If $f\in\mathcal{H}^{1}$, then the problem
\begin{equation} \Delta u=f\quad\text{in}\ \mathbb{R}^{2},\quad\lim\limits_{x\to\infty}u(x)=0, \end{equation}
admits the unique solution \begin{equation} u(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}f(y)\log|x-y|\rm{d}y\in D^{1,2}(\mathbb{R}^{2})\cap D^{2,1}(\mathbb{R}^{2}). \end{equation}
How to prove this proposition?