Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities
Abstract
Let m≥1, k≥0, n=m+k, and write x=(v,z)∈ℝm× ℝk, where v is the active velocity variable and z is passive with respect to Y=v·∇y+∂t. We consider divx(A∇xu)−Y(wu)=0, where w=w(x)∈ A2(ℝn) 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 Lp 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 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.
Related
- Weighted derivative estimates for Kolmogorov equations
- Boundedness and contractive estimates for orthotropic, widely degenerate, doubly nonlinear diffusion equations
- Weighted decoupling inequalities via fractal bilinear Kakeya estimates
- Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains
- Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations
BibTeX
@misc{nystrom2026weighted,
title = {{Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities}},
author = {Kaj Nyström},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21737},
eprint = {2609.21737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21737},
}
RIS
TY - UNPB
TI - Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities
AU - Kaj Nyström
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.21737
UR - https://arxiv.org/abs/2609.21737
AB - Let m≥1, k≥0, n=m+k, and write x=(v,z)∈ℝ^m× ℝ^k, where v is the active velocity variable and z is passive with respect to Y=v·∇_y+∂_t. We consider div_x(A∇_xu)−Y(wu)=0, where w=w(x)∈ A_2(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 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 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.
ER -