发表机构
Uppsala University（乌普萨拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究加权Kolmogorov方程，在一般A2权重下证明传递估计、局部有界性、Harnack不等式及强最小值原理，并涵盖分数幂扩展中的特定权重。
AI 中文摘要
设$m\geq1$，$k\geq0$，$n=m+k$，并记$x=(v,z)\in\mathbb R^m\times \mathbb R^k$，其中$v$是主动速度变量，$z$相对于$Y=v\cdot\nabla_y+\partial_t$是被动变量。我们考虑\\[ \operatorname{div}_x(A\nabla_xu)-Y(wu)=0，\\] 其中$w=w(x)\in A_2(\mathbb R^n)$，可测矩阵$A$的椭圆性和大小由$w$控制。对于任意这样的权重，包括依赖于主动变量的权重，我们证明了加权次椭圆传递估计、动力学Sobolev不等式、局部有界性、弱Harnack和Harnack不等式、局部Hölder连续性以及强最小值原理。Harnack论证使用了全变量加权测度-数据紧致性定理。一个$\mathrm L^p$速度平均估计给出了$wu$的光滑矩的紧致性，所有扩散变量中的加权Poincaré不等式重构了$u$。在二相极限中，用非负测试函数产生由加权平均速度给出的传输方向。将$\mathrm{BV}$论证应用于极限子解不等式，得到扩展正性所需的无跳跃原理和加权动力学墨迹引理。该理论包括Garofalo和Tralli对Kolmogorov算子分数幂的扩展中的权重$|\lambda|^{1-2s}$。
英文摘要
Let $m\geq1$, $k\geq0$, $n=m+k$, and write $x=(v,z)\in\mathbb R^m\times \mathbb R^k$, where $v$ is the active velocity variable and $z$ is passive with respect to $Y=v\cdot\nabla_y+\partial_t$. We consider \[ \operatorname{div}_x(A\nabla_xu)-Y(wu)=0, \] where $w=w(x)\in A_2(\mathbb R^n)$ and the measurable matrix $A$ has ellipticity and size controlled by $w$. For arbitrary such weights, including weights depending on the active variables, we prove a weighted hypoelliptic transfer estimate, a kinetic Sobolev inequality, local boundedness, the weak Harnack and Harnack inequalities, local Hölder continuity, and the strong minimum principle. The Harnack argument uses a full-variable weighted measure-data compactness theorem. An $\mathrm L^p$ velocity-averaging estimate gives compactness of smooth moments of $wu$, and weighted Poincaré inequalities in all diffusive variables reconstruct $u$. At a two-phase limit, testing with non-negative profiles produces transport directions given by weighted mean velocities. A $\mathrm{BV}$ argument applied to the limiting subsolution inequality yields the no-jump principle needed for expansion of positivity and a weighted kinetic ink-spots lemma. The theory includes the weight $|λ|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.
CommentsTitle capitalization corrected; manuscript unchanged
