Decay estimates for the wave and Dirac equations with a magnetic potential

We study the electromagnetic wave equation and the perturbed massless Dirac equation on ℝ t × ℝ 3 :$$u_{tt}-(\nabla+iA(x))^{2}u+B(x)u=0, \;\;\;\; iu_{t}-{\cal D}u+V(x)u=0,$$ where the potentials A ( x ), B ( x ), and V ( x ) are assumed to be small but may be rough. For both equations, we prove the expected time decay rate of the solution$$|u(t,x)|\leq {1 \over {t}}\|f\|_X$$ where the norm ‖ f ‖ X can be expressed as the weighted L 2 ‐norm of a few derivatives of the data f . © 2006 Wiley Periodicals, Inc.