On the local Hölder continuity of the inverse of the $p$-Laplace operator

We prove an interpolation type inequality between $C^\al$, $L^\infty$ and $L^p$ spaces and use it to establish the local Hölder continuity of the inverse of the $p$-Laplace operator: $\Vert(-\Delta_p)^{-1}(f) - (-\Delta_p)^{-1}(g)\Vert _{C^{1}(\bar{\Omega})} \leq C \Vert f - g \Vert^r_{L^\infty(\Om)}$, for any $f$ and $g$ in a bounded set in $L^\infty(\Omega)$.