We all know the classic Hausdorff Young inequality for fourier series
$\|\widehat{f}\|_{\ell^q}\leq\|f\|_{L^p}$,
where $\frac{1}{p}+\frac{1}{q}=1$ and $p\in[1,2]$.
My question is: is there any result related to the "inverse" inequality $\|f\|_{L^p}\leq\|\widehat{f}\|_{\ell^q}$ with $q\in[1,2]$ ?
Any help or advice would be helpful.
First you need to note that Hausdorff-Young holds for $1\le p\le 2$, not for all $p$. If it held for all $p\in[1,\infty]$ the "inverse" inequality would be a trivial consequence.
The inverse inequality is false, even with a constant factor added. For example let $f=\sin(t)/t$. Then (if you put the $2\pi$'s in the right place in your definition of the Fourier transform) $\hat f=\chi_{[-1,1]}$. So $\hat f\in L^\infty$ but $ f\notin L^1$, so the inequality $||f||_1\le c||\hat f||_\infty$ is false.