Level Sets of Certain Subclasses of α-analytic Functions

Abstract

For an open set V ⊂Cⁿ, denote by Mα(V) the family of α-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded “harmonically fat” domain Ω ⊂ Cⁿ, a function f ∈ M_α(Ω\ f⁻¹(0)) automatically satisfies f ∈ M_α(Ω), if it is C^αj−1-smooth in the zj variable, α ∈ Zⁿ₊ up to the boundary. For a submanifold U⊂Cⁿ, denote by M_α(U), the set of functions locally approximable by α-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C³-smooth hypersurface, Ω, a member of M_α(Ω), 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. The result can be partially generalized to C⁴-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.