Let $\Omega_1 \subset \mathbb R^2$ be a bounded simply-connected Lipschitz domain, and $f: \bar \Omega_1 \rightarrow \bar \Omega_2$ be a homeomorphism, which is a diffeomorphism on $\Omega_1$ such that $f$ and its inverse have first derivatives in $L^p$ with $p>2$.

The Morrey inequality implies that $f$ is Hölder. What can be said about its inverse provided that $p$ is large enough? The Morrey inequality cannot be directly applied to the inverse since the boundary of $\Omega_2$ might be not Lipschitz, but could it be salvaged with the additional information we have?