Level Sets of Certain Subclasses of α-analytic Functions (Journal of Partial Differential Equations, 2020)
For an open set V ⊂Cn, denote by Mα(V) the family of α-analytic functionsthat 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 satisfiesf ∈ 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 approximableby α-analytic functions where each approximating member and its reciprocal (off thesingularities) obey the boundary maximum modulus principle. We prove, that for aC3-smooth hypersurface, Ω, a member of Mα(Ω), cannot have constant modulus neara point where the Levi form has a positive eigenvalue, unless it is there the trace ofa polyanalytic function of a simple form. The result can be partially generalized toC4-smooth submanifolds of higher codimension, at least near points with a Levi conecondition.