Kuznecov formulae for fractal measures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917208974688256 |
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| author | Xi, Yakun |
| author_facet | Xi, Yakun |
| contents | 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 $-Δ_g e_j=λ_j^2 e_j$. Given a finite Borel measure $μ$ on $M$, consider the Kuznecov sum \[
N_μ(λ):=\sum_{λ_j\le λ}\Bigl|\int_M e_j\,dμ\Bigr|^2. \] Assume that $μ$ admits an averaged $s$-density constant $A_μ$ with correlation dimension $s\in(0,n)$. We prove that \[N_μ(λ)= (2π)^{-(n-s)}\,{\rm vol}(B^{\,n-s})\,A_μ\,λ^{n-s}+ o(λ^{n-s})\qquad (λ\to\infty). \] The averaged $s$-density condition is necessary for such a one-term asymptotic, and in general, the remainder $o(λ^{n-s})$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18379 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kuznecov formulae for fractal measures Xi, Yakun Analysis of PDEs Classical Analysis and ODEs 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 $-Δ_g e_j=λ_j^2 e_j$. Given a finite Borel measure $μ$ on $M$, consider the Kuznecov sum \[ N_μ(λ):=\sum_{λ_j\le λ}\Bigl|\int_M e_j\,dμ\Bigr|^2. \] Assume that $μ$ admits an averaged $s$-density constant $A_μ$ with correlation dimension $s\in(0,n)$. We prove that \[N_μ(λ)= (2π)^{-(n-s)}\,{\rm vol}(B^{\,n-s})\,A_μ\,λ^{n-s}+ o(λ^{n-s})\qquad (λ\to\infty). \] The averaged $s$-density condition is necessary for such a one-term asymptotic, and in general, the remainder $o(λ^{n-s})$ 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. |
| title | Kuznecov formulae for fractal measures |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2512.18379 |