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Quantitative absolute continuity of planar measures with two independent Alberti representations
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Bate, David and Orponen, Tuomas (2020) Quantitative absolute continuity of planar measures with two independent Alberti representations. Calculus of Variations and Partial Differential Equations, 59 . 72. doi:10.1007/s00526-020-1714-x ISSN 0944-2669.
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Official URL: https://doi.org/10.1007/s00526-020-1714-x
Abstract
We study measures μ on the plane with two independent Alberti representations. It is known, due to Alberti, Csörnyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A–C–P. Assuming that the representations of μ are bounded from above, in a natural way to be defined in the introduction, we prove that μ∈L2. If the representations are also bounded from below, we show that μ satisfies a reverse Hölder inequality with exponent 2, and is consequently in L2+ϵ by Gehring’s lemma. A substantial part of the paper is also devoted to showing that both results stated above are optimal.
Item Type: | Journal Article | |||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Journal or Publication Title: | Calculus of Variations and Partial Differential Equations | |||||||||
Publisher: | Springer | |||||||||
ISSN: | 0944-2669 | |||||||||
Official Date: | 11 March 2020 | |||||||||
Dates: |
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Volume: | 59 | |||||||||
Article Number: | 72 | |||||||||
DOI: | 10.1007/s00526-020-1714-x | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Access rights to Published version: | Open Access (Creative Commons) | |||||||||
Date of first compliant deposit: | 12 February 2020 | |||||||||
Date of first compliant Open Access: | 12 March 2020 | |||||||||
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