A quantative Sobolev regularity for absolute minimizers involving Hamiltonian $H(p)\in C^0({\mathbb{R}}^2)$ in plane

Suppose that $H\in C^0({\mathbb{R}}^2)$ satisfies \begin{enumerate} \item[(H1)] $H$ is locally strongly convex and locally strongly concave in ${\text{rr}}^2$, \item[(H2)] $H(0)={\min }_{p\in {\text{rr}}^2}H(p)=0$. \end{enumerate} Let $\mathrm{\Omega }\subset {\text{rr}}^2$ be any domain. For any $u$ absolute minimizer for $H$ in $\mathrm{\Omega }$, or if $H\in C^1({\text{rr}}^2)$ additionally, for any viscosity solution to the Aronsson equation \mathscr A_H[u]=\sum_{i,j=1}^2 H_{p_i}(Du) H_{p_j}(Du)u_{x_ix_j}=0 \quad in $\Omega$, the following are proven in this paper: \begin{enumerate} \item[(i)] We have $[H(Du){]}^{\alpha }\in W_{\text{loc}}^{1,2}(\mathrm{\Omega })$ whenever $\alpha >1\mathrm{/}2-{\tau }_H(0)$; some quantative upper bounds are also given. Here ${\tau }_H(0)=1\mathrm{/}2$ when $H\in C^2(\rr^2)$, and $0<{\tau }_H(0)\le 1\mathrm{/}2$ in general. \item[(ii)] If $H\in C^1({\text{rr}}^2)$, then the distributional determinant $-\mathrm{d}\mathrm{e}\mathrm{t}{D}^{2}udx$ is a nonnegative Radon measure in $\mathrm{\Omega }$ and enjoys some quantative lower/upper bounds. \item[(iii)] If $H\in C^1({\text{rr}}^2)$, then for all $\alpha >\frac{1}{2}-{\tau }_H(0)$, we have $\langle D [H(Du )]^\alpha ,D_p H(Du )\rangle=0 $ almost everywhere in $\Omega$. \end{enumerate}