Local extension of boundary holomorphic forms on real hypersurfaces in \(\mathbb C^{n}\)
Let \(M\) be real hypersurface in \(\mathbb{C}^n\), with a smooth defining function \(\rho\); for \(p \in M\), a fixed point and \(U\) a neighbourhood of \(p\) in \(\mathbb{C}^n\) set \(U^- = \{z\in U \mid p(z)\leq 0\}\), \(U^+ = \{z \in U \mid p(z) \geq 0\}\). The main result of the authors is: if \(M\) is as above, \(n \geq 3\) and \(p \in M\), and if we suppose that the Levi form of \(\rho\) at \(p\) has \(q + 1\) positive eigenvalues \((1 \leq q \leq n - 2)\) then there is a neighbourhood \(U\) of \(p\) such that for any \( \alpha \in \Lambda^{0, r}(M \cap U)\), \(1 \leq r \leq q\) satisfying \(\overline\partial_{b}\alpha = 0\) on \(M \cap U\) there exists \(\widetilde \alpha \in \Lambda^{0, r} (U^-)\) such that \(\overline\partial \widetilde\alpha = 0\) in \(U^- \backslash M\) and \((\widetilde\alpha - \alpha) \wedge\overline\partial \rho = 0\) on \(M \cap U\). Here \(\Lambda^{0, r}(U^-)\) denotes the space of smooth \((p, q)\) forms on \(U^-\). Thus the local extension problem is solved generalizing a result of \textit{J. Xu} [Ph. D thesis, Purdue (1999)] which proved the local one sided \(\overline\partial\)-closed extension problem for \((0, 1)\) forms, except some bad set \(B \subset M\). The method of proof is based on the following result which has an independent interest. Namely, let \(\{\Omega_t\}_{t \in I}\) be a smooth family of strongly pseudoconvex domain in \(\mathbb{C}^{k_1}\) and \(N_t\) be the \(\overline\partial\)-Neumann operator in \(\Omega_t\), for each \(t \in I\), \(I\) being an open subset of \(\mathbb{C}^{k_2}\). Then \(N_t\) depends smoothly on \(t \in I\), and if \(u(z, t) = N_t \alpha_t(z)\) (which is proved to be a smooth form in \({\text{di}}\cup_{t \in I} \Omega_t\) which is open in \(\mathbb{C}^{k_1 + k_2}\)) it verifies \(\| u\| _{m(U \Omega_t)} \leq C_m \| \alpha\| _{3m} (U \Omega_t)\) provided \(\alpha \in H_{3m} (U \Omega_t)\) where the \(H_m\) are the Sobolev norms. This proof of this result is a slight modification of \textit{S. Cho} [Int. J. Math. 13, No. 10, 1027--1042 (2002; Zbl 1109.32302)]. Then the authors find for a fixed point \(p \in M\) an open neighbourhood of \(U\) of \(p\) in \(\mathbb{C}^n\) such that \(U^- = U \cap \{ z \mid p(z) \leq 0 \}\) is decomposed into a smooth family of strongly psudoconvex domains in \(\mathbb{C}^k\), supposing the Levi form of the defining function \(\rho\) has \(k\) positive eigenvalues, \(2 \leq k \leq n - 1\). This is achieved using in an ingenious way a convenient representation of \(\rho\), and the implicit function theorem. The authors prove, first, the main result in a weak form, i.e. the local extension \(U^- \backslash B\) where \(B\) is a bad set; it turns out that \(B\) is a \(2n - 2k -1\) real dimension variety where \(\{z" \in \mathbb{C}^k, p (z', z") \leq 0\}\) is reduced to a one point set. This extension result is proved using classical results of Kohn. Finally, the authors get rid of this bad set, proving that the form \(\widetilde\alpha \in \Lambda^{0, r} (U^- \backslash B)\) given by the ``weak version'' is in fact in \(\Lambda^{0, r}(U^-)\), by a careful analysis of the construction of \(\widetilde \alpha\), which shows that in fact \(\| \widetilde \alpha\| _{m (U^-\backslash M)}< \infty\). As a corollary the authors obtain a condition sufficient to solve locally the equation \(\overline\partial_b u = \alpha\), supposing that \(M\) as above satisfies the condition that at \(p \in M\) the Levi form of \(\rho\) has \(\max(q + 1, n - q)\) positive eigenvalues \(1 \leq q \leq n - 2\). This result is contained in the results of \textit{A. Andreotti} and \textit{C. D. Hill} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26, 325--363 (1972; Zbl 0256.32007) and ibid. 26, 747--806 (1972; Zbl 0283.32013)] and \textit{M.-C. Shaw} [Integral representations for \(\overline\partial_b\) in CR manifolds. Geometric complex analysis. Proceedings of the conference held at the 3rd International Research Institute of the Mathematical Society of Japan, Hayama, March 19--29, 1995. Singapore: World Scientific. 535--549 (1996; Zbl 0947.32024)].
scientific article; zbMATH DE number 5078925