A solution operator for the \overline∂ equation in Sobolev spaces of negative index

Abstract

Let Ω \Omega be a strictly pseudoconvex domain in C n \mathbb {C}^n with C k + 2 C^{k+2} boundary, k ≥ 1 k \geq 1 . We construct a ∂ ¯ \overline \partial solution operator (depending on k k ) that gains 1 2 \frac 12 derivative in the Sobolev space H s , p ( Ω ) H^{s,p} (\Omega ) for any 1 > p > ∞ 1>p>\infty and s > 1 p − k s>\frac {1}{p} -k . If the domain is C ∞ C^{\infty } , then there exists a ∂ ¯ \overline \partial solution operator that gains 1 2 \frac 12 derivative in H s , p ( Ω ) H^{s,p}(\Omega ) for all s ∈ R s \in \mathbb {R} . We obtain our solution operators via the method of homotopy formula. A novel technique is the construction of “anti-derivative operators” for distributions defined on bounded Lipschitz domains.