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

TL;DR

This work establishes global anisotropic maximal $L^p$-regularity for the degenerate Ornstein–Uhlenbeck operator $\mathcal{A}=\Delta_y - y\cdot \nabla_x$ by analyzing the non-stationary problem $L=\partial_t-\mathcal{A}$ and deriving sharp pointwise bounds for its fundamental solution $\Gamma$. The authors prove that, for $p,q\in(1,\infty)$ and suitable $f$, there exists a weak solution $u$ to $\mathcal{A}u=f$ with $\Delta_y u$ and $|\partial_x|^{2/3}u$ belonging to the anisotropic space $L_y^qL_x^p$, with a bound $\|\Delta_y u\|_{L_y^qL_x^p}+\|\,|\partial_x|^{2/3}u\|_{L_y^qL_x^p} \le C\|f\|_{L_y^qL_x^p}$. The approach combines explicit kernel estimates, Hörmander-type conditions, and vector-valued Calderón–Zygmund theory, and uses a time-averaging technique to extend non-stationary results to the stationary problem. In addition to $L^p$-estimates, a weak $(1,1)$ bound for the non-stationary operator is established. The work advances anisotropic regularity theory for hypoelliptic degenerate PDEs and has potential applications to nonlinear kinetic-type problems and boundary-layer analyses in fluid mechanics, where anisotropic smoothing in different variables is essential.

Abstract

We consider the maximal regularity of a specific Vlasov-Fokker-Planck equation $\mathcal{A}u=f$ in the Euclidean space. The operator $\mathcal{A}=Δ_{y}u-y\cdot \nabla_x{u}$ 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. These results rely on the pointwise estimates of the fundamental solution of $L$.

Theorems & Definitions (27)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• proof
• Corollary 5