The $$\overline{\partial }$$ ∂ ¯ -equation on variable strictly pseudoconvex domains

Abstract

Abstract. We investigate regularity properties of the ∂-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.

Families of Strictly Pseudoconvex Domains and Peak Functions

“…As noticed by the referee, our result can be strengthened in the spirit of Theorem 5.1 from [6]. It gives the construction of Henkin-Ramírez functions for variable strictly pseudoconvex open sets (with boundaries of class C 2+a, j ; see Definition 2.5 therein) depending C 1+a, j -smoothly on a parameter.…”

“…It gives the construction of Henkin-Ramírez functions for variable strictly pseudoconvex open sets (with boundaries of class C 2+a, j ; see Definition 2.5 therein) depending C 1+a, j -smoothly on a parameter. Under similar assumptions as in [6], and by merging the method of proof of our Theorem 1.3 with the method of proof of Theorem 5.1 from [6], we can get similar regularity for the dependence of our peak functions on the parameter.…”

“…The author is very grateful to the referees for valuable comments, especially for drawing his attention to the paper [6] and for Remark 1.5.…”

Splitting Lemma for Biholomorphic Mappings with Smooth Dependence on Parameters

“…This requires a parametric solution of the ∂-problem with extremely nice behaviour with respect to parameters. The one we need is given in [8,Theorem 4.10] (the space of parameters there is just [0, 1], but the analysis of the proof reveals that it can be adjusted to our situation). Now, the interpolation condition with regular behaviour with respect to parameters is also nontrivial.…”

“…The loss of regularity in Theorem 1.3 comes from two reasons: one is the use of (a variant of) in [8,Theorem 4.10]. Here, one would get somehow more precise result in terms of the families of functions of class C l, j on the so called total space of the family of domains depending on a parameter, introduced in the quoted paper, but even in this language one needs to have j ≤ k −1.…”