Suppose $\Omega \subseteq \mathbb{R}^3$ is a smooth domain. Let $-\Delta_\Omega$ denote the Dirichlet Laplacian on this domain (considered as a densely defined self-adjoint operator).
Question: For $f\in L^2(\Omega)$, is the function $V_f := (-\Delta_\Omega)^{-1/2}f$ representable as a sum of a $L^2(\Omega)$ and a $L^\infty(\Omega)$ function?
Motivation: I am interested into spectral properties of Hamiltonians $H_f = -\Delta_\Omega + V_f$. For this I obviously have to ensure self-adjointness of $H_f$ first, which should be possible via the Kato-Rellich theorem. We could apply the latter if we know $V_f \in (L^2\! +\! L^\infty)(\Omega)$.
Ideas: One may use kernel estimates $(-\Delta_\Omega)^{-1/2}(x,y) \lesssim \lvert x-y \rvert^{-2}$ in combination with the Hardy-Littlewood-Sobolev inequality to obtain for example $\| V_f \|_2 \lesssim \| f \|_{6/5}$, which would imply the claim in the case of a bounded domain. However, I am also interested into the unbounded case. Probably one should use linearity of $V_f$ in $f$ in order to truncate $f$ into a compactly supported part plus rest and then...?