The A∞-property of the Kolmogorov measure

Concerning $ψ$ and $Ω$ , we assume that $Ω$ is what we call an (unbounded) admissible ${Lip}_{K}$ -domain: $ψ$ satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $K$ , 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

Milestones

Received: 1 February 2017

Accepted: 17 June 2017

Published: 1 August 2017

Authors

Kaj Nyström
Department of Mathematics Uppsala University Uppsala Sweden

Vol. 10, No. 7, 2017

Kaj Nyström

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors