Lets consider the PDE $\Delta_{\infty}f=0$.
The function $f(x,y)=x^{4/3}-y^{4/3}$ is a viscosity solution of that PDE.
It has been shown that viscosity solutions of $\Delta_{\infty}f=0$ in an open set $U$ of $\mathbb{R}^{2}$ belongs to $C^{1}(U)$.
In particular, it is said that $f$ belongs to $C^{1,1/3}(\mathbb{R}^{2})$, but I'm not able to demostrate this fact.
Does anyone know how to do that? In every article I read, it's written that it's an "easy matter" to prove it.
But for me it's not!
Thanks a lot.