Whitney-type extensions in quasi-metric spaces

Ryan Alvarado; Irina Mitrea; Marius Mitrea

doi:10.3934/cpaa.2013.12.59

Article Contents

2013, Volume 12, Issue 1: 59-88. Doi: 10.3934/cpaa.2013.12.59

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Whitney-type extensions in quasi-metric spaces

1.
Department of Mathematics, University of Missouri at Columbia, Columbia, MO 65211
2.
Department of Mathematics, Temple University, Philadelphia, PA 19122
3.
Department of Mathematics, University of Missouri, Columbia, MO 65211

Received: April 30, 2011

Revised: February 29, 2012

Published: December 2012

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Abstract

We discuss geometrical scenarios guaranteeing that functions defined on a given set may be extended to the entire ambient, with preservation of the class of regularity. This extends to arbitrary quasi-metric spaces work done by E.J. McShane in the context of metric spaces, and to geometrically doubling quasi-metric spaces work done by H. Whitney in the Euclidean setting. These generalizations are quantitatively sharp.

Keywords:

Quasi-metric space,
geometrically doubling quasi-metric space,
extension of Hölder functions,
extension of Lipschitz functions,
quasi-metric space,
metrization,
Hölder functions,
Lipschitz functions,
partition of unity,
Whitney extension,
quantitative Urysohn lemma,
Whitney decomposition.

Mathematics Subject Classification: Primary: 26A16, 26B35; Secondary 26B05, 54E35.

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