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

Marianna Chatzakou, Uwe Kahler, Michael Ruzhansky

TL;DR

The paper addresses the decay of zero modes for the Dirac operator with a matrix-valued potential in Clifford-valued function spaces, without imposing regularity on the potential. It develops explicit-constant Dirac-Sobolev and Dirac-Sobolev-L^p/L^q inequalities, along with logarithmic-Sobolev, Nash, and Gaussian versions, to build a robust analytic framework. These tools enable sharp decay estimates for zero modes, improving and generalizing prior results (notably BES08) by removing regularity requirements on the potential and delivering stronger integrability results. The findings enrich the functional-analytic toolkit for Dirac-type operators and have potential implications for stability analyses in mathematical physics and related PDE contexts, with connections to Gaussian-measure inequalities and Clifford-algebraic operator theory.

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 Poincaré inequalities in this setting, with explicit values for constants.

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

TL;DR

The paper addresses the decay of zero modes for the Dirac operator with a matrix-valued potential in Clifford-valued function spaces, without imposing regularity on the potential. It develops explicit-constant Dirac-Sobolev and Dirac-Sobolev-L^p/L^q inequalities, along with logarithmic-Sobolev, Nash, and Gaussian versions, to build a robust analytic framework. These tools enable sharp decay estimates for zero modes, improving and generalizing prior results (notably BES08) by removing regularity requirements on the potential and delivering stronger integrability results. The findings enrich the functional-analytic toolkit for Dirac-type operators and have potential implications for stability analyses in mathematical physics and related PDE contexts, with connections to Gaussian-measure inequalities and Clifford-algebraic operator theory.

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 - Dirac Sobolev inequality with explicit constant, as well as the - Dirac-Sobolev inequalities. We prove its logarithmic counterpart for , extending it to its Gaussian version of Gross, as well as show Nash and Poincaré inequalities in this setting, with explicit values for constants.
Paper Structure (7 sections, 16 theorems, 120 equations)

This paper contains 7 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Clifford algebras
  4. Clifford-Hilbert modules
  5. Sobolev inequalities
  6. Logarithmic Sobolev and Poincaré inequalities
  7. Zero modes

Key Result

Theorem 3.1

Let $1<p,q<\infty$, $1/p+1/q+\lambda/n=2$ and $0<\lambda<n$. Let $m\geq n\geq 1$. Then for any $f\in L^p(\mathbb{R}^n,\mathbb{R}_{0,m}) ,g\in L^q(\mathbb{R}^n, \mathbb{R}_{0,m})$ there exists a constant $C_{p,\lambda,n,m}>0$ such that The same result holds for $f\in L^p(\mathbb{R}^n,\mathbb{C}_{m})$ and $g\in L^q(\mathbb{R}^n, \mathbb{C}_{m})$.

Theorems & Definitions (30)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • Remark 3.7
  • proof : Proof of Theorem \ref{['THM:Sob.p']}
  • ...and 20 more