Vol. 10, No. 7, 2017

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The $A_\infty$-property of the Kolmogorov measure

Kaj Nyström

Vol. 10 (2017), No. 7, 1709–1756
Abstract

We consider the Kolmogorov–Fokker–Planck operator

$\mathsc{K}:=\sum _{i=1}^{m}{\partial }_{{x}_{i}{x}_{i}}+\sum _{i=1}^{m}{x}_{i}{\partial }_{{y}_{i}}-{\partial }_{t}$

in unbounded domains of the form

$\Omega =\left\{\left(x,{x}_{m},y,{y}_{m},t\right)\in {ℝ}^{N+1}\mid {x}_{m}>\psi \left(x,y,t\right)\right\}.$

Concerning $\psi$ and $\Omega$, we assume that $\Omega$ is what we call an (unbounded) admissible ${Lip}_{K}$-domain: $\psi$ satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathsc{K}$, as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible ${Lip}_{K}$-domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon–Nikodym derivative defines an ${A}_{\infty }$ weight with respect to this surface measure. Our result is sharp.

Keywords
Kolmogorov equation, ultraparabolic, hypoelliptic, Lipschitz domain, doubling measure, parabolic measure, Kolmogorov measure, $A_\infty$
Mathematical Subject Classification 2010
Primary: 35K65, 35K70
Secondary: 35H20, 35R03