The Dirac-Hardy and Dirac-Sobolev inequalities in $L^1$

TL;DR

This paper establishes Dirac-Sobolev and Dirac-Hardy inequalities in the weak L^p spaces for L^1 functions, and provides counterexamples using zero modes of Pauli operators to show the failure of classical analogues.

Contribution

It introduces new Dirac inequalities in weak L^p spaces and demonstrates the limitations of classical inequalities through explicit counterexamples.

Findings

Dirac-Sobolev and Dirac-Hardy inequalities are valid in weak L^p spaces.

Counterexamples are constructed using zero modes of Pauli operators.

Classical inequalities do not hold in the L^1 setting without modifications.

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.