For $q<r \leq p' <2$, how to construct a function $f \in \mathscr{F} L^p \cap L^q$, but does not belong to $L^r$? Where $\mathscr{F}$ is the space of the Fourier transform of all $L^p$ functions with the norm $ \|f\|_{\mathscr{F} L^p } = \|\hat{f}\|_p$. Or how can we prove the embedding $\mathscr{F} L^p \cap L^q \hookrightarrow L^r$?
I met this problem in the study of modulation space $M_{p,q}$, I believe the embedding is not true but I have not constructed a function as desired.
By Hausdorff-Young inequality we have $\mathscr{F} L^p \cap L^q \hookrightarrow \mathscr{F} L^p \cap \mathscr{F} L^{q'} \hookrightarrow \mathscr{F} L^{r'}$, so the function we need must belong to $\mathscr{F} L^{r'} \setminus L^r$, which exists by Hausdorff-Young inequality, but I do not know the specific counterexample of it.
Any idea will be helpful. Thanks a lot!