A Weak Weyl's Law on compact metric measure spaces

TL;DR

This paper establishes a weak form of Weyl's law on certain compact metric measure spaces with a self-adjoint operator, linking eigenvalue counts to covering numbers under specific inequalities.

Contribution

It proves a weak Weyl's law on metric measure spaces using Poincaré inequalities and heat kernel estimates, extending classical results beyond Riemannian manifolds.

Findings

Eigenvalue count controlled by covering number under Poincaré inequality

Opposite inequality holds with Gaussian heat kernel estimates

Equivalence of eigenvalue count and covering number in certain Dirichlet spaces

Abstract

The well known Weyl's Law (Weyl's asymptotic formula) gives an approximation to the number $\mathcal{N}_{\omega}$ of eigenvalues (counted with multiplicities) on a large interval $[0,\>\omega]$ of the Laplace-Beltrami operator on a compact Riemannian manifold ${\bf M}$. In this paper we prove a kind of a weak version of the Weyl's law on certain compact metric measure spaces ${\bf X}$ which are equipped with a self-adjoint non-negative operator $\mathcal{L}$ acting in $L_{2}({\bf X})$. Roughly speaking, we show that if a certain Poincar\'e inequality holds then $\mathcal{N}_{\omega}$ is controlled by the cardinality of an appropriate cover $\mathcal{B}_{\omega^{-1/2}}=\{B(x_{j},\omega^{-1/2})\},\>\>\>x_{j}\in {\bf X},$ of ${\bf X}$ by balls of radius $\omega^{-1/2}$. Moreover, an opposite inequality holds if the heat kernel that corresponds to $\mathcal{L}$ satisfies short time Gaussian estimates. It is known that in the case of the so-called strongly local regular with a complete intrinsic metric Dirichlet spaces the Poincar\'e inequality holds iff the corresponding heat kernel satisfies short time Gaussian estimates. Thus for such spaces one obtains that $\mathcal{N}_{\omega}$ is essentially equivalent to the cardinality of a cover $\mathcal{B}_{\omega^{-1/2}}$.