We investigate $L^1\to L^\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac{1}{2}}$ decay rate, which may be improved to $t^{-\frac{1}{2} - \gamma}$ for any $0\leq \gamma<\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being composed of a two dimensional space of p-wave resonances and a finite dimensional space of eigenfunctions at zero energy. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate except for a finite-rank piece. While in the case of a threshold eigenvalue only, the natural decay rate is preserved. In both cases we show that the decay rate may be improved at the cost of spatial weights.

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二维无质量狄拉克方程：零能量障碍物和色散估计

我们研究了具有势的无质量二维狄拉克方程的 $L^1\to L^\infty$ 色散估计。特别是，我们表明狄拉克演化满足自然的 $t^{-\frac{1}{2}}$ 衰减率，可以改进为 $t^{-\frac{1}{2} - \ gamma}$ 为任何 $0\leq \gamma<\frac{3}{2}$ 以空间权重为代价。我们将阈值障碍物的结构分类为由 p 波共振的二维空间和零能量下的本征函数的有限维空间组成。我们表明，在存在阈值共振的情况下，狄拉克演化满足自然衰减率，但有限秩块除外。而在仅阈值特征值的情况下，自然衰减率得以保留。在这两种情况下，我们都表明可以以空间权重为代价来提高衰减率。