Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator

Kazuhiro Hirao

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arXiv · 2602.17378

Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator

Kazuhiro Hirao

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Abstract

We consider the maximal regularity of a specific Vlasov-Fokker-Planck equation $\mathcal{A}u=f$ in the Euclidean space. The operator $\mathcal{A}=\Delta_{y}u-y\cdot \nabla_x{u}$ is an example of the Ornstein-Uhlenbeck operators. We prove the existence of a solution that satisfies the anisotropic maximal regularity estimates. To prove this we also show a similar estimates and a weak (1, 1) estimate for $L=\partial_t-\mathcal{A}$, which is of independent interest. Moreover, we show a maximal regularity estimate containing a fractional transport operator. These results rely on the pointwise estimates of the fundamental solution of $L$.

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Kazuhiro Hirao. 2026-02-19. Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator. https://arxiv.org/abs/2602.17378

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