# Quantitative absolute continuity of planar measures with two independent Alberti representations

## TL;DR

This paper quantifies the absolute continuity of planar measures with two independent Alberti representations, showing they belong to certain L^p spaces and satisfy reverse Hölder inequalities, with optimality results.

## Contribution

It provides the first quantitative bounds on measures with two Alberti representations, establishing their membership in L^2 and L^{2+ε} spaces under boundedness conditions.

## Key findings

- Measures with two independent Alberti representations are in L^2.
- Under additional bounds, these measures satisfy reverse Hölder inequalities.
- The results are proven to be optimal.

## Abstract

We study measures $\mu$ on the plane with two independent Alberti representations. It is known, due to Alberti, Cs\"ornyei, 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 H\"older 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.