Polynomial Fourier decay for fractal measures and their pushforwards

Abstract: We prove that the pushforwards of a very general class of fractal measures $\mu$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,\eta>0$ such that $|\widehat{F\mu}(\xi)|\leq C|\xi|^{-\eta}$ for all $\xi\neq 0$. Using this, we prove that if $\Phi = { \varphi_a \colon [0,1] \to [0,1] }_{a \in \mathcal{A}}$ is an iterated function system consisting of analytic contractions, and there exists $a \in \mathcal{A}$ such that $\varphi_a$ is not an affine map, then every non-atomic self-conformal measure for $\Phi$ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets.