Bounding Fourier transform of characteristic functions Let $A$ be a compact convex (and even nicer if needed) subset of $\mathbb{R}^n$. Do we have a result saying that the characteristic function $\mathbf{1}_A$ is of moderate growth, i.e. that there is a constant $c > 0$ such that
$$ \widehat{\mathbf{1}_A} (x) \ll x^{-c} $$
when $x$ growth to infinity?
 A: For $n=1$ we have that $A$ is an interval, say $[a, b]$ (open or closed doesn't matter). As a distribution the derivative $\mathbf{1}_A'(x) = \delta(x-b) - \delta(x-a)$ so
$i\xi \, \widehat{\mathbf{1}_A}(\xi) = \widehat{\mathbf{1}_A'}(\xi) = e^{-ib\xi} - e^{-ia\xi}.$ Hence $\widehat{\mathbf{1}_A}(\xi) = O(\xi^{-1}).$
In higher dimensions $\nabla \mathbf{1}_A = -\delta_{\partial A} \mathbf{n}$ where $\delta_{\partial A}$ is the distribution that gives the values on the boundary of $A$ and $\mathbf{n}$ is the outwards-pointing unit normal on the boundary. For $\mathbf{n}$ to be defined it's required that the boundary is $C^1.$ Then we have
$i\xi \, \widehat{\mathbf{1}_A}(\xi) = \widehat{\nabla \mathbf{1}_A}(\xi) = -\int_{\partial A} \mathbf{n} \, e^{-i\xi\cdot x} \, dx.$ Taking the norm gives
$|\xi| |\widehat{\mathbf{1}_A}(\xi)| = |\int_{\partial A} \mathbf{n} \, e^{-i\xi\cdot x} \, dx| \leq \int_{\partial A} |\mathbf{n}| \, |e^{-i\xi\cdot x}| \, |dx| = \int_{\partial A} |dx| = \mathrm{length}(\partial A).$ Hence, if $\partial A$ is of finite length, $|\widehat{\mathbf{1}_A}(\xi)| = O(|\xi|^{-1}).$
