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 $\mu$ on ${\mathbb{R}}^k$ under sufficiently nonlinear real-analytic maps $f:{\mathbb{R}}^k\to {\mathbb{R}}^d$. 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 $\mu$ 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 $\mu$ 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.

45 pages