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Sharp Variational estimates for Schrödinger equation in ℝ2

  • Yanping Chen
  • Yongrui Tang
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Abstract

For p≥2 and r≥ 1, we prove the sharp strong r-variation estimate for solutions to the free Schrödinger equation |Vr[0,1]{eitΔf}|Lp(ℝ2) ≲ |f|Hs(ℝ2). The estimate holds if s> s(p,r) and fails if s< s(p,r), where s(p,r)=max{ 2/r+1/p−1/2, 2/r−2/p, 1/p+1/r, 2/p−1/3, 1−2/p }. In particular, s(3,∞)=1/3. Thus our result yields a sharp variational extension of the two-dimensional Schrödinger maximal estimate of Du, Guth, and Li [{Ann. Math.}, 186, 607--640 (2017)]. The proof employs fractal restriction estimates to eliminate the regularity loss originating from the summation in the variation norm.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026sharp,
  title = {{Sharp Variational estimates for Schrödinger equation in $\mathbbR^2$}},
  author = {Yanping Chen and Yongrui Tang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31099},
  eprint = {2610.31099},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31099},
}
RIS
TY  - UNPB
TI  - Sharp Variational estimates for Schrödinger equation in ℝ^2
AU  - Yanping Chen
AU  - Yongrui Tang
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.31099
UR  - https://arxiv.org/abs/2610.31099
AB  - For p≥2 and r≥ 1, we prove the sharp strong r-variation estimate for solutions to the free Schrödinger equation |V^r_[0,1]{e^itΔf}|_L^p(ℝ^2) ≲ |f|_H^s(ℝ^2). The estimate holds if s> s(p,r) and fails if s< s(p,r), where s(p,r)=max{ 2/r+1/p−1/2, 2/r−2/p, 1/p+1/r, 2/p−1/3, 1−2/p }. In particular, s(3,∞)=1/3. Thus our result yields a sharp variational extension of the two-dimensional Schrödinger maximal estimate of Du, Guth, and Li [{Ann. Math.}, 186, 607--640 (2017)]. The proof employs fractal restriction estimates to eliminate the regularity loss originating from the summation in the variation norm.
ER  -