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Residually many BV homeomorphisms map a null set onto a set of full measure

Part of: Linear function spaces and their duals Functions of several variables

Published online by Cambridge University Press:  27 December 2018

Andrea Marchese
Andrea Marchese*
Affiliation:
Institut für Mathematik, Mathematisch-naturwissenschaftliche Fakultät, Universität Zürich Winterthurerstrasse 190, Zürich CH-8057, Switzerland (andrea.marchese@math.uzh.ch)
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Abstract

Let Q be the open unit square in ℝ2. We prove that in a natural complete metric space of BV homeomorphisms f : Q → Q with f|∂Q = Id, residually many homeomorphisms (in the sense of Baire categories) map a null set onto a set of full measure, and vice versa. Moreover, we observe that for 1 ⩽ p < 2, the family of W1,p homemomorphisms satisfying the above property is of the first category.

Keywords

Sobolev homeomorphismBaire categoriespiecewise affine homeomorphism

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 26B35: Special properties of functions of several variables, Hölder conditions, etc.
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 1047 - 1059
Copyright
Copyright © Royal Society of Edinburgh 2018 

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