Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schrödinger Equation
Abstract
We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, |e−itΔ/2f|LtqLxr(ℝ1+d)≤ Cq,rlVert frVertL2(ℝd), with 2/q+d/r=d/2, q,r≥ 2, and thus r≤ 2d/(d−2) if d≥ 3. We show that, in low dimensions 1≤ d≤ 5, the thresholds ρ1=10, ρ2=6, ρ3=4√7−6≈ 4.583, ρ4=2√(15)−4≈ 3.746, and ρ5=10/3≈ 3.333 are such that gaussians are stable local maximizers for 2<r<ρd, and fail to be local maximizers for ρd<r≤ 2d/(d−2) (with the conventions there is no upper bound on r when d∈{1,2} and that r=∞ is excluded when d=2). In the cases (q,r,d)∈{(6,6,1),(8,4,1),(4,4,2)}, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.
Related
- Dispersive and Strichartz Estimates for the Schrödinger Equation Inside Cylindrical Domains
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Strichartz estimates to fractional Schrödinger equations
- Sharp critical Strichartz estimates for a resonant hyperbolic Schrödinger flow on the 3D waveguide
- Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces
BibTeX
@misc{goncalves2026gaussians,
title = {{Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schrödinger Equation}},
author = {Felipe Gonçalves and Giuseppe Negro and Diogo Oliveira e Silva},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11644},
eprint = {2609.11644},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11644},
}
RIS
TY - UNPB
TI - Gaussians Do Not Always Maximize Mixed-Norm Strichartz Inequalities for the Schrödinger Equation
AU - Felipe Gonçalves
AU - Giuseppe Negro
AU - Diogo Oliveira e Silva
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11644
UR - https://arxiv.org/abs/2609.11644
AB - We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, |e^−itΔ/2f|_L_t^qL_x^r(ℝ^1+d)≤ C_q,rlVert frVert_L^2(ℝ^d), with 2/q+d/r=d/2, q,r≥ 2, and thus r≤ 2d/(d−2) if d≥ 3. We show that, in low dimensions 1≤ d≤ 5, the thresholds ρ_1=10, ρ_2=6, ρ_3=4√7−6≈ 4.583, ρ_4=2√(15)−4≈ 3.746, and ρ_5=10/3≈ 3.333 are such that gaussians are stable local maximizers for 2<r<ρ_d, and fail to be local maximizers for ρ_d<r≤ 2d/(d−2) (with the conventions there is no upper bound on r when d∈{1,2} and that r=∞ is excluded when d=2). In the cases (q,r,d)∈{(6,6,1),(8,4,1),(4,4,2)}, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.
ER -