Prove that a homogeneous distribution is tempered. Suppose $F$ is a homogeneous distribution of degree $λ$.Prove that $F$ is tempered,i.e.$F$ is continuous in the Schwartz space $\mathcal{S}$.
It seems that it's an easy result in distribution theory,but I really don't know how to prove a distribution is tempered in a simpler way.
 A: You have asked another question here, whose result can be used to prove the claim in this question.
Also, let us fix some notations once and for all. We define $\varphi_a(x) = \varphi(ax)$, $F_a(\varphi) = F(\varphi^a)$, and $\varphi^a = a^{−d} \varphi_{a^{−1}}$. (These are the notations used in Stein and Shakarchi's Functional Analysis, which seems to be the textbook from which your questions are from.)
Define $\eta_R$ as in the answer that I provided there. (Notice that this notation is unfortunately kind of opposite to that of $\varphi_a$ above.) We can find an integer $N$ and constant $c > 0$, so that $$|\eta_1 F(\varphi) | \le c \|\varphi\|_N$$, all for $\varphi \in \mathcal{S}$. This is because $\eta_1 F$ is of compact support and thus automatically tempered, allowing us to use Proposition 1.4 in Chapter 3 of Stein and Shakarch.
We have thus
$$\eta_R R^{-\lambda} F(\varphi) = \eta_R F_{1/R}(\varphi) = F_{1/R}(\eta_R \varphi) = R^d F(\eta_1 \varphi_R) = R^d \eta_1 F(\varphi_R),$$ and therefore for $R \ge 1$ $$|(η_RF)(\varphi)| \le c R^{d+\lambda} \|\varphi_R\|_N \le c R^{N+d+\lambda} \|\varphi\|_N,$$ where the last inequality between $\|\varphi_R\|_N$ and $\|\varphi\|_N$ can be easyly shown from the definition of $\|\cdot\|_N$.
Now we can specialize to $\varphi \in \mathcal{D}$ supported in $|x| \le R$. The inequality becomes $$|F(\varphi)| = |\eta_RF(\varphi)| \le c R^{N+d+\lambda} \sup_{|\alpha|,|\beta| \le N} |x^\beta\partial_x^\alpha \varphi(x)| \le c R^{2N+d+\lambda} \sup_{|\alpha| \le N} |\partial_x^\alpha \varphi(x)|.$$
This is where we use the result of your previous question (with $N$ there being $2N+d+\lambda$ here), and conclude that $F$ is tempered.
