On Boman's theorem on partial regularity of mappings

Tejinder S. Neelon

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 3, page 349-357
  • ISSN: 0010-2628

Abstract

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Let and be a positive integer. Let be a locally bounded map such that for each , the derivatives , , exist and are continuous. In order to conclude that any such map is necessarily of class it is necessary and sufficient that be not contained in the zero-set of a nonzero homogenous polynomial which is linear in and homogeneous of degree in . This generalizes a result of J. Boman for the case . The statement and the proof of a theorem of Boman for the case is also extended to include the Carleman classes and the Beurling classes (Boman J., Partial regularity of mappings between Euclidean spaces, Acta Math. 119 (1967), 1–25).

How to cite

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Neelon, Tejinder S.. "On Boman's theorem on partial regularity of mappings." Commentationes Mathematicae Universitatis Carolinae 52.3 (2011): 349-357. <http://eudml.org/doc/246863>.

@article{Neelon2011,
abstract = {Let $\Lambda \subset \mathbb \{R\}^\{n\}\times \mathbb \{R\}^\{m\}$ and $k$ be a positive integer. Let $f:\mathbb \{R\}^\{n\}\rightarrow \mathbb \{R\}^\{m\}$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda $, the derivatives $D_\{\xi \}^\{j\}f(x):= \frac\{d^\{j\}\}\{dt^\{j\}\}f(x+t\xi ) \Big \vert _\{t=0\}$, $j=1,2,\dots k$, exist and are continuous. In order to conclude that any such map $f$ is necessarily of class $C^\{k\}$ it is necessary and sufficient that $\Lambda $ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =(\eta _\{1\},\eta _\{2\},\dots ,\eta _\{m\})$ and homogeneous of degree $k$ in $\xi =(\xi _\{1\},\xi _\{2\},\dots ,\xi _\{n\})$. This generalizes a result of J. Boman for the case $k=1$. The statement and the proof of a theorem of Boman for the case $k=\infty $ is also extended to include the Carleman classes $C\lbrace M_\{k\}\rbrace $ and the Beurling classes $C(M_\{k\})$ (Boman J., Partial regularity of mappings between Euclidean spaces, Acta Math. 119 (1967), 1–25).},
author = {Neelon, Tejinder S.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$C^\{k\}$ maps; partial regularity; Carleman classes; Beurling classes; -mapping; Carleman class; Beurling class},
language = {eng},
number = {3},
pages = {349-357},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On Boman's theorem on partial regularity of mappings},
url = {http://eudml.org/doc/246863},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Neelon, Tejinder S.
TI - On Boman's theorem on partial regularity of mappings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 3
SP - 349
EP - 357
AB - Let $\Lambda \subset \mathbb {R}^{n}\times \mathbb {R}^{m}$ and $k$ be a positive integer. Let $f:\mathbb {R}^{n}\rightarrow \mathbb {R}^{m}$ be a locally bounded map such that for each $(\xi ,\eta )\in \Lambda $, the derivatives $D_{\xi }^{j}f(x):= \frac{d^{j}}{dt^{j}}f(x+t\xi ) \Big \vert _{t=0}$, $j=1,2,\dots k$, exist and are continuous. In order to conclude that any such map $f$ is necessarily of class $C^{k}$ it is necessary and sufficient that $\Lambda $ be not contained in the zero-set of a nonzero homogenous polynomial $\Phi (\xi ,\eta )$ which is linear in $\eta =(\eta _{1},\eta _{2},\dots ,\eta _{m})$ and homogeneous of degree $k$ in $\xi =(\xi _{1},\xi _{2},\dots ,\xi _{n})$. This generalizes a result of J. Boman for the case $k=1$. The statement and the proof of a theorem of Boman for the case $k=\infty $ is also extended to include the Carleman classes $C\lbrace M_{k}\rbrace $ and the Beurling classes $C(M_{k})$ (Boman J., Partial regularity of mappings between Euclidean spaces, Acta Math. 119 (1967), 1–25).
LA - eng
KW - $C^{k}$ maps; partial regularity; Carleman classes; Beurling classes; -mapping; Carleman class; Beurling class
UR - http://eudml.org/doc/246863
ER -

References

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  9. Neelon T.S., 10.1090/S0002-9939-99-04759-0, Proc. Amer. Math. Soc. 127 (1999), 2099–2104. MR1487332DOI10.1090/S0002-9939-99-04759-0
  10. Neelon T.S., 10.4153/CMB-2006-026-9, Canad. Math. Bull. 49 (2006), 256–264. MR2226248DOI10.4153/CMB-2006-026-9
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