Abstract
We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(Łambda-łambda_k)_+^\sigma$ of the eigenvalues $łambda_k$ of the Dirichlet Laplacian in a domain if $\sigma3/2$. It contains a correction term of the order of the standard second term in the Weyl asymptotics. The result is based on an application of sharp Lieb-Thirring inequalities with operator valued potential to spectral estimates of the Dirichlet Laplacian in domains.
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