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 $$N_{\Omega }(\lambda )=\sum _{{\lambda }_j\le \lambda }{\left|{\int }_{\Omega }{e}_{j}d{V}_g\right|}^2.$$ 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 $$N_{\Omega }(\lambda )=\mathrm{vol}(\Omega )+O\left({\lambda }^{s-d}\right),$$ 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 ${\mathcal{V}}^s(\partial \Omega )$, in which case $$N_{\Omega }(\lambda )=\mathrm{vol}(\Omega )-C_{d,s}{\mathcal{V}}^s(\partial \Omega ){\lambda }^{s-d}+o\left({\lambda }^{s-d}\right).$$ 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.

arXiv comment30 pages