Hausdorff–Young inequality The Hausdorff–Young inequality says that If $ f \in L^p(\mathbb{R}^d), \quad 1\leq p\leq 2$ then
\begin{equation*}
 \| \hat{f}\|_{L^{p'}(\mathbb{R})}\leq \|f\|_{L^p(\mathbb{R})}, \quad \frac{1}{p}+\frac{1}{p'}=1.
\end{equation*}
where $\hat{f}$ is the Fourier transform of $f$.
I would like to know if it is possibile, by assuming some hypothesis on function $f$, to get for $p>2$, $ \|\hat{f}\|_{L^{p'}(\mathbb{R})}$ bounded.
In other words, it is reasonable to have a function $f$ such that $\hat{f}$ is in $L^{p'},\ p'<2$?
 A: In general, you need additional derivatives to control the $L^q$ norm of a Fourier transform with $q<2$.
An easy way to get a sufficient condition is the following. Let $\langle \xi\rangle := (1+|\xi|^2)^{1/2}$ and $n>d\,(\frac{1}{q}-\frac{1}{2})$. Then by Hölder's inequality
$$
\|\widehat f\|_{L^q(\Bbb R^d)} = \|\langle \xi\rangle^{-n}\langle \xi\rangle^{n}\widehat f\|_{L^q(\Bbb R^d)} ≤ C_{n,q}\, \|\langle \xi\rangle^n\widehat f\|_{L^2(\Bbb R^d)}
$$
where $C_{n,q} = \|\langle \xi\rangle^{-n}\|_{L^\frac{2\,q}{2-q}(\Bbb R^d)} < \infty$ with our choice of $n$. Now by Plancherel Theorem,
$$
\|\langle \xi\rangle^n\widehat f\|_{L^2(\Bbb R^d)} = \|(1-\Delta)^{n/2}f\|_{L^2(\Bbb R^d)} = \|f\|_{H^n}
$$
so that
$$
\|\widehat f\|_{L^q(\Bbb R^d)} ≤ C_{n,q}\,\|f\|_{H^n}
$$
A particular case is the case $q=1$, which is lot more studied. A sufficient condition to have $\widehat f∈L^1$ is $f∈ H^n$ with $n>d/2$. A slightly sharper condition in this case is $f∈B^{d/2}_{2,1}$ (where $B^s_{p,q}$ are the Besov spaces). Another easy condition is $\widehat f ≥ 0$ and $f$ continuous, since in this case $\|f\|_{L^1} = \int_{\Bbb R^d} f = \widehat{f}(0)$. A lot more different sufficient conditions of different kind can be found in [1].
As proved for example in [2], to have derivatives is a necessary condition (at least in dimension $d>1$) since if $\widehat f ∈ L^1$ then its radial part $f_0$ satisfies $f_0 ∈ C^{(d-1)/2}(0,\infty)$.

[1] Liflyand, Samko, Trigub, The Wiener algebra of absolutely convergent Fourier integrals: an overview
[2] Liflyand, Necessary Conditions for Integrability of the Fourier Transform
