Dirac--Sobolev Spaces and Sobolev Spaces

TL;DR

This paper investigates the relationship between Dirac-Sobolev spaces and classical Sobolev spaces in three dimensions, revealing their equivalence for p in (1,∞) and proper inclusion at p=1.

Contribution

It clarifies the connection between Dirac-Sobolev and Sobolev spaces, showing their equivalence for p in (1,∞) and the proper subspace relation at p=1.

Findings

Dirac-Sobolev spaces coincide with Sobolev spaces for 1<p<∞.

At p=1, Sobolev spaces are proper subspaces of Dirac-Sobolev spaces.

Abstract

The aim of this work is to study the first order Dirac-Sobolev spaces in $L^p$ norm on an open subset of ${\mathbb R}^3$ to clarify its relationship with the corresponding Sobolev spaces. It is shown that for $1< p <\infty$, they coincide, while for $p=1$, the latter spaces are proper subspaces of the former.