Some Sobolev inequality I was doing some work in spectral theory and it seems to imply the following inequality: on $\mathbb{R}^d$, we have for any $m\ge 1$,
$\|D\psi\|^m_{L^2} \le \|D^m \psi\|_{L^2}\cdot\|\psi\|^{m-1}_{L^2}$
Presumably for any $\psi\in H^m(\mathbb{R}^d)$, but technically I also need some decay at infinity.
Is this inequality true ? Is it known ? Sorry if this is classical. Also note there is no constant in the upper bound.
Thanks !
 A: While a more general form is true, in the case $p=2$ there is a much more elementary argument (as sketched in the wiki page linked in the comments). We will use the equality
$$ \lVert D^k\psi \rVert_{L^2(\Bbb R^d)}^2 := \sum_{|\alpha|=k} \int_{\Bbb R^d} |D^{\alpha}\psi|^2\,\,\mathrm{d}x = \int_{\Bbb R^d} |\xi|^{2k}|\hat{\psi}|^2 \,\mathrm{d} \xi,$$
which follows by the Plancherel theorem. The result is vacuous if $m=1,$ so assume that $m>1.$ Then we can use Hölder's inequality with $p = m,$  and $f(\xi) = |\xi|^2|\hat\psi|^{\frac2m},$ $g(\xi) = |\hat\psi|^{\frac{2m-2}m}$ to get
$$\begin{split} \int_{\Bbb R^n} |\xi|^2|\hat\psi|^2 \,\mathrm{d}\xi &= \int_{\Bbb R^n} f(\xi)g(\xi) \,\mathrm{d}\xi \\
&\leq \left( \int_{\Bbb R^n} f(\xi)^m \,\mathrm{d}\xi\right)^{\frac1m}\left( \int_{\Bbb R^n} g(\xi)^{\frac{m}{m-1}} \,\mathrm{d}\xi\right)^{\frac{m-1}m} \\
&= \left( \int_{\Bbb R^n} |\xi|^{2m} |\hat\psi|^2 \,\mathrm{d}\xi\right)^{\frac1m}\left( \int_{\Bbb R^n} |\hat\psi|^2 \,\mathrm{d}\xi\right)^{\frac{m-1}m}, 
\end{split}$$
so the result follows by combining these results.
