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 $R_t\times R_3$: $$ u_{tt} − (∇ +i A(x))^2u + B(x)u = 0, iu_t − Du + V(x)u = 0, $$ 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 $|u(t, x)| ≤\frac{1}{t}\|f\|_X$, where the norm $\|f\|_X$ can be expressed as the weighted $L^2$-norm of a few derivatives of the data $f$ .