Regularity of $$\log (\partial \phi )$$ for the Solutions of Beltrami Equations with Coefficient in the Sobolev Space $$W^{1,p}_c({\mathbb {C}})$$

In this paper, we study the regularity of $$\log (\partial \phi )$$ when $$\phi $$ is the principal solution of the homogeneous Beltrami equation and the coefficient $$\mu $$ has compact support and is in the Sobolev space $$W^{1,p}_c({\mathbb {C}})$$ with $$1