Quantum $L^p$ inequalities in thermal states

In thermal quantum field theory, the global Liouvillian (the generator of time translations) is passive. How is this reflected in the properties of its local density, a quantum field? We propose that the locally averaged density is bounded below, but not above, with respect to the noncommutative $L^4$ norm. This is analogous to the known quantum energy inequalities in the vacuum situation. Our examples include thermal equilibrium on Minkowski space, and the Unruh effect on the Rindler wedge, both for the real scalar free field.