Kuznecov formulas for domains with fractal boundary
Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{λ_j\}_{j=0}^\infty$. For a domain $Ω\subset M$, consider $$ N_Ω(λ) = \sum_{λ_j\leqλ} \left|\int_Ωe_j\,dV_g\right|^2. $$ Suppose that $\partialΩ$ is $s$-Ahlfors regular, with $s\in{[d-1,d)}$, and that $Ω$ satisfies a two-sided corkscrew condition. We prove that $$ N_Ω(λ) = \operatorname{vol}(Ω)+O\bigl(λ^{s-d}\bigr), $$ 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 $\mathcal V^s(\partialΩ)$, in which case $$ N_Ω(λ) = \operatorname{vol}(Ω) - C_{d,s}\, \mathcal V^s(\partialΩ)\, λ^{s-d} + o\bigl(λ^{s-d}\bigr). $$ 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.