Kuznecov formulas for domains with fractal boundary

Abstract

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 $\{\lambda_j\}_{j=0}^\infty$. For a domain $\Omega\subset M$, consider $$ N_\Omega(\lambda) = \sum_{\lambda_j\leq\lambda} \left|\int_\Omega e_j\,dV_g\right|^2. $$ Suppose that $\partial\Omega$ is $s$-Ahlfors regular, with $s\in{[d-1,d)}$, and that $\Omega$ satisfies a two-sided corkscrew condition. We prove that $$ N_\Omega(\lambda) = \operatorname{vol}(\Omega)+O\bigl(\lambda^{s-d}\bigr), $$ and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order $\lambda^{s-d}$ holds precisely when the boundary admits a natural $s$-dimensional crossing volume $\mathcal V^s(\partial\Omega)$, in which case $$ N_\Omega(\lambda) = \operatorname{vol}(\Omega) - C_{d,s}\, \mathcal V^s(\partial\Omega)\, \lambda^{s-d} + o\bigl(\lambda^{s-d}\bigr). $$ 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.