I'm actually completely mindfucked : I have the Dirichlet Laplcian operator $(-\Delta_\mathcal{D})$ with domain $W^{2,p}(\mathbb{R}^d_+)\cap W^{1,p}_0(\mathbb{R}^d_+)$, it is not invertible but we can construct (using potential theory and good extension operators) a right and left (continuous) inverse $$(-\Delta_\mathcal{D})^{-1}:\mathrm{L}^p\longrightarrow \dot{W}^{2,p}_\mathcal{D}(\mathbb{R}^d_+)$$ where $\dot{W}^{2,p}_\mathcal{D}(\mathbb{R}^d_+)$ is the homogeneous Sobolev space with order 2 including Dirichlet Boundary Condition.
The fact is that for $u\in C_c^\infty(\mathbb{R}^d_+)$ , $$(-\Delta_\mathcal{D})^{-1} \Delta u = (-\Delta_\mathcal{D})^{-1} \Delta_\mathcal{D} u = -u$$ so it must extend as (minus) the identity on $\mathrm{L}^p(\mathbb{R}^d_+)$, but it is no longer true on $W^{2,p}(\mathbb{R}^d_+)$ (no specific Boundary condition) for $$(-\Delta_\mathcal{D})^{-1} \Delta$$ since there is no particular boundary condition.
So I have a map that is (minus) the Identity on a larger space $\mathrm{L}^p(\mathbb{R}^d_+)$ but which cannot be (minus) the identity on the subspace $W^{2,p}(\mathbb{R}^d_+)$, what the heck happened here ?
Thanks to let me know my possible mistakes.