An integral converges- Analysis Fourier

Question

I want to show that if $k\geq2$ and $f\in C_{c}(\mathbb{R})\cap C^k(\mathbb{R})$ then $\int\limits_{\mathbb{R}} |\xi|^{κ-2}|\hat{f}(\xi)|dm(\xi)<\infty$.

I can't understand what can i use for that. Thanks

Answer 1

Integration by parts gives $$ \widehat{f^{(k)}}(\xi)=(i\,\xi)^k\hat f(\xi) $$ Since $f^{(k)}$ is continuous with compact support, its Fourier transform is bounded. Thus there exists a constant $C$ such that $$ |\hat f(\xi)|\le\frac{C}{|\xi|^k}\quad\forall\xi\in\mathbb{R}. $$ Let $R>0$. $$\begin{align} \int_{\mathbb{R}} |\xi|^{κ-2}|\hat{f}(\xi)|\,d\xi&=\int_{|\xi|\le R} |\xi|^{κ-2}|\hat{f}(\xi)|\,d\xi+\int_{|\xi|> R} |\xi|^{κ-2}|\hat{f}(\xi)|\,d\xi\\ &\le2\,R^{k-1}\,\|\hat f\|_\infty+C\int_{|\xi|> R} |\xi|^{-2}\,d\xi\\ &<\infty. \end{align}$$