I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to exist some $C_p$ such that for all $f\in\mathscr{S}(\mathbb R^d)$, $$\lVert T_m(f)\rVert_{L^p} = \lVert (m\hat{f})^\vee\rVert_{L^p}\le C_p\lVert f\rVert_{L^p}$$ which typically require some degree of $C^k$-regularity on $m$.

However, it is not clear to me in what sense the operator $T_m$ can be extended by density. Of course, this can be done formally by Hahn-Banach, but for $f\in L^p$ for $2<p<\infty$ (or even $p=\infty$), do we know that $\hat f\in\mathscr{S}'(\mathbb R^d)$ will be sufficiently regular that $m\hat f$ can actually be defined for, say, $m\in C^d(\mathbb R^d)$? Put more succinctly, what is the worst possible order of the tempered distribution $\hat f$ for $f\in L^p(\mathbb R^d)$, where $2<p\le\infty$? According to this question on MO, in general the distribution $\hat f$ may be of positive order, but I have yet to find further results in this direction.