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<p>We analyze the sharpness of the Sobolev order for smooth vector fields on compact Riemannian manifolds. Utilizing techniques from pseudo-differential operator theory and microlocal analysis, we investigate the asymptotic behavior of eigenvalues associated with these vector fields. As an application, we demonstrate the ill-posedness of a class of Cauchy problems involving left-invariant vector fields on compact Lie groups.</p>

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On the Sobolev boundedness of vector fields on compact Riemannian manifolds

  • 期刊论文

  • 发表日期：

    2025年6月27日

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  • Duván Cardona,
  • Alexandre Kirilov,
  • André Pedroso Kowacs,
  • Wagner A. A. de Moraes

摘要

We analyze the sharpness of the Sobolev order for smooth vector fields on compact Riemannian manifolds. Utilizing techniques from pseudo-differential operator theory and microlocal analysis, we investigate the asymptotic behavior of eigenvalues associated with these vector fields. As an application, we demonstrate the ill-posedness of a class of Cauchy problems involving left-invariant vector fields on compact Lie groups.