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Title: On Boman's theorem on partial regularity of mappings (English)
Author: Neelon, Tejinder S.
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 52
Issue: 3
Year: 2011
Pages: 349-357
Summary lang: English
Category: math
Summary: 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 $C\{M_{k}\}$ and the Beurling classes $C(M_{k})$ (Boman J., Partial regularity of mappings between Euclidean spaces, Acta Math. 119 (1967), 1--25). (English)
Keyword: $C^{k}$ maps
Keyword: partial regularity
Keyword: Carleman classes
Keyword: Beurling classes
MSC: 26B12
MSC: 26B35
idZBL: Zbl 1249.26019
idMR: MR2843228
Date available: 2011-08-15T19:12:16Z
Last updated: 2013-10-14
Stable URL:
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