Weighted Kolmogorov Equations: Transfer Estimates, Local Boundedness, and Harnack Inequalities

Let $m\ge 1$, $k\ge 0$, $n=m+k$, 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 $Y=v\cdot {\mathrm{\nabla }}_y+{\mathrm{\partial }}_t$. We consider $${\mathrm{div}}_x(A{\mathrm{\nabla }}_{x}u)-Y(wu)=0,$$ 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 $\mathrm{B}\mathrm{V}$ 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 $\mathrm{\mid }\lambda {\mathrm{\mid }}^{1-2s}$ in the extension of Garofalo and Tralli for fractional powers of the Kolmogorov operator.