Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:

Let us consider the orientation-preserving homeomorphic solutions $f: D \to D $ of the Beltrami equation $f_\bar{z}=\mu. f_z, ||\mu||_{L^{\infty}(D)}\le k\le 1, D$ is the unit disk in the complex plane $C$, $f_z, f_\bar{z} $ are the partial derivatives of $f$ w.r.t $z, \bar{z} $ respectively. I am looking for the known results on the sufficient , or, necessary and sufficient condition on the Beltrami coefficient $\mu$ such that any solution of the equation is at least $C^1(\bar{D})$, but I would be happier to get $C^k(\bar{D}), 1\le k \le \infty $, or even real-analytic upto the boundary. By $C^k(\bar{D})$, I mean the $k$th derivative exists and is continuous upto the boundary. Holder continuity of solutions is also I would be interested in. ( Please note that I stated 'any' solution , since we know that any two solutions differ by a Mobius transformation of $D$.

In general, I am looking for results for the boundary regularity of the solution to the Beltrami equation.

I would highly appreciate if you state any known results on this topic , thanks in advance !