Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rρ(z) := k (z −λk) ρ +. Here {λk}∞k =1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λk := 1 k k λ , and on the eigenvalue counting function. For example, we prove that for all domains and all k j 1+d/2 1+d/4 , λk λj 2 1+d/4 1+d/2 1+2/d k j 2/d .