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Level Sets of Certain Subclasses of α-analytic Functions

Wikström, Frank LU orcid and Daghighi, Abtin (2017) In Journal of partial differential equations 30(4). p.281-298
Abstract
For an open set V subset of C-n, denote by M-alpha(V) the family of a-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded "harmonically fat" domain Omega subset of C-n, a function f is an element of M-alpha (Omega\f(-1)(0)) automatically satisfies f is an element of M-alpha(Omega), if it is C alpha j-1-smooth in the z(j) variable, alpha is an element of Z(+)(n) up to the boundary. For a submanifold U subset of C-n, denote by M-alpha(U), the set of functions locally approximable by a-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C-3-smooth hypersurface, Omega, a member of... (More)
For an open set V subset of C-n, denote by M-alpha(V) the family of a-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded "harmonically fat" domain Omega subset of C-n, a function f is an element of M-alpha (Omega\f(-1)(0)) automatically satisfies f is an element of M-alpha(Omega), if it is C alpha j-1-smooth in the z(j) variable, alpha is an element of Z(+)(n) up to the boundary. For a submanifold U subset of C-n, denote by M-alpha(U), the set of functions locally approximable by a-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C-3-smooth hypersurface, Omega, a member of M-alpha(Omega), 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-4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition. (Less)
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author
and
organization
alternative title
Nivåmängder för vissa klasser av α-analytiska funktioner
publishing date
type
Contribution to journal
publication status
published
subject
keywords
polyanalytic functions, q -analytic functions, zero sets, level sets, α -analytic functions
in
Journal of partial differential equations
volume
30
issue
4
pages
281 - 298
publisher
Global Science Press
ISSN
1000-940X
DOI
10.4208/jpde.v30.n4.1
language
English
LU publication?
yes
id
de61bb2c-945b-4d76-944c-9940d3b9ceeb
alternative location
http://www.global-sci.org/intro/article_detail.html?journal=jpde&article_id=10675
date added to LUP
2018-11-11 22:42:19
date last changed
2018-11-21 21:43:09
@article{de61bb2c-945b-4d76-944c-9940d3b9ceeb,
  abstract     = {{For an open set V subset of C-n, denote by M-alpha(V) the family of a-analytic functions that obey a boundary maximum modulus principle. We prove that, on a bounded "harmonically fat" domain Omega subset of C-n, a function f is an element of M-alpha (Omega\f(-1)(0)) automatically satisfies f is an element of M-alpha(Omega), if it is C alpha j-1-smooth in the z(j) variable, alpha is an element of Z(+)(n) up to the boundary. For a submanifold U subset of C-n, denote by M-alpha(U), the set of functions locally approximable by a-analytic functions where each approximating member and its reciprocal (off the singularities) obey the boundary maximum modulus principle. We prove, that for a C-3-smooth hypersurface, Omega, a member of M-alpha(Omega), 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-4-smooth submanifolds of higher codimension, at least near points with a Levi cone condition.}},
  author       = {{Wikström, Frank and Daghighi, Abtin}},
  issn         = {{1000-940X}},
  keywords     = {{polyanalytic functions; q -analytic functions; zero sets; level sets; α -analytic functions}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{281--298}},
  publisher    = {{Global Science Press}},
  series       = {{Journal of partial differential equations}},
  title        = {{Level Sets of Certain Subclasses of α-analytic Functions}},
  url          = {{http://dx.doi.org/10.4208/jpde.v30.n4.1}},
  doi          = {{10.4208/jpde.v30.n4.1}},
  volume       = {{30}},
  year         = {{2017}},
}