Kuznecov formulas for domains with fractal boundary
Abstract
Let (M,g) be a compact d-dimensional Riemannian manifold without boundary, and let {ej}j=0∞ be an orthonormal basis of Laplace eigenfunctions with frequencies {λj}j=0∞. For a domain Ω⊂ M, consider NΩ(λ) = ∑λj≤λ |∫Ωej dVg|2. Suppose that ∂Ω is s-Ahlfors regular, with s∈[d−1,d), and that Ω satisfies a two-sided corkscrew condition. We prove that NΩ(λ) = vol(Ω)+O(λs−d), and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order λs−d holds precisely when the boundary admits a natural s-dimensional crossing volume 𝒱s(∂Ω), in which case NΩ(λ) = vol(Ω) − Cd,s 𝒱s(∂Ω) λs−d + o(λs−d). For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.
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BibTeX
@misc{hu2026kuznecov,
title = {{Kuznecov formulas for domains with fractal boundary}},
author = {Xucheng Hu and Yakun Xi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08487},
eprint = {2610.08487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08487},
}
RIS
TY - UNPB
TI - Kuznecov formulas for domains with fractal boundary
AU - Xucheng Hu
AU - Yakun Xi
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08487
UR - https://arxiv.org/abs/2610.08487
AB - Let (M,g) be a compact d-dimensional Riemannian manifold without boundary, and let {e_j}_j=0^∞ be an orthonormal basis of Laplace eigenfunctions with frequencies {λ_j}_j=0^∞. For a domain Ω⊂ M, consider N_Ω(λ) = ∑_λ_j≤λ |∫_Ωe_j dV_g|^2. Suppose that ∂Ω is s-Ahlfors regular, with s∈[d−1,d), and that Ω satisfies a two-sided corkscrew condition. We prove that N_Ω(λ) = vol(Ω)+O(λ^s−d), and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order λ^s−d holds precisely when the boundary admits a natural s-dimensional crossing volume 𝒱^s(∂Ω), in which case N_Ω(λ) = vol(Ω) − C_d,s 𝒱^s(∂Ω) λ^s−d + o(λ^s−d). For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.
ER -