Curved Ingham Inequalities for Dispersive Trigonometric Sums
Abstract
We study L2 restrictions of dispersive trigonometric sums F(t,x)=∑n∈Z cn e2π i(|n|s t+nx) to curves in the (t,x)-plane. Our main result is a curved analogue of Ingham's inequality: for a class of trajectories satisfying quantitative curvature assumptions, we obtain upper and lower L2 estimates for the restriction of F in terms of the ℓ2-norm of its coefficients. %Unlike the affine case, the lower estimate requires sufficiently large time. %We also show that its normalized lower-bound constant cannot remain uniformly positive as the interval shrinks to zero. We then investigate two situations in which this obstruction disappears. First, for measures with polynomial Fourier decay, sufficiently high frequencies form a Riesz sequence; in particular, along smooth curves with non-vanishing curvature this yields strong Ingham inequalities for sufficiently high frequencies when s>2. Second, for every fixed time interval, the full system satisfies such a lower estimate when the dispersion exponent s is sufficiently large. We also study finite dispersive trigonometric sums and prove rigidity results for their zero sets along analytic and meromorphic curves. Finally, we apply these harmonic-analysis results to fractional Schrödinger equations on the torus, including equations with bounded potentials.
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BibTeX
@misc{haak2026curved,
title = {{Curved Ingham Inequalities for Dispersive Trigonometric Sums}},
author = {Bernhard H Haak and Philippe Jaming and Ming Wang and Yunlei Wang},
howpublished = {HAL},
year = {2026},
month = sep,
eprint = {2603.15193},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05549390v2},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Curved Ingham Inequalities for Dispersive Trigonometric Sums
AU - Bernhard H Haak
AU - Philippe Jaming
AU - Ming Wang
AU - Yunlei Wang
PY - 2026
DA - 2026/09/28
PB - HAL
UR - https://hal.science/hal-05549390v2
AB - We study L^2 restrictions of dispersive trigonometric sums F(t,x)=∑_n∈Z c_n e^2π i(|n|^s t+nx) to curves in the (t,x)-plane. Our main result is a curved analogue of Ingham's inequality: for a class of trajectories satisfying quantitative curvature assumptions, we obtain upper and lower L^2 estimates for the restriction of F in terms of the ℓ^2-norm of its coefficients. %Unlike the affine case, the lower estimate requires sufficiently large time. %We also show that its normalized lower-bound constant cannot remain uniformly positive as the interval shrinks to zero. We then investigate two situations in which this obstruction disappears. First, for measures with polynomial Fourier decay, sufficiently high frequencies form a Riesz sequence; in particular, along smooth curves with non-vanishing curvature this yields strong Ingham inequalities for sufficiently high frequencies when s>2. Second, for every fixed time interval, the full system satisfies such a lower estimate when the dispersion exponent s is sufficiently large. We also study finite dispersive trigonometric sums and prove rigidity results for their zero sets along analytic and meromorphic curves. Finally, we apply these harmonic-analysis results to fractional Schrödinger equations on the torus, including equations with bounded potentials.
ER -