Orthonormal Strichartz estimates over fractal time
Abstract
We study Strichartz estimates for orthonormal families of initial data when time is integrated against an α-dimensional Frostman measure μ, 0<α≤1, with coefficients measured in ℓβ. For the fractional Schrödinger, wave and Klein--Gordon equations, we obtain orthogonality gains in the admissible region and necessary conditions on β. For the fractional Schrödinger equation the exponents are sharp up to an endpoint; the optimality of the threshold β=q/2 requires μ to be Ahlfors regular. For the Schrödinger equation in dimensions d≥2, we prove a Lorentz endpoint estimate and, if d>2α, the critical exponent β=q/2 at every pair 2≤ q,r<∞ with d/2(1/2−1/r)≥α/q, r>dq/(d−α); on r=∞ we obtain a restricted weak-type estimate. Since the constants depend on μ only through its Frostman constant, Lebesgue measure yields global-in-time critical estimates in this range when d≥3. The proofs combine Frostman fractional integration, a spectral near--far argument at the endpoint, and asymmetric Hilbert--Schmidt bounds with orthogonal frequency-block summation. We also obtain further estimates in the low-dimensional cases d=1 (fractional Schrödinger) and d=2 (wave and Klein--Gordon) with α>1/2, and formulate the consequences for densities of freely evolving states.
Related
- Critical orthonormal Strichartz estimates for Lebesgue spaces
- Critical Strichartz estimates for orthonormal systems
- Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces
- Orthonormal maximal estimates for fractional Schrödinger operators along Hölder curves
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
BibTeX
@misc{feng2026orthonormal,
title = {{Orthonormal Strichartz estimates over fractal time}},
author = {Guoxia Feng and Jinbong Lee and Sanghyuk Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18042},
eprint = {2610.18042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18042},
}
RIS
TY - UNPB
TI - Orthonormal Strichartz estimates over fractal time
AU - Guoxia Feng
AU - Jinbong Lee
AU - Sanghyuk Lee
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18042
UR - https://arxiv.org/abs/2610.18042
AB - We study Strichartz estimates for orthonormal families of initial data when time is integrated against an α-dimensional Frostman measure μ, 0<α≤1, with coefficients measured in ℓ^β. For the fractional Schrödinger, wave and Klein--Gordon equations, we obtain orthogonality gains in the admissible region and necessary conditions on β. For the fractional Schrödinger equation the exponents are sharp up to an endpoint; the optimality of the threshold β=q/2 requires μ to be Ahlfors regular. For the Schrödinger equation in dimensions d≥2, we prove a Lorentz endpoint estimate and, if d>2α, the critical exponent β=q/2 at every pair 2≤ q,r<∞ with d/2(1/2−1/r)≥α/q, r>dq/(d−α); on r=∞ we obtain a restricted weak-type estimate. Since the constants depend on μ only through its Frostman constant, Lebesgue measure yields global-in-time critical estimates in this range when d≥3. The proofs combine Frostman fractional integration, a spectral near--far argument at the endpoint, and asymmetric Hilbert--Schmidt bounds with orthogonal frequency-block summation. We also obtain further estimates in the low-dimensional cases d=1 (fractional Schrödinger) and d=2 (wave and Klein--Gordon) with α>1/2, and formulate the consequences for densities of freely evolving states.
ER -