Fourier Restriction Beyond Dimension Data: A Multiscale Moment Criterion
Abstract
We establish an exact Fourier restriction criterion for a class of weighted Moran measures. Assuming uniformly bounded one-level frame bounds, we prove that, for every s>2, the measure itself already detects the full extension estimate: L2(μ)→ Ls(ℝ) extension holds if and only if μ̂∈ Ls(ℝ), equivalently, if and only if the one-level Fourier moments Gs,n are summable, where Gs,n is the sth moment of the nonzero Fourier coefficients of the nth digit law. The sufficient direction is obtained from a finite-frame stability estimate and its multiscale iteration, while necessity follows from disjoint resonant frequency grids. This gives, in particular, an exact formula for the critical restriction exponent and a precise criterion for endpoint attainment. The criterion reveals that the optimal restriction range is not determined by standard Fourier or dimensional data. We construct singular continuous spectral Moran measures with the same scales, digit cardinalities, exponential spectrum, and maximal nonzero Fourier coefficients. They also satisfy the same global Fourier decay bound, which is sharp along the same sequence of resonant frequencies, yet their critical restriction exponents are different. More generally, after fixing the summability rate of the maximal Fourier coefficients, we determine the optimal interval of possible restriction exponents and show that, at every interior critical exponent, the same fixed data allow both attainment and failure of the critical endpoint. These measures may simultaneously have $ \dim_{\mathrm F}^θμ=θ$ and $ D_q(μ)=1,$for all admissible θ and q, and pointwise local dimension one everywhere on their supports.
Related
BibTeX
@misc{li2026fourier,
title = {{Fourier Restriction Beyond Dimension Data: A Multiscale Moment Criterion}},
author = {Jinjun Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11227},
eprint = {2610.11227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11227},
}
RIS
TY - UNPB
TI - Fourier Restriction Beyond Dimension Data: A Multiscale Moment Criterion
AU - Jinjun Li
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11227
UR - https://arxiv.org/abs/2610.11227
AB - We establish an exact Fourier restriction criterion for a class of weighted Moran measures. Assuming uniformly bounded one-level frame bounds, we prove that, for every s>2, the measure itself already detects the full extension estimate: L^2(μ)→ L^s(ℝ) extension holds if and only if μ̂∈ L^s(ℝ), equivalently, if and only if the one-level Fourier moments G_s,n are summable, where G_s,n is the sth moment of the nonzero Fourier coefficients of the nth digit law. The sufficient direction is obtained from a finite-frame stability estimate and its multiscale iteration, while necessity follows from disjoint resonant frequency grids. This gives, in particular, an exact formula for the critical restriction exponent and a precise criterion for endpoint attainment. The criterion reveals that the optimal restriction range is not determined by standard Fourier or dimensional data. We construct singular continuous spectral Moran measures with the same scales, digit cardinalities, exponential spectrum, and maximal nonzero Fourier coefficients. They also satisfy the same global Fourier decay bound, which is sharp along the same sequence of resonant frequencies, yet their critical restriction exponents are different. More generally, after fixing the summability rate of the maximal Fourier coefficients, we determine the optimal interval of possible restriction exponents and show that, at every interior critical exponent, the same fixed data allow both attainment and failure of the critical endpoint. These measures may simultaneously have $ \dim_{\mathrm F}^θμ=θ$ and $ D_q(μ)=1,$for all admissible θ and q, and pointwise local dimension one everywhere on their supports.
ER -