Fourier transform of a characteristic function

Question

Let $B$ be the open unit ball in $\mathbb R^d$ and $f=\chi_B$. Show the following statements for the fourier transform $\hat{f}$ of $f$.
a) $\hat{f}\in L^p(\mathbb R^d)$ for all $p\in [2,\infty]$
b) $\hat{f}\not\in L^1(\mathbb R^d)$
c) $\hat{f}\in C^\infty(\mathbb R^d)$
I need some hints so solve this. Do I have to calculate the fourier transform?

Answer 1

Hints:

a) The Fourier transform maps $L^2$ to $L^2$, and $L^1$ to $L^\infty$

b) The Fourier transform maps $L^1$ to $C^0$

c) $f$ is compactly supported