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 $p = p(\epsilon,\sigma) \geq 1$ such that $$\|\hat{\sigma}\|_{L^{p}(B(R))}^{p} \lesssim_{\epsilon,\sigma} R^{\epsilon}, \qquad R \geq 1.$$