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 ℝ^k under sufficiently nonlinear real-analytic maps f : ℝ^k → ℝ^d. For example, we prove that if f is analytic on ℝ^k, its graph does not lie in an affine hyperplane in ℝ^k+d, and μ is not supported in an affine hyperplane in 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 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.

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BibTeX RIS

BibTeX

@misc{banaji2026fourier,
  title = {{Fourier transform of nonlinear images of self-similar measures: qualitative aspects}},
  author = {Amlan Banaji and Han Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.09743},
  eprint = {2606.09743},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.09743},
}

RIS

TY  - UNPB
TI  - Fourier transform of nonlinear images of self-similar measures: qualitative aspects
AU  - Amlan Banaji
AU  - Han Yu
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2606.09743
UR  - https://arxiv.org/abs/2606.09743
AB  - The goal of this paper is to establish polynomial Fourier decay for images of self-similar measures μ on ℝ^k under sufficiently nonlinear real-analytic maps f : ℝ^k → ℝ^d. For example, we prove that if f is analytic on ℝ^k, its graph does not lie in an affine hyperplane in ℝ^k+d, and μ is not supported in an affine hyperplane in 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 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.
ER  -

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