Fourier decay for Weierstrass images of Frostman measures

Prove or disprove that, for every Frostman measure $\\nu$ on the circle and every $0<\\mu<1$, $\\lambda\\geq2$, the image measure $(W_{\\mu,\\lambda})_*\\nu$ has power Fourier decay.

Background

The paper establishes Fourier decay for images of Lebesgue measure under autosimilar Weierstrass maps. The proposed extension replaces the absolutely continuous source measure by an arbitrary Frostman measure, making the source potentially singular while retaining the rigid autosimilar structure of the map. A self-similar Cantor measure invariant under multiplication by λ\lambda is identified as a particularly natural test case.

References

It is natural to ask whether the same mechanism survives for singular source measures: if $\nu$ is a Frostman measure on $\mathbb T$, does the image $(W_{\mu,\lambda})_*\nu$ have power Fourier decay for every $0<\mu<1$ and $\lambda\geq2$?

— Arbitrarily Fast Quantum Dispersion in Long-Range Crystals  (2608.27326 - Leclerc et al., 27 Aug 2026) in Section 3, subsection “Prospects for the theory,” part (c) “A $C^\\alpha$ van der Corput principle for fractal measures”