Suppose that $f \in C^n(\mathbb{R}^d)$ and $\partial^{\alpha} f \in L^1(\mathbb{R}^d)$ for $|\alpha| \le n$. Show that $$ |\hat{f}(k)| \le \frac{C}{(1+|k|)^n} $$ for some constant $C$.
I used the relation $\widehat{\partial^{\alpha}_{x} f}(k) = (-\mathbf{i}k)^{\alpha} \hat{f}(k)$ which gives $$ |\hat{f}(k)| \le \frac{D \| \partial^{\alpha}_{x} f(x) \|_{L^1} }{|k^{\alpha}|} \le \frac{C}{|k^{\alpha}|} $$ But I can not show that $|k^\alpha| \ge (1+|k|)^n$.
Update: Actually, we can show that $$ |k^\alpha| \le max_{i=1,2,\cdots,d}\{ |k_i|^{|\alpha|} \} \le |k|^{|\alpha|} $$ So I guess a better estimate is needed.
$f \in L^1$ and $\partial^\alpha f \in L^1$, so the bound is direct from $$(1+|k|^n)|\hat{f}(k)| \le C \sum_{\alpha \le n}|k^\alpha\hat{f}(k)| \le C\sum_{\alpha} \| \partial^{\alpha} f \|_{L^1} $$