# Article

 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: http://hdl.handle.net/10338.dmlcz/141607 . Reference: [1] Agbor D., Boman J.: On modulus of continuity of mappings between Euclidean spaces.Math. Scandinavica(to appear). Reference: [2] Bierstone E., Milman P.D., Parusinski A.: A function which is arc-analytic but not continuous.Proc. Amer. Math. Soc. 113 (1991), 419–423. Zbl 0739.32009, MR 1072083, 10.1090/S0002-9939-1991-1072083-4 Reference: [3] Bochnak J.: Analytic functions in Banach spaces.Studia Math. 35 (1970), 273–292. Zbl 0199.18402, MR 0273396 Reference: [4] Boman J.: Partial regularity of mappings between Euclidean spaces.Acta Math. 119 (1967), 1–25. Zbl 0186.10001, MR 0220883, 10.1007/BF02392077 Reference: [5] Hörmander L.: The Analysis of Linear Partial Differential Operators I.Springer, Berlin, 2003. MR 1996773 Reference: [6] Korevaar J.: Applications of $\mathbb{C}^{n}$ capacities.Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), Amer. Math. Soc., Providence, RI, 1991, pp. 105–118. MR 1128518 Reference: [7] Krantz S.G., Parks H.R.: A Primer of Real Analytic Functions.second edition, Birkhäuser, Boston, MS, 2002. Zbl 1015.26030, MR 1916029 Reference: [8] Neelon T.S.: On separate ultradifferentiability of functions.Acta Sci. Math. (Szeged) 64 (1998), 489–494. Zbl 0927.46023, MR 1666030 Reference: [9] Neelon T.S.: Ultradifferentiable functions on lines in $\mathbb{R}^{n}$.Proc. Amer. Math. Soc. 127 (1999), 2099–2104. MR 1487332, 10.1090/S0002-9939-99-04759-0 Reference: [10] Neelon T.S.: A Bernstein–Walsh type inequality and applications.Canad. Math. Bull. 49 (2006), 256–264. MR 2226248, 10.4153/CMB-2006-026-9 Reference: [11] Neelon T.S.: Restrictions of power series and functions to algebraic surfaces.Analysis (Munich) 29 (2009), no. 1, page 1–15. Zbl 1179.26088, MR 2524101, 10.1524/anly.2009.0929 Reference: [12] Rudin W.: Principles of Mathematical Analysis.3rd edition, McGraw-Hill, New York, 1976. Zbl 0346.26002, MR 0385023 Reference: [13] Siciak J.: A characterization of analytic functions of $n$ real variables.Studia Mathematica 35 (1970), 293–297. Zbl 0197.05801, MR 0279263 .

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