Level sets of certain classes of $\alpha$-analytic functions

Abstract

For an open set $V\subset\mathbb{C}^n$, denote by $\mathscr{M}_{\alpha}(V)$ the family of $\alpha$-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded domain $\Omega\subset \mathbb{C}^n$, with continuous boundary (that in each variable separately allows a solution to the Dirichlet problem), a function $f \in \mathscr{M}_{\alpha}(\Omega\setminus f^{-1}(0))$ automatically satisfies $f\in \mathscr{M}_{\alpha}(\Omega)$, if it is $C^{\alpha_j-1}$-smooth, in the $z_j$ variable, $\alpha\in \mathbb{Z}^n_+$, up to the boundary. For a submanifold $U\subset \mathbb{C}^n$, denote by $\mathfrak{M}_{\alpha}(U)$ the set of functions locally approximable by $\alpha$-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a $C^3$-smooth hypersurface, $\Omega$, a member of $\mathfrak{M}_{\alpha}(\Omega)$, cannot have constant modulus near a point where the Levi form has a positive eigenvalue, unless it is there the trace of a polyanalytic function of a simple form.