I'm struggling with this question and was hoping someone could help. There are 4 parts which I think lead on from one another.
We have Nash's inequality for $f\in\mathcal{S}(\mathbb{R})$ of the following form \begin{equation} \|f\|_2<C\|f\|_1^\alpha\|f'\|_2^\beta \end{equation} (1) What is the only possible pair $(\alpha,\beta)$ for the inequality to be true? And write the two relations that they must satisfy. \begin{equation} \left.\right. \end{equation} (2) Let $f\in\mathcal{S}(\mathbb{R})$, show that for any $\lambda>0$, we have \begin{equation} \int_{|k|\geq\lambda}|\widehat{f}(k)|^2dk\leq \frac{1}{4\pi^2\lambda^2}\|f'\|_2^2. \end{equation} \begin{equation} \left.\right. \end{equation} (3) Show that for any $f\in \mathcal{S}(\mathbb{R}^d)$ and any $\lambda>0$, we have \begin{equation} \|f\|_2^2\leq 2\lambda\|f\|_1^2+\frac{1}{4\pi^2\lambda^2}\|f'\|_2^2. \end{equation} \begin{equation} \left.\right. \end{equation} (4) Prove Nash's inequality given at the start for the correct exponents $(\alpha,\beta)$.
Hint: For 1), try a scaling argument. i.e. $g(x) = a f(bx) $ to find the pair $(\alpha, \beta)$. For 2), use Plancherel's Theorem. For 3), use something like $$ ab \leq \lambda a^2 + b^2 / \lambda $$ for $\lambda >0, a >0$. Now use 2) and 3) for 4).