Problem: Let us denote $\mathbb{W}^{2,p}(\mathbb{R}^2)$ the space of Sobolev functions in the plane. Let us denote with $\Delta$ the classic Laplacian operator. We know that there exists a constant $C>0$, depending only on $p$ and $d$, such that if $f \in \mathbb{L}^p(\mathbb{R}^2)$, $u \in \mathbb{W}^{2,p}(\mathbb{R}^2)$ and: $$ -\Delta u = f $$ in the weak sense, then it holds: $$ \|\partial_i \partial_j u \|_{\mathbb{L}^{p}(\mathbb{R}^2)} \leq C \|f \|_{\mathbb{W}^{2,p}(\mathbb{R}^2)} \quad \text{ for all } i,j \in \{1,2\} $$ I want to find the sharp constant $C$ with this property. In other words I want to find the least possible constant $C$ such that the property above holds.
Attempt: I tried using Fourier theory and the Hardy–Littlewood–Sobolev theorem but this does not work if $\alpha=2$ is equal to the dimension $d=2$, where $\alpha$ is the power associated to the $\Delta$ operator seen as a Fourier multiplier.
Any help or any reference will be appreciated.