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

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 $p=p(\epsilon ,\sigma )\ge 1$ such that $$\mathrm{\parallel }\widehat{\sigma }{\mathrm{\parallel }}_{L^p(B(R))}^p{\lesssim }_{\epsilon ,\sigma }{R}^{\epsilon },R\ge 1.$$