Solyanik Estimates for Ergodic Flows
Abstract
Let {Tt}t∈ℝ be a measure-preserving flow on a probability space (Ω,Σ,μ) and let T* be the associated one-sided ergodic maximal operator. We show that the sharp Tauberian constant C(α)=supμ(E)>0μ({T*χE>α})/μ(E),0<α<1, equals 1/α for every α∈(0,1), unless every measurable set is invariant modulo null sets, in which case C(α)=1 for every α. In particular the Solyanik estimate limα→1-C(α)=1 holds for every measure-preserving flow, and for ergodic flows on nonatomic spaces the estimate C(α)−1≤ 1/α−1 holds with equality. This answers the one-parameter case of a question of Hagelstein and Parissis. The main point is the lower bound, which is obtained from a lacunary cross-section built out of the upcrossings of a probe function along the orbits of the flow.
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BibTeX
@misc{huang2026solyanik,
title = {{Solyanik Estimates for Ergodic Flows}},
author = {Kesen Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41151},
eprint = {2610.41151},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41151},
}
RIS
TY - UNPB
TI - Solyanik Estimates for Ergodic Flows
AU - Kesen Huang
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41151
UR - https://arxiv.org/abs/2610.41151
AB - Let {T_t}_t∈ℝ be a measure-preserving flow on a probability space (Ω,Σ,μ) and let T^* be the associated one-sided ergodic maximal operator. We show that the sharp Tauberian constant C(α)=sup_μ(E)>0μ({T^*χ_E>α})/μ(E),0<α<1, equals 1/α for every α∈(0,1), unless every measurable set is invariant modulo null sets, in which case C(α)=1 for every α. In particular the Solyanik estimate lim_α→1^-C(α)=1 holds for every measure-preserving flow, and for ergodic flows on nonatomic spaces the estimate C(α)−1≤ 1/α−1 holds with equality. This answers the one-parameter case of a question of Hagelstein and Parissis. The main point is the lower bound, which is obtained from a lacunary cross-section built out of the upcrossings of a probe function along the orbits of the flow.
ER -