On Boman's theorem on partial regularity of mappings

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 -