Sharp critical Strichartz estimates for a resonant hyperbolic Schrödinger flow on the 3D waveguide
Abstract
We study critical Strichartz estimates for the resonant hyperbolic Schrödinger flow on the waveguide ℝ × 𝕋2. For initial data localized to frequencies at most N, we prove that the unit-time L2→ L10/3 operator norm has sharp order N1/5. We further track its dependence on the observation length, obtaining sharp estimates in the short- and long-time regimes and characterizing the frequency-dependent intervals on which the critical estimate remains lossless. The precise dependence on T in the intermediate regime N−1<T<1 remains open. The upper bounds are obtained by lifting the global endpoint estimate on the elliptic cylinder ℝ × 𝕋 to Hilbert-valued data, preserving its time summability, and combining this with periodic Bernstein and interpolation. The lower bounds exploit an exact integer null direction in the periodic frequency lattice together with a suitably scaled Euclidean wave packet. Finally, using a Bloch--Floquet transference argument, we recover the known elliptic unit-time estimate from compact torus theory and obtain a lossless elliptic estimate on frequency-dependent short time intervals.
Related
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Bilinear Strichartz estimates on the hyperbolic space and the cubic NLS
- Strichartz estimates to fractional Schrödinger equations
- Strichartz estimates for Klein-Gordon equations on asymptotically hyperbolic manifolds with applications
- Critical orthonormal Strichartz estimates for Lebesgue spaces
BibTeX
@misc{cen2026sharp,
title = {{Sharp critical Strichartz estimates for a resonant hyperbolic Schrödinger flow on the 3D waveguide}},
author = {Xi Cen and Zhezhi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30437},
eprint = {2609.30437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30437},
}
RIS
TY - UNPB
TI - Sharp critical Strichartz estimates for a resonant hyperbolic Schrödinger flow on the 3D waveguide
AU - Xi Cen
AU - Zhezhi Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30437
UR - https://arxiv.org/abs/2609.30437
AB - We study critical Strichartz estimates for the resonant hyperbolic Schrödinger flow on the waveguide ℝ × 𝕋^2. For initial data localized to frequencies at most N, we prove that the unit-time L^2→ L^10/3 operator norm has sharp order N^1/5. We further track its dependence on the observation length, obtaining sharp estimates in the short- and long-time regimes and characterizing the frequency-dependent intervals on which the critical estimate remains lossless. The precise dependence on T in the intermediate regime N^−1<T<1 remains open. The upper bounds are obtained by lifting the global endpoint estimate on the elliptic cylinder ℝ × 𝕋 to Hilbert-valued data, preserving its time summability, and combining this with periodic Bernstein and interpolation. The lower bounds exploit an exact integer null direction in the periodic frequency lattice together with a suitably scaled Euclidean wave packet. Finally, using a Bloch--Floquet transference argument, we recover the known elliptic unit-time estimate from compact torus theory and obtain a lossless elliptic estimate on frequency-dependent short time intervals.
ER -