Let Ω be a strictly pseudoconvex domain in C n with C k+2 boundary, k ≥ 1. We construct a ∂ solution operator (depending on k) that gains 1 2 derivative in the Sobolev space H s,p (Ω) for any 1 < p < ∞ and s > 1 p − k. If the domain is C ∞ , then there exists a ∂ solution operator that gains 1 2 derivative in H s,p (Ω) for all s ∈ R. We obtain our solution operators through the method of homotopy formula; a new feature is the construction of "anti-derivative operators" on distributions defined on bounded Lipschitz domains.

Given a bounded Lipschitz domain Ω ⊂ R n , Rychkov showed that there is a linear extension operator E for Ω which is bounded in Besov and Triebel-Lizorkin spaces. In this paper we introduce a class of operators that generalize E which are more versatile for applications. We also derive some quantitative smoothing estimates of the extended function and all its derivatives in Ω c up to boundary.

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We give an example of a C k C^k -integrable almost complex structure that does not admit a corresponding C k + 1 C^{k+1} -complex coordinate system.

We construct homotopy formulas for ∂-equations on convex domains of finite type that have optimal Sobolev and Hölder estimates. For a bounded smooth finite type convex domain Ω ⊂ C n that has q-type mq for 1 ≤ q ≤ n, our ∂ solution operator Tq on (0, q)-forms has (fractional) Sobolev boundedness Tq : H s,p → H s+1/mq ,p and Hölder-Zygmund boundedness Tq : C s → C s+1/mq for all s ∈ R and 1 < p < ∞. We also show the L p -boundedness Tq : H s,p → H s,prq /(rq −p) for all s ∈ R and 1 < p < rq, where rq := (n − q + 1)mq + 2q.