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Quantitative absolute continuity of planar measures with two independent Alberti representations
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-03-11 , DOI: 10.1007/s00526-020-1714-x
David Bate , Tuomas Orponen

We study measures \(\mu \) 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 \(\mu \) are bounded from above, in a natural way to be defined in the introduction, we prove that \(\mu \in L^{2}\). If the representations are also bounded from below, we show that \(\mu \) satisfies a reverse Hölder inequality with exponent 2, and is consequently in \(L^{2 + \epsilon }\) by Gehring’s lemma. A substantial part of the paper is also devoted to showing that both results stated above are optimal.



中文翻译:

具有两个独立的Alberti表示的平面测度的定量绝对连续性

我们用两个独立的Alberti表示研究平面上的度量\(\ mu \)。众所周知,由于阿尔贝蒂(Alberti),切尔涅伊(Csörnyei)和普雷斯(Preiss),此类措施相对于勒贝格措施绝对是连续的。本文的目的是量化A–C–P的结果。假设\(\ mu \)的表示形式是从上方定界的,以引言中定义的自然方式,我们证明\(\ mu \ in L ^ {2} \)。如果这些表示也从下面定界,则表明\(\ mu \)满足指数为2的逆Hölder不等式,因此位于\(L ^ {2 + \ epsilon} \)由盖林的引理。本文的大部分内容还致力于表明上述两个结果都是最佳的。

更新日期:2020-04-20
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