Hilbert-valued forms and barriers on weakly pseudoconvex domains (Q1281291)

Let \(\Omega\) and \(\Omega^*\) be bounded domains in \(\mathbb{C}^n\) with \(\text{Lip}^1\) boundaries, such that \(\Omega\) is weakly pseudoconvex and compactly supported in \(\Omega^*\). The author proves the existence of functions \(w_1(z,\zeta), \dots, w_n(z,\zeta)\in C^k(\Omega \times\overline U)\), \(U:=\Omega^* \setminus\Omega\), holomorphic in \(z\in\Omega\) for any fixed \(\zeta \in\overline U\), such that \(\sum_jw_j (z,\zeta) (z_j-\zeta_j) =1\) and \(|D^\ell_\zeta w_j (z,\zeta) |\leq C_k \text{dist}(z,\partial \Omega)^{-((n+1) k+n^2+ 5n+2)}\) for each derivation \(D^\ell_\zeta\) in \(\zeta\) of order \(\ell\leq k\), where the constant \(C_k\) depends on \(k,\Omega\), and \(\Omega^*\) only. Comparing it with the construction of a barrier map from [\textit{J. Michel} and \textit{M.-C. Shaw}, Math. Z. 230, No. 1, 1-19 (1999; Zbl 0956.32031), see below ], this result relaxes the regularity condition on \(\Omega\) and improves the growth control for \(D^\ell_\zeta w_j (z,\zeta)\). Its proof, quite different from Michel-Shaw's, is based on Hörmander-type \(L^2\) techniques applied to functions with values in a Sobolev space.