The zero modes and zero resonances of massless Dirac operators
  The zero modes and zero resonances of the Dirac operator $H=\alpha\cdot D + Q(x)$ are discussed, where   $\alpha= (\alpha_1, \alpha_2, \alpha_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with   $| q_{jk}(x) | \le C < x >^{-\rho} $, $\rho >1$.   We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$ and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $\rho > 3/2$. 
Yoshimi Saito (University of Alabama at Birmingham), Tomio Umeda   (University of Hyogo)