A.A. Balinsky, W.D. Evans and T. Umeda The Dirac-Hardy and Dirac-Sobolev inequalities in $L^1$ (31K, LaTeX2e) ABSTRACT. Dirac-Sobolev and Dirac-Hardy inequalities in $L^1$ are established in which the $L^p$ spaces which feature in the classical Sobolev and Hardy inequalities are replaced by weak $L^p$ spaces. Counter examples to the analogues of the classical inequalities are shown to be provided by zero modes for appropriate Pauli operators constructed by Loss and Yau.