Kuznecov formulae for fractal measures

2025/12/20 by Yakun Xi, Xi, Yakun
Mathematics · #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations #Mathematical Approximation and Integration

paper · doi:10.48550/arxiv.2512.18379

Abstract

Let (M,g) be a compact, connected Riemannian manifold of dimension n≥ 2, and let \e_j\_j=0^∞ be an orthonormal basis of Laplace eigenfunctions -Δ_g e_j=λ_j² e_j. Given a finite Borel measure μ on M, consider the Kuznecov sum N_μ(λ):=∑_{λj≤ λ}|∫_M e_j dμ|². Assume that μ admits an averaged s-density constant A_μ with correlation dimension s∈(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) (λ→∞). 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.

Citations

• Kuznecov remainders and generic metrics
• Refined L^p restriction estimate for eigenfunctions on Riemannian surfaces
• Can you hear your location on a manifold?
• Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
• The spectral function of an elliptic operator

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