Can $\mathcal{F}$ map $L^p$ into a space of locally integrable functions when $p>2$?

Question

We have covered the function analytic theory for the convergence of partial Fourier Integrals in $L^p$ when $1 < p \le 2$ and I was wondering what could be said for $\mathcal{F}$ when $p > 2$ without invoking the theory of (tempered) distributions. I imagine in general the transform would not be a function, but what are some good references for proof?

Thanks.