Holomorphic dependence for the Beltrami equation in Sobolev spaces

We prove that, given a path of Beltrami differentials on $\mathbb{C}$ that live in and vary holomorphically in the Sobolev space $W_{loc}^{l,\mathrm{\infty }}(\mathrm{\Omega })$ of an open subset $\mathrm{\Omega }\subset \mathbb{C}$, the canonical solutions to the Beltrami equation vary holomorphically in $W_{loc}^{l+1,p}(\mathrm{\Omega })$ for admissible $p>2$. This extends a foundational result of Ahlfors and Bers (the case $l=0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data.