Abstract: Assume that $u$ is planar infinity hamonic funcfunctions, we prove  that $|Du|^\alpha\

in W^{1,2}_\loc$ for all $\alpha>0$, which is   sharp as $\alpha\to0$. We also show that the distributional

determinant $-\det D^2u$ is a nonnegative Radon maesure with some quatative estiamtes from above and below.

Moreover, for viscosity solutions $u$ to inhomogeneous infnity Laplace equaiton in plane where the

inhomogeneous term $f \in BV_\loc$, we prove  that $|Du|^\alpha\in W^{1,2}_\loc$ for all $\alpha>3/2$,

which is   sharp as $\alpha\to3/2$. For $\alpha\in(0,3/2)$,  $|Du|^\alpha\in W^{1,p}_\loc$ for $p<3/(3-\alpha)$, which is sharp.  

The proofs rely on a fundamental structural identity for infnity Laplace operator as we established.