A pure smoothness condition for Radó’s theorem for α -analytic functions

Abtin Daghighi; Frank Wikström

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 1, page 57-62
  • ISSN: 0011-4642

Abstract

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Let Ω ⊂ ℂ n be a bounded, simply connected ℂ -convex domain. Let α ∈ ℤ + n and let f be a function on Ω which is separately C 2 α j - 1 -smooth with respect to z j (by which we mean jointly C 2 α j - 1 -smooth with respect to Re z j , Im z j ). If f is α -analytic on Ω ∖ f - 1 ( 0 ) , then f is α -analytic on Ω . The result is well-known for the case α i = 1 , 1 ≤ i ≤ n , even when f a priori is only known to be continuous.

How to cite

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Daghighi, Abtin, and Wikström, Frank. "A pure smoothness condition for Radó’s theorem for $\alpha $-analytic functions." Czechoslovak Mathematical Journal 66.1 (2016): 57-62. <http://eudml.org/doc/276792>.

@article{Daghighi2016,
abstract = {Let $\Omega \subset \mathbb \{C\}^n$ be a bounded, simply connected $\mathbb \{C\}$-convex domain. Let $\alpha \in \mathbb \{Z\}_+^n$ and let $f$ be a function on $\Omega $ which is separately $C^\{2\alpha _j-1\}$-smooth with respect to $z_j$ (by which we mean jointly $C^\{2 \alpha _j-1\}$-smooth with respect to $\mathop \{\rm Re\} z_j$, $ \mathop \{\rm Im\} z_j$). If $f$ is $\alpha $-analytic on $\Omega \setminus f^\{-1\}(0)$, then $f$ is $\alpha $-analytic on $\Omega $. The result is well-known for the case $\alpha _i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.},
author = {Daghighi, Abtin, Wikström, Frank},
journal = {Czechoslovak Mathematical Journal},
keywords = {$\alpha $-analytic function; polyanalytic function; zero set; Radó’s theorem},
language = {eng},
number = {1},
pages = {57-62},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A pure smoothness condition for Radó’s theorem for $\alpha $-analytic functions},
url = {http://eudml.org/doc/276792},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Daghighi, Abtin
AU - Wikström, Frank
TI - A pure smoothness condition for Radó’s theorem for $\alpha $-analytic functions
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 57
EP - 62
AB - Let $\Omega \subset \mathbb {C}^n$ be a bounded, simply connected $\mathbb {C}$-convex domain. Let $\alpha \in \mathbb {Z}_+^n$ and let $f$ be a function on $\Omega $ which is separately $C^{2\alpha _j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2 \alpha _j-1}$-smooth with respect to $\mathop {\rm Re} z_j$, $ \mathop {\rm Im} z_j$). If $f$ is $\alpha $-analytic on $\Omega \setminus f^{-1}(0)$, then $f$ is $\alpha $-analytic on $\Omega $. The result is well-known for the case $\alpha _i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.
LA - eng
KW - $\alpha $-analytic function; polyanalytic function; zero set; Radó’s theorem
UR - http://eudml.org/doc/276792
ER -

References

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