A weak Weyl’s law on compact metric measure 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 $\mathbf{M}$. In this paper we prove a kind of a weak version of the Weyl’s law on certain compact metric measure spaces $\mathbf{X}$ which are equipped with a self-adjoint non-negative operator ${\mathcal {L}}$ acting in $L_{2}(\mathbf{X})$. Roughly speaking, we show that if a certain Poincaré inequality holds then ${\mathcal {N}}_{\omega }$ is controlled by the cardinality of an appropriate cover ${\mathcal {B}}_{\omega ^{-1/2}}=\{B(x_{j},\omega ^{-1/2})\},\quad x_{j}\in \mathbf{X},$ of $\mathbf{X}$ 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 Poincaré 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 ${\mathcal {B}}_{\omega ^{-1/2}}$.