Mean value formulas for solutions of some degenerate elliptic equations and applications

We prove a mean value formula for weak solutions of $div(\mathrm{\mid }y{\mathrm{\mid }}^a\text{grad}u)=0$ in ${\mathbb{R}}^{n+1}=\{(x,y):x\in {\mathbb{R}}^n,y\in \mathbb{R}\}$, $-1<a<1$ and balls centered at points of the form $(x,0)$. We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-\mathrm{△}{)}^{s}f=0$ on a domain $D$ of ${\mathbb{R}}^n$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of $(-\mathrm{△}{)}^{s}f=0$.