On the extension of Lipschitz functions from boundaries of subvarieties to strongly pseudoconvex domains (Q811607)

Let \(D=\{z:\;\rho(z)<0\}\) be a bounded strongly pseudoconvex domain in \(\mathbb{C}^ n\) with \(C^ \infty\) boundary and let \(h_ 1,\dots,h_ p\) be holomorphic functions in a neighbourhood \(\tilde D\) of \(\overline D\). Define \[ \tilde V=\{z\in\tilde D:\;h_ 1(z)=\dots=h_ p(z)=0\}, \qquad V=\tilde V\cap D. \] Assume that \(\partial h_ 1\land\dots\land\partial h_ p\land\partial\rho\neq 0\) on \(\partial V\). We define the principal value integral on \(\partial V\) using the support function obtained by \textit{G. M. Henkin} [Math. USSR, Sb. 7, 597-616 (1970); translation from Mat. Sb., n. Ser. 78(120), 611-632 (1969; Zbl 0206.09004)] and \textit{E. Ramírez de Arellano} [Math. Ann. 184, 172-187 (1970; Zbl 0189.09702)]. The main result is the following: Theorem. Let \(f\in\hbox{Lip}(\alpha,\partial V)\), \(0<\alpha\leq 1\). If \(f\) satisfies for any \(z\in\partial V\) \[ P.V.\int_{\partial V}f(\zeta)\Omega(\zeta,z)={1\over 2}f(z), \] then there exists a function \(F\in{\mathcal O}(D)\cap C(\overline D)\) such that \(F\mid_{\partial V}=f\).