Sobolev $\frac{1}{2}$ estimates for $\overline{\partial}$ equations on strictly pseudoconvex domains with $C^2$ boundary

Abstract

We construct a solution operator for ∂ equation that gains 1 2 derivative in the fractional Sobolev space H s,p on bounded strictly pseudoconvex domains in C n with C 2 boundary, for all 1 < p < ∞ and s > 1 p .

Sobolev and Hölder Estimates for Homotopy Operators of $\overline\partial$-Equations on Convex Domains of Finite Multitype

“…Recall that these two results are not comparable since the Sobolev embedding H s+ 1 2 ,p ֒→ H s, 4np 4n−p is not contained in H s, (2n+2)p 2n+2−p . By keeping check of the proof using regularized distance functions, one could show that the results for non-smooth domains are true: if k ≥ 0 is an integer and bΩ ∈ C k+2 , then T q : H s,p → H s, (2n+2)p 2n+2−p is still true for all 1 < p < 2n + 2 and s > 1 p − k. We refer [SY21c,SY21a] to reader. To prove Theorem 7.1 we repeat the arguments in Section 3.…”

“…In our case we choose E to be the Rychkov's extension operator, which works on Lipschitz domain and extends H s,p and C s for all s (including s < 0), see (4.6) and (4.14). The Rychkov's extension operator was first introduced to ∂-solutions in [SY21c].…”

Existence of Solutions for $\overline\partial$ Equation in Sobolev Spaces of Negative Index

“…More recently, Gong [Gon19] constructed a ∂ solution operator on strictly pseudoconvex domains with C 2 boundary which gains 1/2 derivative for any ϕ ∈ Λ r (Ω), r > 1. Assuming C 2 boundary still, the authors in a recent preprint [SY21b] found a ∂ solution operator which gains 1/2 derivative for any ϕ ∈ H s,p (Ω), with 1 < p < ∞ and s > 1 p ; it was further shown in [SY21b] that the same solution operator gains 1/2 derivative for any ϕ ∈ Λ r (Ω), r > 0. All the results mentioned hitherto assume the index is non-negative.…”