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2008/2009 An Ontology of Directional Regularity Implying Joint Regularity
Steven G. Krantz
Real Anal. Exchange 34(2): 255-266 (2008/2009).


It is an old idea to consider whether a function on $\mathbb{R}^N$ that is smooth in each variable separately is in fact jointly smooth. It turns out that some uniformity of estimates in each variable is necessary for such a result. More recently, there have been studies of functions that are smooth along integral curves of certain vector fields. Depending on the commutator properties of the vector fields, different types of results may be obtained. Another recent idea is that if one has smoothness along all curves then the uniformity hypothesis may be dropped. In the present paper we explore all these approaches to the problem in a variety of new norms. We present new, simpler proofs of some classical results. We also explore new theorems in the real analytic category.


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Steven G. Krantz. "An Ontology of Directional Regularity Implying Joint Regularity." Real Anal. Exchange 34 (2) 255 - 266, 2008/2009.


Published: 2008/2009
First available in Project Euclid: 29 October 2009

zbMATH: 1186.26008
MathSciNet: MR2569187

Primary: 26B05
Secondary: 26A16 , 26A24 , 35B65

Keywords: infinitely differentiable , joint smoothness , Lipschitz , real analytic , regularity , smoothness

Rights: Copyright © 2008 Michigan State University Press

Vol.34 • No. 2 • 2008/2009
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