The inequality I am trying to prove is the following: For all $k,m \in \mathbb{N}$, there exists constants $C > 0$, $k', m' \in \mathbb{N}$ such that \begin{align} q_{k,m}(\hat{\phi}) \leq C q_{k',m'}(\phi),\end{align} where $q_{k,m}(f) := \sup\limits_{\stackrel{x \in \mathbb{R}^n}{|\alpha| \leq m}} (1+|x|^2)^{k/2}|\partial^\alpha f(x)|$ (the seminorm on the Schwartz space and assuming $\phi$ is in the Schwartz space [if it's not the inequality is trivial]). This is what I have so far: \begin{align*} q_{k,m}(\hat{\phi}) &= \sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} (1+|\xi|^2)^{k/2}|\partial^\alpha \hat{\phi}(\xi)| \\ &=\sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} \frac{(1+|\xi|^2)^{k/2}}{(2\pi)^{n/2}} |\partial^\alpha_\xi \int_{\mathbb{R}^n} e^{-ix\cdot\xi}\phi(x) dx| \\ &= \sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} \frac{(1+|\xi|^2)^{k/2}}{(2\pi)^{n/2}} |\int_{\mathbb{R}^n} \partial^\alpha_\xi e^{-ix\cdot\xi}\phi(x) dx|\\ &= \sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} \frac{(1+|\xi|^2)^{k/2}}{(2\pi)^{n/2}} |\int_{\mathbb{R}^n} \partial^\alpha_x \frac{x^{\alpha}}{\xi^{\alpha}} e^{-ix\cdot\xi}\phi(x) dx| \\ &= \sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} \frac{(1+|\xi|^2)^{k/2}}{(2\pi)^{n/2}} |(-1)^{|\alpha|}\int_{\mathbb{R}^n} \frac{x^{\alpha}}{\xi^{\alpha}} e^{-ix\cdot\xi}\partial^\alpha_x\phi(x) dx| \\ &\leq \sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} \frac{(1+|\xi|^2)^{k/2}}{(2\pi)^{n/2}} \sup\limits_{z\in\mathbb{R}^n} |\partial^\alpha\phi(z)| \frac{x^{\alpha}}{\xi^{\alpha}} e^{-ix\cdot\xi}| dx \\ &= \sup\limits_{\stackrel{\xi \in \mathbb{R}^n}{|\alpha| \leq m}} \frac{(1+|\xi|^2)^{k/2}}{(2\pi)^{n/2}} \sup\limits_{z\in\mathbb{R}^n} |\partial^\alpha\phi(z)| \int_{\mathbb{R}^n} |\frac{x^{\alpha}}{\xi^{\alpha}}| dx, \end{align*} but this last line is most definitely infinite. I was thinking of trying not to isolate the $\sup\limits_{z\in\mathbb{R}^n} |\partial^\alpha\phi(z)|$ too soon, but I don't see how to bring that piece out without bringing all the absolute values inside, causing the integral to blow up (also a reason for why Holder's inequality doesn't appear to be useful). I'm thinking I might have to head in a different direction, but I'm not sure where to go.
Any hints would be greatly appreciated!