An ontology of directional regularity implying joint regularity
type: Item
instance of: scholarly article
title: An ontology of directional regularity implying joint regularity
zbMATH Open document ID: 1186.26008
author: Steven G. Krantz
published in: Real Analysis Exchange
publication date: 2010-01-04T00:00:00Z
review text: This is a survey on results which state that a function on a subset of \(\mathbb R^n\) has a certain smoothness (Hölder, Lipschitz, higher-order smoothness, belongs to a Sobolev space, or is (real or complex) analytic) if it has this smoothness in each of the variables separately and satisfies some extra conditions. One of these extra conditions is a uniform bound in the corresponding norms, and moreover, it then suffices to assume smoothness along a few paths (even less than \(n\) paths if the paths can be chosen appropriately). Another variant of such results which requires no knowledge of a uniform bound is to assume smoothness along each path from an appropriate class. For complex analytic functions even stronger results hold true. Besides being a survey, the paper also contains shorter new proofs for some of the results and new results in the class of real analytic functions.
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5652536
zbMATH Keywords: joint smoothness, coordinatewise smoothness
zbMATH Keywords: Hölder condition
zbMATH Keywords: Lipschitz condition
zbMATH Keywords: Sobolev space
zbMATH Keywords: analytic
zbMATH Keywords: real-analytic
zbMATH Keywords: infinitely differentiable
zbMATH Keywords: regularity
MaRDI profile type: Publication
OpenAlex ID: W1575878314
DOI: 10.14321/REALANALEXCH.34.2.0255
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description: scientific article; zbMATH DE number 5652536