log|x| is a tempered distribution

Question

How do I prove that $\log|x|$ is a tempered distribution on $\mathcal{S}(\mathbb{R}^n)$, ie., I need to prove that the linear functional $$\phi \in \mathcal{S}(\mathbb{R}^n) \mapsto \int_{\mathbb{R}^n} \phi(x)\log|x|dx $$ is continuous. It's suficcient to prove that $\phi_k \rightarrow 0$ in $\mathcal{S}(\mathbb{R}^n)$ implies that $\int_{\mathbb{R}^n} \phi_k(x)\log|x| \rightarrow 0$ as $k \rightarrow \infty$, but I don't know estimate $\phi_k(x)\log|x|$ by an integrable function, so I could use dominated convergence theorem, or estimate it by a sum seminorms $\|\phi_k\|_{\alpha,\beta}$. (this is the Example 2.3.5 (7) from Grafakos's Book - Classical Fourier Analysis - third edition)

Answer 1

Suppose $\phi_k \to 0$ in $\mathcal S.$ Then

$$\sup_x (1+|x|)^{n+1}|\phi_k(x)| =M_k \to 0$$

as $k\to \infty.$ Thus the sequence $M_k$ is bounded by some $M.$ We then have

$$|\phi_k(x)||\ln |x|| \le M\frac{|\ln |x||}{(1+|x|)^{n+1}}$$

for all $k$ and $x.$ The function on the right is in $L^1(\mathbb R^n)$ (verify by going to polar coordinates in $\mathbb R^n$). Since $\phi_k\to 0$ pointwise, the DCT gives the desired result.