TITLE: Codomain of the Fourier transform
QUESTION [6 upvotes]: Among other things, the Fourier transform maps functions from $L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $L^1(\mathbb{R}^n) \to C_0(\mathbb{R}^n)$ (continuous functions vanishing at infinity), and $\mathcal{S}(\mathbb{R}^n)\to\mathcal{S}(\mathbb{R}^n)$ (Schwartz space, or the space of rapidly decreasing functions). 
I'm interested in looking more closely at the codomain of the second mapping. Since $C_0 \subset L^\infty$, every $L^1$ function is mapped by the Fourier transform into an $L^\infty$ function. However, I was wondering if it is easy to find specific examples of functions that are only in $L^1$, but are mapped into $L^1$ or $L^2$ (or possibly any other $L^p$ for $1 \leq p < \infty$).

REPLY [4 votes]: By only in $L^1$ I understand that the function $f$ is in $L^1$ but not in $L^p$ for any $p>1$. Such a function is for instance $f(x)=\min(|x|^{-a},|x|^{-b})$, where $0<a<1<b$ (and $f(0)=0$). Then $\hat f\in L^\infty$, but $\hat f\not\in L^p$ for $1\le p\le2$. This a consequence of the inversion formula and the Hausdorff Young inequality: if $g\in L^p$, $1\le p\le2$, then $\|\hat g\|_q\le C\|g\|_p$ for a constant $C$ independent of $g$ and $p^{-1}+q^{-1}=1$. However, it could still happen $\hat f\in L^p$ for some $p\in(2,\infty)$.