Lp-based L-infinity estimate for the normalized infinity Laplacian

Determine whether an $L^\infty$ estimate for solutions of the normalized infinity-Laplacian Poisson equation $\Delta_\infty^N u=f$ can be obtained in terms only of $\|f\|_{L^p}$ for some exponent $p>n$.

Background

The paper discusses the normalized infinity Laplacian $$\Delta_\infty^N u=|\nabla u|^{-2}\langle D^2u\nabla u,\nabla u\rangle=f$$ as an extreme degenerate elliptic equation for which the classical Aleksandrov–Bakelman–Pucci mechanism fails. Earlier work obtained $L^\infty$ bounds depending on $\|f\|_{L^\infty}$, but raised the unresolved issue of whether weaker integrability information on $f$, specifically an $L^p$ norm with $p>n$, suffices for such an estimate. The paper states that no result resolving this question was known to the author at the time of writing.