  In the present paper, Kato's distributional inequality with magnetic field is
generalized to vector-valued functions and operator-valued vector potentials.
This result is then used in non-relativistic quantum electrodynamics (QED) to
show that eigenstates of the Pauli-Fierz Hamiltonian satisfy a subsolution
estimate, and hence that any $L^2$-exponential bound in terms of a Lipschitz
function implies the corresponding pointwise exponential bound. Similar
pointwise bounds are also established for the one-particle density of states
that are not eigenstates.
