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arXiv · 2509.09354

Uniformly perfect measures on strictly convex planar graphs are $L^{2}$-flattening

Amir Algom·Tuomas Orponen
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Abstract

Uniformly perfect measures are a common generalisation of Ahlfors regular measures, self-conformal measures on the line, and their push-forwards under sufficiently regular maps. We show that every uniformly perfect measure $\sigma$ on a strictly convex planar $C^{2}$-graph is $L^{2}$-flattening. That is, for every $\epsilon>0$, there exists $p = p(\epsilon,\sigma) \geq 1$ such that $$\|\hat{\sigma}\|_{L^{p}(B(R))}^{p} \lesssim_{\epsilon,\sigma} R^{\epsilon}, \qquad R \geq 1.$$

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Amir Algom, Tuomas Orponen. 2025-09-11. Uniformly perfect measures on strictly convex planar graphs are $L^{2}$-flattening. https://arxiv.org/abs/2509.09354

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