Kuznecov formulas for domains with fractal boundary

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j{\}}_{j=0}^{\mathrm{\infty }}$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{{\lambda }_j{\}}_{j=0}^{\mathrm{\infty }}$. For a domain $\mathrm{\Omega }\subset M$, consider $$N_{\mathrm{\Omega }}(\lambda )=\sum _{{\lambda }_j\le \lambda }{\mid {\int }_{\mathrm{\Omega }}{e}_{j}d{V}_g\mid }^2\mathrm{.}$$ Suppose that $\mathrm{\partial }\mathrm{\Omega }$ is $s$-Ahlfors regular, with $s\in [d-1,d)$, and that $\mathrm{\Omega }$ satisfies a two-sided corkscrew condition. We prove that $$N_{\mathrm{\Omega }}(\lambda )=\mathrm{vol}(\mathrm{\Omega })+O\left({\lambda }^{s-d}\right),$$ 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 ${\mathcal{V}}^s(\mathrm{\partial }\mathrm{\Omega })$, in which case $$N_{\mathrm{\Omega }}(\lambda )=\mathrm{vol}(\mathrm{\Omega })-C_{d,s}{\mathcal{V}}^s(\mathrm{\partial }\mathrm{\Omega }){\lambda }^{s-d}+o\left({\lambda }^{s-d}\right)\mathrm{.}$$ 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.