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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> Dirac Sobolev inequality with explicit constant, as well as the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^q\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>q</mi> </msup> </math></EquationSource> </InlineEquation> Dirac-Sobolev inequalities. We prove its logarithmic counterpart for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(q=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, extending it to its Gaussian version of Gross, as well as show Nash and Poincaré inequalities in this setting, with explicit values for constants.</p>

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

  • 期刊论文

  • 发表日期：

    2026年6月27日

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  • Marianna Chatzakou,
  • Uwe Kähler,
  • Michael Ruzhansky

摘要

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 p - \(L^2\) L 2 Dirac Sobolev inequality with explicit constant, as well as the \(L^p\) L p - \(L^q\) L q Dirac-Sobolev inequalities. We prove its logarithmic counterpart for \(q=2\) 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.