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.
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.