If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally integrable of moderate growth, so it represents by integral pairing a tempered distribution and then it is well defined the convolution operator: $$K_n*:\mathcal{S}\to C^\infty_\mathcal{M}$$ where $\mathcal{S}$ is the space of Schwartz test functions and $C^\infty_\mathcal{M}$ is the subspace of tempered distributions represented by $C^\infty$-functions of moderate growth with all derivatives of moderate growth.
It can be proved that if $1<p <\frac{n}{2}$ then $K_n*$ extends continuously to an operator: $$T_p:L^p\to \mathcal{S'},$$ where $\mathcal{S}'$ is the space of tempered distribution equipped with the weak*-topology.
Also it holds that $$\forall f\in L^p, \forall\varphi\in\mathcal{S}, \int_{\mathbb{R}^n}T_p(f)(x)(-\Delta\varphi(x))\operatorname{d}x = \int_{\mathbb{R}^n}f(x)\varphi(x)\operatorname{d}x,$$ i.e. the distributional laplacian of $T_p(f)$ is $f$ (hence the title of this question).
Furthermore, we can prove that: $$\forall f\in L^p, T_p(f)\in L^{\frac{np}{n-2p}}$$ and that: $$\exists C_p>0, \forall f\in L^p, \|T_p(f)\|_{\frac{np}{n-2p}}\le C_p\|f\|_p,$$ so, actually, we have that $T_p$ is a continuous operator from $L^p$ to $L^\frac{np}{n-2p}$.
What if $p\ge \frac{n}{2}$? I.e.:
If $p\ge \frac{n}{2}$, is it possible to continuously extend the operator $K_n*$ to an operator from $L^p$ to $\mathcal{S'}$? If we can, can we refine the result in such a way that we can guarantee that the extension is actually continuous from $L^p$ to some Banach space $V\subset \mathcal{S}'$ to be identified?