A sufficient and necessary condition for a distribution to be tempered.

Question

Show that the distribution $F$ is tempered if and only if there is an integer $N$ and a constant $A$,so that for all $R\geq 1$, $$F(\varphi)\leq AR^N sup_{|x|\leq R,0\leq |\alpha|\leq N}|\partial_x^\alpha\varphi(x)|$$ for all $\varphi\in\mathcal{D}$ supported in $|x|\leq R$. The 'only if' part is obvious by the necessary and sufficient condition that $F$ is tempered iff $F(\varphi)\leq c||\varphi||_N$ for all $\varphi\in\mathcal{S}$.But I don't know how the prove the converse.

Answer 1

First we need to extend $F$ from $\mathcal{D}^*$ to $\mathcal{S}^*$ as follows.

Fix $\eta \in \mathcal{D}$, $\eta(x) = 1$, for $| x | \le 1$, $\eta$ supported in $| x | \le 2$. Let $\eta_k(x) = \eta(x/k)$.

Note that $$\sup_{0\le|\alpha|\le N, k\ge 1} |\partial_x^\alpha \eta_k(x)| < \infty.$$

(It is obviously true when we fix $k=1$ because $\eta \in \mathcal{D}$. Furthermore, $\sup_{0\le|\alpha|\le N} |\partial_x^\alpha \eta_k(x)|$ decreases as $k$ increases.)

For any $\varphi \in \mathcal{S}$, define $\varphi_k(x) = \varphi(x) \eta_k(x)$. Since $\varphi_k \in \mathcal{D}$, $F(\varphi_k)$ is well defined.

Now we can show that $$\lim_{k\rightarrow\infty} F(\varphi_{2^k}) = F(\varphi_1) + \sum_{k=0}^\infty F(\varphi_{2^{k+1}}) - F(\varphi_{2^k})$$ absolutely converges.

Indeed, since $\varphi_{2^{k+1}} - \varphi_{2^k}$ is supported in $2^k\le|x|\le 2^{k+2}$, we have $$ \begin{equation} \begin{split} && |F(\varphi_{2^{k+1}}) - F(\varphi_{2^k})| \\ &=& |F\left(\varphi_{2^{k+1}} - \varphi_{2^k}\right)| \\ &\le& A \cdot 2^{(k+2)N} \sup_{2^k\le|x|\le 2^{k+2}\\ 0\le|\alpha|\le N} \left(|\partial_x^\alpha\varphi_{2^k}(x)|+|\partial_x^\alpha\varphi_{2^{k+1}}(x)|\right) \\ &\le& A \cdot 2^{-k} 2^{k(N+1)+2N} \sup_{2^k\le|x|\le 2^{k+2}\\ 0\le|\alpha|\le N} \left(|\partial_x^\alpha\varphi_{2^k}(x)|+|\partial_x^\alpha\varphi_{2^{k+1}}(x)|\right) \\ &\le& A' 2^{-k} \sup_{2^k\le|x|\le 2^{k+2}\\ 0\le|\alpha|,|\beta|\le N+1} \left(|x^\beta \partial_x^\alpha\varphi_{2^k}(x)|+|x^\beta \partial_x^\alpha\varphi_{2^{k+1}}(x)|\right) \\ &\le& A'' 2^{-k} \sup_{2^k\le|x|\le 2^{k+2}\\ 0\le|\alpha|,|\beta|\le N+1} |x^\beta \partial_x^\alpha\varphi(x)| \\ &\le& A'' 2^{-k} \|\varphi\|_{N+1}. \end{split} \end{equation} $$

(The first inequality is the assumption given in the problem. The third is because $|x| \ge 2^k$. The fourth is because $\partial_x^\alpha \eta_k(x)$ is bounded. The last is from the definition of $\|\varphi\|_{N+1}$.)

Now we can simply define $F(\varphi) = \lim_{k\rightarrow\infty} F(\varphi_{2^k})$.

The derivation above also shows that $F(\varphi_k) \rightarrow F(\varphi)$ whenever $\varphi_k \rightarrow \varphi$. This proves that $F$ is a tempered distribution.