Kuznecov formulae for fractal measures

Abstract

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$, consider the Kuznecov sum \[ N_\mu(\lambda):=\sum_{\lambda_j\le \lambda}\Bigl|\int_M e_j\,d\mu\Bigr|^2. \] Assume that $\mu$ admits an averaged $s$-density constant $A_\mu$ with correlation dimension $s\in(0,n)$. We prove that \[N_\mu(\lambda)= (2\pi)^{-(n-s)}\,{\rm vol}(B^{\,n-s})\,A_\mu\,\lambda^{n-s}+ o(\lambda^{n-s})\qquad (\lambda\to\infty). \] The averaged $s$-density condition is necessary for such a one-term asymptotic, and in general, the remainder $o(\lambda^{n-s})$ is sharp in the sense that it cannot be improved uniformly to a power-saving error term. This extends the classical Kuznecov formula of Zelditch for smooth submanifold measures to a broad class of singular and fractal measures.