TY - JOUR
T1 - Level Sets of Certain Subclasses of α-analytic Functions
AU - Daghighi , Abtin
AU - Wikström , Frank
JO - Journal of Partial Differential Equations
VL - 4
SP - 281
EP - 298
PY - 2017
DA - 2017/11
SN - 30
DO - http://doi.org/10.4208/jpde.v30.n4.1
UR - https://global-sci.org/intro/article_detail/jpde/10675.html
KW - Polyanalytic functions
KW - q-analytic functions
KW - zero sets
KW - level sets
KW - α-analytic functions.
AB - <p>For an open set V ⊂C<sup>n</sup>, 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 ⊂ C<sup>n</sup>, a function f ∈ M<sub>α</sub>(Ω\ f<sup>−1</sup>(0)) automatically satisfies
f ∈ M<sub>α</sub>(Ω), if it is C<sup>αj−1</sup>-smooth in the zj variable, α ∈ Z<sup>n</sup><sub>+</sub> up to the boundary.
For a submanifold U⊂C<sup>n</sup>, denote by M<sub>α</sub>(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
C<sup>3</sup>-smooth hypersurface, Ω, a member of M<sub>α</sub>(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
C<sup>4</sup>-smooth submanifolds of higher codimension, at least near points with a Levi cone
condition.</p>