Construct $f\in L^p (p>2)$, but $\hat{f}\notin L^{p^{\prime}}$ .

Question

From Hausdorff-Young inequality, for $1\leqslant p\leqslant 2$, if $f\in L^p$, then $\hat{f}\in L^{p^{\prime}}$, where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. If $2<p\leqslant \infty$, can we construct $f\in L^p$, but $\hat{f}\notin L^{p^{\prime}}$? For $p=\infty$, Let $f=\chi_{(-1,1)}$, then $\hat{f}(\xi)=-\frac{sin(2\pi \xi)}{\pi \xi}\notin L^1$, but I have no idea for $2<p<\infty$.