We consider the following to operators acting over regular functions in $\mathbb{R}^2$: $$\Delta u(\overline{x})=\frac{\partial^2 u}{\partial x^2_1}(\overline{x})+\frac{\partial^2 u}{\partial x^2_2}(\overline{x}); \qquad \overline{x}=(x_1,x_2),$$ $$Lu(\overline{x})=\lim_{r\to 0}\left(\frac{1}{r^2}\int_{|\overline{y}|<r}(u(\overline{x}+\overline{y})-u(\overline{x}))d\overline{y}\right); \qquad r=0,$$ Is there any relationship between the operators $\Delta$ and $L$? In affirmative case, is this relationship restrited to the two dimensional case?
I've studied a course of harmonic functions and partial differential equations, but I found this exercise and I have no idea how to begin to solve it. I will appreciate any help or hint because I'm completely lost.