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.