Fourier transform of nonlinear images of self-similar measures: qualitative aspects

Abstract

The goal of this paper is to establish polynomial Fourier decay for images of self-similar measures $μ$ on $\mathbb{R}^k$ under sufficiently nonlinear real-analytic maps $f \colon \mathbb{R}^k \to \mathbb{R}^d$. For example, we prove that if $f$ is analytic on $\mathbb{R}^k$, its graph does not lie in an affine hyperplane in $\mathbb{R}^{k+d}$, and $μ$ is not supported in an affine hyperplane in $\mathbb{R}^k$, then the image measure has polynomial Fourier decay. Key steps in the proof include establishing a uniform Lojasiewicz-type inequality for self-similar measures, and using the decay of the Fourier transform of $μ$ outside a very small exceptional set of frequencies. As an application of our results, we prove polynomial Fourier decay for self-conformal measures on $\mathbb{C}$ for a large class of complex analytic IFSs which are not self-similar but are conjugate to a linear IFS via an analytic map.