Holomorphic dependence for the Beltrami equation in Sobolev spaces

We prove that, given a family of Beltrami forms on C C with L ∞ L^∞ norm at most η > 1 η >1 and that live in and vary holomorphically in the Sobolev space W l o c l , ∞ ( Ω ) Wlocl,∞(Ω ) of an open subset Ω ⊂ C Ω ⊂ C , the canonical solutions to the Beltrami equation vary holomorphically in W l o c l + 1 , p ( Ω ) Wlocl+1,p(Ω ) , for some p = p ( η ) > 2 p=p(η )>2 . This extends a foundational result of Ahlfors and Bers (the case l = 0 l=0 ). As an application, we deduce that Bers metrics depend holomorphically on their input data.