  The main aim of the paper is to present a general version of the Fourier
Tauberian theorem for monotone functions. This result, together with Berezin's
inequality, allows us to obtain a refined version the Li-Yau estimate for the
counting function of the Dirichlet Laplacian which implies the Li-Yau estimate
itself and, at the same time, the asymptotic results obtained by the
variational method.
