Berezin--Li--Yau inequalities for Aharonov--Bohm operators

We prove the sharp Berezin--Li--Yau inequality for Riesz means of order at least one for Aharonov--Bohm operators on bounded planar domains with Dirichlet boundary conditions. This confirms a conjecture posed by Frank and Hansson in 2008. We also obtain an improved lower bound for individual eigenvalues and extend it to the case of magnetic potentials with bounded magnetic field. These results are then applied to deduce estimates for Riesz means of orders between zero and one, and for Aharonov--Bohm eigenvalues on product domains.