Like this (I only have to fix all the n+1's, they should be gammas): $$\begin{aligned}
& \|\mathcal{F}f \|_{\alpha, \beta} = \sup_x \left | x^\alpha D^\beta \mathcal{F}f(x) \right | = \sup_x \left | (-2 \pi i)^\beta \mathcal{F} x^\beta D^\alpha f(x) \right | \leq \\
& \max(K, K^\prime) \left ( \| x^\beta D^\alpha f(x)\|_\infty + \| x^{n+1} x^\beta D^\alpha f(x) \|_\infty \right ) < \infty
\end{aligned}$$

$$\begin{aligned}
& \|\mathcal{F}f \|_{\alpha, \beta} = \sup_x \left | x^\alpha D^\beta \mathcal{F}f(x) \right | = \sup_x \left | (-2 \pi i)^\beta \mathcal{F} x^\beta D^\alpha f(x) \right | \leq \\
& \max(K, K^\prime) \left ( \| x^\beta D^\alpha f(x)\|_\infty + \| x^{n+1} x^\beta D^\alpha f(x) \|_\infty \right ) < \infty
\end{aligned} $$

Didn't think anyone was here... Consider the equivalence relation on $\mathbb{R}^{2}$ defined by $(x_{1},x_{2})\sim
(w_{1},w_{2})$ if $x_{1}^{2}+x_{2}^{2}=w_{1}^{2}+w_{2}^{2}$. I'm trying to describe the resulting quotient space. I know that the equivalence classes are circles, but that's about it...