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A SHORT NOTE ON ENHANCED DENSITY SETS

Published online by Cambridge University Press:  01 August 2011

SILVANO DELLADIO
SILVANO DELLADIO*
Affiliation:
Department of Mathematics, University of Trento, via Sommarive 14, Povo, 38123 Trento, Italy e-mail: delladio@science.unitn.it
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Abstract

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We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana).

Keywords

Primary 28Axx28A7549Q15Secondary 26A45

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 631 - 635
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

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