Fourier Tauberian Theorems and Applications

The main aim of the paper is to present a general version of the Fourier Tauberian theorem for monotone functions. This result, together with Berezinʹs inequality, allows us to obtain a refined version of 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.