Dispersive estimates for Dirac Operators in dimension four with obstructions at threshold energies

We investigate $L^1\to L^\infty$ dispersive estimates for the Dirac equation with a potential in four spatial dimensions. We classify the structure of the obstructions at the thresholds as being composed of an at most two dimensional space of resonances per threshold, and finitely many eigenfunctions. Similar to the Schr\"odinger evolution, we prove the natural $t^{-2}$ decay rate when the thresholds are regular. When there is a threshold resonance or eigenvalue, we show that there is a time dependent, finite rank operator satisfying $\|F_t\|_{L^1\to L^\infty}\lesssim (\log t)^{-1}$ for $t>2$ such that $$\|e^{it\mathcal H}P(\mathcal H)-F_t\|_{L^1\to L^\infty}\lesssim t^{-1} \quad \text{for } t>2,$$ with $P$ a projection onto a subspace of the absolutely continuous spectrum in a small neighborhood of the thresholds. We further show that the operator $F_t=0$ if there is a threshold eigenvalue but no threshold resonance. We pair this with high energy bounds for the evolution and provide a complete description of the dispersive bounds.