Dirac-Sobolev inequalities and estimates for the zero modes of massless Dirac operators

TL;DR

This paper investigates the decay properties of zero modes of massless Dirac operators with matrix potentials, establishing new Dirac-Sobolev inequalities and embedding theorems for spinor functions.

Contribution

It introduces novel Dirac-Sobolev inequalities and embedding theorems that analyze zero modes of massless Dirac operators with decay conditions.

Findings

Decay estimates for zero modes of Dirac operators

Embedding theorems for Dirac-Sobolev spaces

Inversion techniques with respect to the unit sphere

Abstract

The paper analyses the decay of any zero modes that might exist for a massless Dirac operator $H:= \ba \cdot (1/i) \bgrad + Q, $ where $Q$ is $4 \times 4$-matrix-valued and of order $O(|\x|^{-1})$ at infinity. The approach is based on inversion with respect to the unit sphere in $\R^3$ and establishing embedding theorems for Dirac-Sobolev spaces of spinors $f$ which are such that $f$ and $Hf$ lie in $(L^p(\R^3))^4, 1\le p<\infty.$