Develops a quantitative flatness framework to obstruct Fourier restriction, L^p-improving, and Fourier decay estimates for measures, applied to bound Fourier dimensions of surfaces, curves, Patterson-Sullivan measures, and self-affine sets.
Fourier dimension of the graph of fractional Brownian motion with $H \ge 1/2$
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We prove that the Fourier dimension of the graph of fractional Brownian motion with Hurst index greater than $1/2$ is almost surely 1. This extends the result of Fraser and Sahlsten (2018) for the Brownian motion and confirms part of the conjecture of Fraser, Orponen and Sahlsten (2014). We introduce a combinatorial integration by parts formula to compute the moments of the Fourier transform of the graph measure. The proof of our main result is based on this integration by parts formula together with Fa\`a di Bruno's formula and strong local nondeterminism of fractional Brownian motion. We also show that the graph of a symmetric $\alpha$-stable process has Fourier dimension 1 almost surely when $\alpha \in [1,2]$ and is a Salem set when $\alpha = 1$.
fields
math.CA 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
A new restriction theorem is established that uses L^q-dimensions to obtain a continuum of Fourier restriction estimates recovering Stein-Tomas at q=∞ via complex interpolation, with improvements shown for certain multifractal measures.
citing papers explorer
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Quantitative flatness and obstructions in Fourier analysis
Develops a quantitative flatness framework to obstruct Fourier restriction, L^p-improving, and Fourier decay estimates for measures, applied to bound Fourier dimensions of surfaces, curves, Patterson-Sullivan measures, and self-affine sets.
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Fourier restriction estimates based on $L^q$-dimensions: beyond Stein--Tomas
A new restriction theorem is established that uses L^q-dimensions to obtain a continuum of Fourier restriction estimates recovering Stein-Tomas at q=∞ via complex interpolation, with improvements shown for certain multifractal measures.