Regularity up to the boundary of conformal mappings Consider any Lipschitz simply connected domain $\Omega \subset \mathbb R^2$ and a conformal mapping $\phi : D \rightarrow \Omega$ which we know it extends continuously to the boundary by Caratheodory's theorem.
Can we say something more besides continuity, for example that $\bar \phi : \bar {D} \rightarrow \bar\Omega$ is globally Holder continuous?
I've seen Kellogg-Warschawski's theorem in another Stack Exchange question but it requires $C^{1,\alpha}$ boundary at least.

I've had to use this result in the last year. The answer is that for chord-arc domains (which Lipschitz domains are an example of) then $\bar \phi$ is continuous, but the derivative $d \bar \phi$ may only be in $L^p$ for some $p > 1$, you cannot ask for more than this. For any $\varepsilon > 0$ you can get $d \bar \phi \not \in L^{1 + \varepsilon}$ by taking a polygon with acute enough angles.
The proof that the derivative will always be at least $L^p$ can be found in the following paper of Baratchard, Bourgeois, Leblond https://hal.inria.fr/hal-01084428v1 on p. 19-20. They show along the proof of their Lemma 5.1 that $d \bar \phi$ must be in the Muckenhoupt class $A_2$ and  give references from which it can be deduced that $d \bar \phi \in L^{1 + \varepsilon}$ for some $\varepsilon > 0$.
I hope that a year later this answer is still useful.
