The Dirac-Hardy and Dirac-Sobolev inequalities in $L^1$
  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. 
A. A. Balinsky, W. D. Evans and T. Umeda
math.SP math.FA