A Weak Weyl's Law on compact metric measure spaces

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