Consider $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$ and $T_f$ is the regular distribution associated to $f$.
- Assume that there exists $m>0$ such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right)$, prove then $T_f \in$ $\mathscr{S}^{\prime}\left(\mathbb{R}^n\right)$(all the linear functionals on Schwartz space).
- Consider a cut-off function $\eta \in \mathscr{D}\left(B_2\right)$(smooth functuons with compact support on a ball with radius 2) such that $\eta \equiv 1$ in $B_1$, and $\eta_R(x)=\eta(x / R)$ for $R$ $R>0$. Prove that for any $k \in \mathbb{N}$, there is $C>0$ such that $$ (1+|x|)^k \sum_{|\alpha| \leq k}\left|\partial^\alpha \eta_R(x)\right| \leq C(1+R)^k, \quad \forall R \geq 1 $$
- Assume now $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$, nonnegative satisfying $T_f \in \mathscr{S}^{\prime}\left(\mathbb{R}^n\right)$. Prove that there exists $m>0$ such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right) $
- Let $g(t)=\sin \left(e^t\right)$ in $\mathbb{R}$, explain quickly that $g^{\prime}(t) \in \mathscr{S}^{\prime}(\mathbb{R})$. Show that for any $m>0$, $(1+|t|)^{-m} g^{\prime}(t) \notin L^1(\mathbb{R})$. This means that the inverse claim of the first question is false in general.
I want to tackle the first question by using a convolution and cut-off function, then we can transfer the derivatibe on $f$ to derivative of the cutoff function, I also want to use this on problem 3 because this can transfer the derivative to the cut-off function and the derivative of the cut-off is nontrivial only on a bounded interval, hence we can use the condition of local integrability, but I can only tackle question 2, how to do this?