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The Berezin inequality gives an upper bound on the Riesz means of the magnetic Schrödinger operator on a set of finite volume. We find an analogous inequality for the magnetic operator with homogeneous magnetic field on sets whose complement in $$\mathbb{R }^2$$ has finite measure. Similar bounds are obtained for the Heisenberg sub-Laplacian.

The Berezin inequality on domains of infinite measure