Zero modes and Dirac-(logarithmic) Sobolev-type inequalities

TL;DR

This paper investigates decay rates of zero modes for the Dirac operator with matrix-valued potentials, establishing new Sobolev, Nash, and Poincaré inequalities with explicit constants, extending classical results to less regular settings.

Contribution

It introduces novel $L^p$-$L^2$ and $L^p$-$L^q$ Dirac-Sobolev inequalities, including their logarithmic and Gaussian versions, for Dirac operators with minimal regularity assumptions.

Findings

Established explicit constants for Dirac Sobolev inequalities.

Extended Sobolev inequalities to Gaussian and logarithmic versions.

Proved Nash and Poincaré inequalities in the Dirac operator context.

Abstract

We study the decay rate of the zero modes of the Dirac operator with a matrix-valued potential that is considered here without any regularity assumptions, compared to the existing literature. For the Dirac operator and for Clifford-valued functions we prove the $L^p$-$L^2$ Dirac Sobolev inequality with explicit constant, as well as the $L^p$-$L^q$ Dirac-Sobolev inequalities. We prove its logarithmic counterpart for $q=2$, extending it to its Gaussian version of Gross, as well as show Nash and Poincar\'e inequalities in this setting, with explicit values for constants.