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

TL;DR

This paper investigates the boundary behavior of $eta$-analytic functions, establishing conditions under which such functions extend smoothly and characterizing their modulus near points with positive Levi form eigenvalues.

Contribution

It proves that $eta$-analytic functions with boundary maximum modulus principles extend smoothly to the boundary and characterizes their modulus near certain boundary points.

Findings

Functions extend smoothly up to the boundary under certain smoothness conditions.

Modulus of functions near points with positive Levi eigenvalues is constrained to specific forms.

Provides conditions for functions to be traces of polyanalytic 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.