I am struggling in solving the second part of this problem.
Let $g$ be a continuous function in $L^1(\mathbb{R})$ whose Fourier transform is the function $F$. Suppose $|F(x)|\leq (1+x^2)^{-2}$. Prove that $g$ is a $C^1$ function such that $d g/d x\in L^\infty(\mathbb{R})\cap L^2(\mathbb{R})$.
Does anyone have any idea?
The following gives the main idea. \begin{eqnarray*} |F(k)| &\leqslant &\frac{1}{(1+k^{2})^{2}}\Rightarrow F(k)\in L^{1}\cap L^{\infty } \\ |kF(k)| &\leqslant &|k|\frac{1}{(1+k^{2})^{2}}\Rightarrow H(k)=kF(k)\in L^{1}\cap L^{\infty } \end{eqnarray*} It follows that $g(x)\in L^{1}$ and $g(x)\in L^{2}\cap L^{\infty }$ so $% g(x)\in L^{1}\cap L^{\infty }$. Also $H(k)\in L^{1}\cap L^{2}$ so $$h(x)=\int dk\exp [-ikx]H(k)\in L^{2}\cap L^{\infty } $$ But $$ h(x)=\int dk\exp [-ikx]kF(k)=i\partial _{x}\int dk\exp [-ikx]F(k)=i\partial _{x}g(x) $$