I'm searching for an inequality in the form $$\forall s>2,\quad\forall u\in H^{s-2}(\mathbb{R}^{2}),\quad\|\Delta^{-1}u\|_{H^{s}(\mathbb{R}^{2})}\lesssim\|u\|_{H^{s-2}(\mathbb{R}^{2})}$$ where $H^s$ is the nonhomogeneous Sobolev space defined by $$H^{s}(\mathbb{R}^{2})=\big\{u\in\mathcal{S}'(\mathbb{R}^{2})\quad\mbox{s.t.}\quad\|u\|_{H^{s}(\mathbb{R}^{2})}:=\left(\int_{\mathbb{R}^{2}}(1+|\xi|^2)^s|\widehat{u}(\xi)|^{2}d\xi\right)^{\frac{1}{2}}<+\infty\big\}$$ and where $\Delta^{-1}$ has symbol $-|\xi|^{-2}$. Such an inequality in homogeneous Sobolev spaces $\dot{H}^s$ (without the addition by $1$ in the norm) seems trivial, but I do not manage to prove the above one.
Such an estimate means in particular that the operator $\Delta^{-1}$ is bounded in $H^s.$ In bounded domains, there exists an elliptic regularity result for the Dirichlet problem, but the proof strongly uses the fact that the domain is bounded (at least in one direction), which is not the case here since we work in the full space.
For me this result may be false, but I do no manage to disprove it (with for instance dilation arguments) and it would be great if it reveals true..
Thank you for your help.
