Local Wegner and Lifshitz tails estimates for the density of states for continuous random Schr\"odinger operators

We introduce and prove local Wegner estimates for continuous generalized Anderson Hamiltonians, where the single-site random variables are independent but not necessarily identically distributed. In particular, we get Wegner estimates with a constant that goes to zero as we approach the bottom of the spectrum. As an application, we show that the (differentiated) density of states exhibits the same Lifshitz tails upper bound as the integrated density of states.