A criterion for the existence of holomorphic extension of functions in \(\mathbb{C}^ 2\) (Q1842485)

For a domain \(D \subset \mathbb{C}^ n\) with smooth boundary denote by \(G(D)\) the class of functions harmonic in \(D\) having a finite growth order near \(\partial D\), i.e. for any \(f \in G(D)\) there are constants \(C > 0\) and \(m > 0\) such that \[ \bigl | f(z) \bigr | \leq C \bigl[ \text{dist} (z, \partial D) \bigr]^{-m} \quad \text{for} \quad z \in D. \] The authors rewrite the normal \(\overline \partial\)-equation for \(f\) in \(D \subset \mathbb{C}^ 2\) in the form \[ *\overline \partial f = a df \wedge dz \tag{1} \] on \(\partial D\). Here \(*\) stands for the Hodge operator in \(\mathbb{C}^ 2\) and \(a(z) = a_ 1(z) + ia_ 2 (z)\) -- some complex function. The main result of the paper is the following Theorem. Let \(D\) be a bounded, simply connected and smooth domain in \(\mathbb{C}^ 2\). If for some real \(k\) one has \(a_ 1(\zeta) + ka_ 2 (\zeta) \neq 0\) for all \(\zeta \in \partial D\), then every \(f \in G(D)\) that satisfies (1) in the sense of distributions, is in fact holomorphic in \(D\).