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Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators

  • Kesen Huang
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Abstract

Let U1,…,Un be commuting invertible measure-preserving transformations of a probability space, and let C*HL(α) denote the sharp Tauberian constant of the associated ergodic uncentered Hardy--Littlewood maximal operator. Hagelstein and Parissis showed that C*HL(α)−1≲n(1/α−1)1/(n(n+1)) as α→1- and asked whether this exponent is sharp. We prove that C*HL(α)−1≲n(1/α−1)1/n for all such transformations, and that this is sharp: equality of the ergodic and discrete Tauberian constants holds for every action admitting arbitrarily large Rokhlin towers (in particular for every free action, ergodic or not), and the discrete constant satisfies C̃HL(α)−1asympn(1/α−1)1/n. For n≥2 the sharp ergodic exponent 1/n is strictly smaller than the sharp Euclidean exponent 2/(n+1) recently obtained by Mukherjee; the discrepancy is caused by a lattice ``first layer'' effect. We also show that the discrete Tauberian constants of the Hardy--Littlewood and strong maximal operators dominate their Euclidean counterparts; for the strong maximal operator this yields C*S=CS for every free action.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026sharp,
  title = {{Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators}},
  author = {Kesen Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41156},
  eprint = {2610.41156},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41156},
}
RIS
TY  - UNPB
TI  - Sharp Solyanik Estimates for Discrete and Ergodic Hardy--Littlewood Maximal Operators
AU  - Kesen Huang
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41156
UR  - https://arxiv.org/abs/2610.41156
AB  - Let U_1,…,U_n be commuting invertible measure-preserving transformations of a probability space, and let C^*_HL(α) denote the sharp Tauberian constant of the associated ergodic uncentered Hardy--Littlewood maximal operator. Hagelstein and Parissis showed that C^*_HL(α)−1≲_n(1/α−1)^1/(n(n+1)) as α→1^- and asked whether this exponent is sharp. We prove that C^*_HL(α)−1≲_n(1/α−1)^1/n for all such transformations, and that this is sharp: equality of the ergodic and discrete Tauberian constants holds for every action admitting arbitrarily large Rokhlin towers (in particular for every free action, ergodic or not), and the discrete constant satisfies C̃_HL(α)−1asymp_n(1/α−1)^1/n. For n≥2 the sharp ergodic exponent 1/n is strictly smaller than the sharp Euclidean exponent 2/(n+1) recently obtained by Mukherjee; the discrepancy is caused by a lattice ``first layer'' effect. We also show that the discrete Tauberian constants of the Hardy--Littlewood and strong maximal operators dominate their Euclidean counterparts; for the strong maximal operator this yields C^*_S=C_S for every free action.
ER  -