We know that a function in $L^1(\mathbb{R}^d)\cap L^2(\mathbb{R}^d)$ and its Fourier transform have the same $L_2$ norm. I wonder if there is any result about the function's Fourier transform's $L_p$ norm, for $p \in [1,\infty)$. We may assume that the function is $L_p$ integrable for any $p$.
In case no exact equality is known, any inequality that bounds the $L_p$ norm from above and below would be great.
Edit: David has given the result for upper bounding the Fourier transform in $p\in(1,2]$, a result known as the Hausdorff-Young inequality. It would be great to have some result for $p>2$ and lower bounds as well.
Thanks in advance.
The Hausdorff-Young inequality, namely $$ \|\hat{f}\|_{p'}\le C_p \|f\|_{p}, $$ where $1\le p\le 2$ and $1/p+1/p'=1$, is the most you can get. For any $p>2$ there exists $f\in L^p(\mathbb R^n)$ such that $\hat{f}$ (taken in distributional sense) is not a function: more precisely, $\hat{f}$ is a distribution with strictly positive order; see Hörmander's first volume (of his 4 volume series), Theorem 7.6.6.
In particular, if $p>2$ then there is no function space $X$ such that $\mathcal F\colon L^p(\mathbb R^n)\to X$ is well-defined.
Remark. This can be considered as the starting point of the restriction problem for the Fourier transform.