Consider the infinity harmonic functions, i.e. solutions of the equation
$$ \Delta_\infty u = \langle Du, D^2 u \, Du \rangle = 0. $$
It is known that the solutions are everywhere differentiable (continuous differentiability is an open question), and in the plane it is known that the solutions are $C^{1,\alpha}$.
My question is: Is anything known about the second derivative of the solutions, or even its trace, the laplacian? Do we have, for example, that $D^2u \in L^1$, or even Radon measure? Or do we have that $\Delta u$ is a Radon measure?
These properties do hold for the Aronsson solution $$u(x,y) = x^{4/3}-y^{4/3}$$ which is taken as the archetypal solution to infinity Laplace equation.
(I asked the question earlier on Mathoverflow, https://mathoverflow.net/q/162046/1445, and got no attention whatsoever.)
