PreprintCC BY 4.0arXiv
A higher-dimensional Schrödinger maximal estimate along families of bi-Lipschitz curves
Abstract
Let Θ⊂ℝn, n≥ 1 be compact with upper Minkowski dimension β, and let γ(x,t,θ) be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate |sup(τ,θ)∈[0,1)×Θ|eiτΔf(γ( · ,τ,θ))||L2(Bn1)≲s|f|Hs(ℝn), for every s>(n(β+1))/(2(n+1)), in the range 0≤β≤ n−1. In the range n−1<β≤ n we prove the threshold s>min{β/2 + 1/(2(n+1)),n/2}. The proof relies on a fractal L2 estimate of Du--Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.
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BibTeX
@misc{minguillon2026higherdimensional,
title = {{A higher-dimensional Schrödinger maximal estimate along families of bi-Lipschitz curves}},
author = {Javier Minguillón},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12565},
eprint = {2610.12565},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12565},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A higher-dimensional Schrödinger maximal estimate along families of bi-Lipschitz curves
AU - Javier Minguillón
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12565
UR - https://arxiv.org/abs/2610.12565
AB - Let Θ⊂ℝ^n, n≥ 1 be compact with upper Minkowski dimension β, and let γ(x,t,θ) be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate |sup_(τ,θ)∈[0,1)×Θ|e^iτΔf(γ( · ,τ,θ))||_L^2(B^n_1)≲_s|f|_H^s(ℝ^n), for every s>(n(β+1))/(2(n+1)), in the range 0≤β≤ n−1. In the range n−1<β≤ n we prove the threshold s>min{β/2 + 1/(2(n+1)),n/2}. The proof relies on a fractal L^2 estimate of Du--Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.
ER -