Hölder estimates for homotopy operators on strictly pseudoconvex domains with $$C^2$$ C 2 boundary

Abstract

We derive a new homotopy formula for a strictly pseudoconvex domain of C 2 boundary in C n by using a method of Lieb and Range and obtain estimates in Lipschitz spaces for the homotopy operators. For r > 1 and q > 0, we obtain a r +1/2 solution u to ∂u = f for a ∂-closed (0, q)-form f of class r in the domain. We apply the estimates to obtain boundary regularities of D-solutions for a domain inthat admits a derivative estimate. Here ϕ is a (0, q)-form in D and ϕ, ∂ϕ are in C 1 (D). We will prove the following C r +1/2 estimate. Communicated by Ngaiming Mok.

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

“…We also prove an analogous result when ϕ is in Hölder-Zygmund space Λ r (Ω) which improves an earlier theorem of Gong [Gon19].…”

“…Together with the commutator estimate and Hardy-Littlewood lemma, this leads to the proof of Theorem 1.1. Unlike in [Gon19] and [Shi21], no integration by parts is used in our proof.…”

“…More recently, Gong [Gon19] used the integral formula method to construct a ∂ solution operator for any C 2 strictly pseudoconvex domains, and the solution u lies in Λ r+ 1 2 (Ω) if ϕ is any (0, q) form (q ≥ 1) in the class Λ r (Ω), for all r > 1.…”

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

“…When the boundary is only finitely smooth, we need the following regularized Henkin-Ramirez function constructed by Gong [Gon19].…”

“…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.…”