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Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations

  • Wontae Kim
  • Juha Kinnunen
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Abstract

We study parabolic Poincaré inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse Hölder inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincaré inequalities and self-improvement results on time intervals at fixed spatial points. These results are based on a pointwise maximal function estimate. As an application, we obtain regularity results in the time direction for solutions to nonlinear systems.

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Cite

BibTeXRIS
BibTeX
@misc{kim2026parabolic,
  title = {{Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations}},
  author = {Wontae Kim and Juha Kinnunen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.03830},
  eprint = {2609.03830},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03830},
}
RIS
TY  - UNPB
TI  - Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations
AU  - Wontae Kim
AU  - Juha Kinnunen
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.03830
UR  - https://arxiv.org/abs/2609.03830
AB  - We study parabolic Poincaré inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse Hölder inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincaré inequalities and self-improvement results on time intervals at fixed spatial points. These results are based on a pointwise maximal function estimate. As an application, we obtain regularity results in the time direction for solutions to nonlinear systems.
ER  -