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Volume 30, Issue 4
Level Sets of Certain Subclasses of α-analytic Functions

Abtin Daghighi & Frank Wikström

DOI: 10.4208/jpde.v30.n4.1

J. Part. Diff. Eq., 30 (2017), pp. 281-298.

Published online: 2017-11

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  • Abstract

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.

  • Keywords

Polyanalytic functions q-analytic functions zero sets level sets α-analytic functions.

  • AMS Subject Headings

35G05, 32A99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

abtindaghighi@gmail.com (Abtin Daghighi)

Frank.Wikstrom@math.lth.se (Frank Wikström)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-30-281, 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 = {

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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v30.n4.1}, url = {http://global-sci.org/intro/article_detail/jpde/10675.html} }
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 -

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.

Daghighi , Abtin and Wikström , Frank. (2017). Level Sets of Certain Subclasses of α-analytic Functions. Journal of Partial Differential Equations. 30 (4). 281-298. doi:10.4208/jpde.v30.n4.1
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