Year: 2026
Type: preprint
Abstract: We prove an $L\log L$-endpoint Cwikel estimate on quantum tori $\mathbb T_θ^d$, i.e., for every $d\geq 2$ and $x\in L\log L(\mathbb T_θ^d)$, \begin{equation*} \big\|(1-Δ)^{-\frac{d}{4}}λ(x)(1-Δ)^{-\frac{d}{4}}\big\|_{\mathcal{L}_{1,\infty}} \leq C_d \Norm{x}_{L\log L(\mathbb T_θ^d)}, \end{equation*} 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 $W_2^{d/2}\hookrightarrow\exp(L_2)$ on quantum tori. When $θ=0$, our method also gives a simpler proof of the $L\log L$ Cwikel estimate on classical tori. As an application, we use the $C^*$-algebraic principal symbol map to prove Weyl's laws for sandwich operators \[ S(1-Δ)^{-d/4}λ(x)(1-Δ)^{-d/4}T, \] where $x\in L\log L(\mathbb T_θ^d)$ and $S,T\inΠ$, the $C^*$-algebra generated by $λ(C(\mathbb T_θ^d))$ and Fourier multipliers with continuous homogeneous symbols of degree zero. We also prove a semiclassical Weyl's law for critical Schrödinger operators with $L\log L$ potentials.
Source: arXiv (Cornell University)
Authors: Liu, Siyu, Zhang, Xinyu
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222226143