On Boman's theorem on partial regularity of mappings

Commentationes mathematicae Universitatis Carolinae

Tejinder S. Neelon¹ 

关键词: $C^{k}$ maps;    partial regularity;    Carleman classes;    Beurling classes;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

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 {\it 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., {\it Partial regularity of mappings between Euclidean spaces\/}, Acta Math. {\bf 119} (1967), 1--25).

【 授权许可】

CC BY