}=1$.
$\mathcal{S}(\mathbb R^n)$ is the Schwartz space defined as the class of all $C^\infty$ functions $\varphi:\Bbb R^n\rightarrow \mathbb C$ such that $$\sup_{x\in\mathbb R^n}|x^\beta \partial^\alpha \varphi(x)|<\infty,$$ for all multi-indices $\alpha, \beta\in\mathbb N^n$
Prove that all Schwartz functions $f$ on $R^n$ satisfy this inequality
$$\|f\|_{\infty}^2 \leq \sum_{|\alpha + \beta|=n} \|\partial^\alpha f\|_p\|\partial^\beta f\|_{p^{'}}$$
where the sum is taken over all pairs of multi-indices $\alpha$ and $\beta$ whose sum has size $n$.
My try:
I have already proved that $\|f\|_{\infty}^2 \leq 2\|f\|_p \|f\|_{p^{'}}$ but I don't know how to continue. Could someone help me?
I am allowed to use only the theorems a definitions of that chapter of the book.
Best Answer
Proof in one sentence: it follows from Sobolev's inequality for $f^2$ and the multinomial theorem.
Details: Let me use the multi-index notation. For any $\gamma = (\gamma_1,\dots,\gamma_n)\in\Bbb N^n$, let $|\gamma| = \gamma_1+\dots+\gamma_n$. It follows from Sobolev's inequalities that $$ \|f\|_{L^\infty}^2 = \|f^2\|_{L^\infty} \leq \int_{\Bbb R^n} |\nabla^n(f^2)|\leq \int_{\Bbb R^n} \sum_{|\gamma|=n}|\partial^\gamma(f^2)|. $$ On the other hand, it follows from the Leibniz formula that $$ \sum_{|\gamma|=n}|\partial^\gamma(f^2)| = \sum_{|\gamma|=n}\left|\sum_{\alpha+\beta=\gamma} \partial^\alpha f \, \partial^\beta f\right| \leq \sum_{|\alpha+\beta|=n}\left|\partial^\alpha f \, \partial^\beta f\right|. $$ To conclude, use the Fubini theorem and Hölder's inequality to get $$ \|f\|_{L^\infty}^2 \leq \sum_{|\alpha+\beta|=n} \int_{\Bbb R^n}\left|\partial^\alpha f \, \partial^\beta f\right|\leq \sum_{|\alpha+\beta|=n} \|\partial^\alpha f\|_{L^p}\, \|\partial^\beta f\|_{L^{p'}}. $$