$\Delta f$ on all of $\mathbb{R}^2$. In fact, $\Delta f$ may not even exist as a function on $\mathbb{R}^2$. But it does exist as a distribution . Let us see how $\Delta f$, the distribution, compares with $g$. Let $\varphi$ be a smooth test function. Then $$ \langle\Delta f,\varphi\rangle = \langle