Parabolic Poincar\'e inequalities and maximal function estimates for systems of partial differential equations

Wontae Kim; Juha Kinnunen

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arXiv · 2609.03830

Parabolic Poincar\'e inequalities and maximal function estimates for systems of partial differential equations

Wontae Kim·Juha Kinnunen

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Abstract

We study parabolic Poincar\'e 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 H\"older inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincar\'e 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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Wontae Kim, Juha Kinnunen. 2026-09-03. Parabolic Poincar\'e inequalities and maximal function estimates for systems of partial differential equations. https://arxiv.org/abs/2609.03830

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