(untitled) (Journal of Partial Differential Equations, 2020)
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 ∈ Mα(Ω f−1(0)) automatically satisfies f ∈ Mα(Ω), if it is Cαj−1-smooth in the zj variable, α ∈ Zn+ up to the boundary. For a submanifold U⊂Cn, denote by Mα(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 Mα(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.