Cwikel estimates and Weyl's laws for Llog L potentials on quantum tori
Abstract
We prove an Llog L-endpoint Cwikel estimate on quantum tori 𝕋θd, i.e., for every d≥ 2 and x∈ Llog L(𝕋θd), |(1−Δ)−d/4λ(x)(1−Δ)−d/4|ℒ1,∞ ≤ Cd NormxLlog L(𝕋θd), where λ(x) is the left multiplication by x. Our argument actually gives an abstract weak Hilbert--Schmidt estimate for multiplication operators in finite von Neumann algebras whose L2-spaces admit orthonormal bases of unitaries. As a consequence, it yields a critical Sobolev embedding W2d/2↪exp(L2) on quantum tori. When θ=0, our method also gives a simpler proof of the Llog L Cwikel estimate on classical tori. As an application, we use the C*-algebraic principal symbol map to prove Weyl's laws for sandwich operators S(1−Δ)−d/4λ(x)(1−Δ)−d/4T, where x∈ Llog L(𝕋θd) and S,T∈Π, the C*-algebra generated by λ(C(𝕋θd)) and Fourier multipliers with continuous homogeneous symbols of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with Llog L potentials.
Related
- Pólya's Conjecture Up to ε‐loss and Quantitative Estimates for the Remainder of Weyl's Law
- Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, II
- Nonclassical Weyl laws and Connes' Integration for weak Lorentz ideals, I
- Symplectic capacities and Weyl laws from Abelian Higgs min max theory
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
BibTeX
@misc{liu2026cwikel,
title = {{Cwikel estimates and Weyl's laws for $L\log L$ potentials on quantum tori}},
author = {Siyu Liu and Xinyu Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20373},
eprint = {2610.20373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20373},
}
RIS
TY - UNPB
TI - Cwikel estimates and Weyl's laws for Llog L potentials on quantum tori
AU - Siyu Liu
AU - Xinyu Zhang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.20373
UR - https://arxiv.org/abs/2610.20373
AB - We prove an Llog L-endpoint Cwikel estimate on quantum tori 𝕋_θ^d, i.e., for every d≥ 2 and x∈ Llog L(𝕋_θ^d), |(1−Δ)^−d/4λ(x)(1−Δ)^−d/4|_ℒ_1,∞ ≤ C_d Normx_Llog L(𝕋_θ^d), where λ(x) is the left multiplication by x. Our argument actually gives an abstract weak Hilbert--Schmidt estimate for multiplication operators in finite von Neumann algebras whose L_2-spaces admit orthonormal bases of unitaries. As a consequence, it yields a critical Sobolev embedding W_2^d/2↪exp(L_2) on quantum tori. When θ=0, our method also gives a simpler proof of the Llog L Cwikel estimate on classical tori. As an application, we use the C^*-algebraic principal symbol map to prove Weyl's laws for sandwich operators S(1−Δ)^−d/4λ(x)(1−Δ)^−d/4T, where x∈ Llog L(𝕋_θ^d) and S,T∈Π, the C^*-algebra generated by λ(C(𝕋_θ^d)) and Fourier multipliers with continuous homogeneous symbols of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with Llog L potentials.
ER -