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$C^{k}$ maps; partial regularity; Carleman classes; Beurling classes
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).
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