I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?

This seems to be a well-known result, but I can not find it in standard references such as Stein's Harmonic Analysis and Grafakos's classical Fourier Analysis.

Thanks in advance.