Two-parameter variational estimates for averages over tori

@misc{lee2026twoparameter,
  title = {{Two-parameter variational estimates for averages over tori}},
  author = {Juyoung Lee and Sanghyuk Lee and Feng Zhang and Shuijiang Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07422},
  eprint = {2609.07422},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07422},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - Two-parameter variational estimates for averages over tori
AU  - Juyoung Lee
AU  - Sanghyuk Lee
AU  - Feng Zhang
AU  - Shuijiang Zhao
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07422
UR  - https://arxiv.org/abs/2609.07422
AB  - One-parameter variational inequalities are well developed, whereas their multi-parameter counterparts remain much less understood. We explore two-parameter variational inequalities for averages over tori in ℝ^3. To capture the underlying two-parameter structure, we introduce a local two-parameter r-variation norm that combines rectangular increments with variations along the boundary. The resulting variation operator pointwise dominates the corresponding two-parameter local maximal function and, unlike the maximal function, also captures oscillation across the two parameters. We establish sharp L^p--L^q bounds for this variation operator up to endpoints. For comparison, we also obtain sharp L^p bounds up to endpoints for the corresponding local one-parameter variation operator, revealing a genuine difference between the one- and two-parameter boundedness regions. The proof combines square function estimates for two-parameter propagators with local smoothing estimates through mixed-norm interpolation.
ER  -