Kuznecov formulae for fractal measures

TL;DR

This paper extends the classical Kuznecov formula to fractal and singular measures on Riemannian manifolds, providing asymptotic behavior of spectral sums involving these measures and demonstrating the necessity of the averaged s-density condition.

Contribution

It introduces a generalized Kuznecov formula for fractal measures, broadening the scope from smooth submanifold measures to singular and fractal measures, with sharp remainder estimates.

Findings

Asymptotic formula for spectral sums with fractal measures

Necessity of averaged s-density condition for asymptotics

Sharpness of the remainder term in the asymptotic expansion

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.