Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only on the dimension and the product $\sqrt\Lambda |\Omega|^{1/d}$, where $\Lambda$ is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when $|\Omega|^{1/d}$ is replaced by a generalized inradius of $\Omega$. Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.