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 Poincare 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 Poincare 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 $${\mathcal {B}}_{\omega ^{-1/2}}$$ .