Weighted Sobolev $$L^{p}$$ Estimates for Homotopy Operators on Strictly Pseudoconvex Domains with $$C^{2}$$ boundary

Abstract

We prove a homotopy formula which yields almost optimal estimates in all (positiveindexed) Sobolev and Hölder-Zygmund spaces for the ∂ equation on finite type domains in C 2 , extending the earlier results of Fefferman-Kohn (1988), Chang-Nagel-Stein (1992), and Range (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain, which generalizes Range's method.

Regularity of a $\bar\partial$-solution operator for strongly $\mathbf C$-linearly convex domains with minimal smoothness

“…When ∂D ∈ C 2 is strongly pseudoconvex, the statement analogous to Proposition 3.3 has been proved recently by Shi [22] when ϕ, ∂ϕ are in the Sobolev space W 1,1 (D). Indeed, note that the estimate (4.29) is trivial when x or y is the origin, so in the sequel we assume that neither of x, y is the origin.…”

“…The study of regularity of the solutions of the ∂-problem via integral representations has a long and rich history. A detailed review of the existing literature may be found in [20] and, for the most recent results, [6] and [22]. Here we briefly recall that for smooth, strongly pseudoconvex domains the optimal 1/2-estimate of Proposition 1.3 was achieved by Henkin-Romanov [11] for ∂-closed forms after Grauert-Lieb [7], Henkin [9], Kerzman [12] proved that a C β -estimate holds for any β < 1/2.…”

“…Proposition 1.3 for forms that are not necessarily ∂-closed is due to Range-Siu [21]. The C k+1/2 solutions for ∂u ∈ C k were obtained by Siu [23] for q = 1 and by Lieb-Range [16] for all degrees, and both require ∂D ∈ C k+2 and k ∈ N. The results in the aforementioned [6] were recently extended to weighted L p Sobolev spaces by Shi [22]. A survey of the extensive literature on the solutions of the ∂-problem with methods other than integral formulas may be found in e.g., Harrington [8].…”

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

“…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.…”

“…Furthermore our operator allows us to obtain 1 2 estimate when the right-hand side is Λ r (Ω), for all r > 0, which improves the above result of Gong. See also [Shi21] for estimates on a certain class of weighted Sobolev space.…”