A decomposition problem on weakly pseudoconvex domains (Q1297924)

The main result of the paper is the following decomposition theorem. Let \(\Omega\subset\mathbb C^n\) be a bounded pseudoconvex domain with \(\mathcal C^K\)-defining function \(\widetilde\rho\) (\(2\leq K\leq\infty\)). Let \(\nu\in[1,+\infty)\) be such that \(\rho:=-(-\widetilde\rho)^{1/\nu}\) is a strictly plurisubharmonic exhaustion function for \(\Omega\). Let \(\Omega^0\) be a bounded open set with \(\overline\Omega\subset\Omega^0\), and let \(S:=\overline\Omega^0\setminus\Omega\). Then for any non-negative integer \(k\) there exist a \(\mathcal C^k\)-map \(W\:\Omega\times S\longrightarrow\mathbb C^n\) and a constant \(C\) such that: (i) \(W(\cdot,\zeta)\in\mathcal O(\Omega,\mathbb C^n)\) for \(\zeta\in S\) and \(\sum_{j=1}^nW_j(z,\zeta)(\zeta_j-z_j)=1\) for \((z,\zeta)\in\Omega\times S\); (ii) \(|D^I_\zeta W(z,\zeta)|\leq C\big(\)dist\((z,\partial\Omega)\big)^ {-[(5t^2+3|I|+1)/\nu+2n]}\) for \(|I|\leq k\) and \((z,\zeta)\in\Omega\times S\), where \(t:=[2\nu\max\{4+3k,(n-1)/5\}+1]\), and \([x]\) denotes the integer part of \(x\).