  The paper analyses the decay of any zero modes that might exist for a
massless Dirac operator $H:= \ba \cdot (1/i) \bgrad + Q, $ where $Q$ is $4
\times 4$-matrix-valued and of order $O(|\x|^{-1})$ at infinity. The approach
is based on inversion with respect to the unit sphere in $\R^3$ and
establishing embedding theorems for Dirac-Sobolev spaces of spinors $f$ which
are such that $f$ and $Hf$ lie in $(L^p(\R^3))^4, 1\le p<\infty.$
