Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities

Abstract

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 H\"older 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 Poincar\'e 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 $|\lambda|^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.