@Article{JPDE-30-4,
author = {Daghighi, Abtin and Wikström, Frank},
title = {Level Sets of Certain Subclasses of α-analytic Functions},
journal = {Journal of Partial Differential Equations},
year = {2017},
volume = {30},
number = {4},
pages = {281--298},
abstract = {<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>},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v30.n4.1},
url = {https://global-sci.com/article/88236/level-sets-of-certain-subclasses-of-a-analytic-functions}
}