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Spatial regularity for a class of degenerate Kolmogorov equations

Abstract

We establish spatial a priori estimates for the solution u to a class of dilation invariant Kolmogorov equation, where u is assumed to only have a certain amount of regularity in the diffusion’s directions, i.e. \(x_{1}, \ldots , x_{m_{0}}\). The result is that u is also regular with respect to the remaining directions. The approach we propose is based on the commutators identities and allows us to obtain a Sobolev exponent that does not depend on the integrability assumption of the right-hand side. Lastly, we provide a new proof for the optimal spatial regularity.

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Acknowledgements

The author is grateful to the anonymous refeerees for their valuable suggestions.

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Correspondence to Francesca Anceschi.

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Anceschi, F. Spatial regularity for a class of degenerate Kolmogorov equations. Ricerche mat 71, 271–281 (2022). https://doi.org/10.1007/s11587-022-00685-6

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  • DOI: https://doi.org/10.1007/s11587-022-00685-6

Keywords

  • Kolmogorov equation
  • Ultraparabolic
  • Sobolev spaces
  • Hypoelliptic
  • Regularity
  • A priori estimates

Mathematics Subject Classification

  • 35K70
  • 35B45
  • 35Q84