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

We consider the Kolmogorov–Fokker–Planck operator

K := i=1m xixi + i=1mx iyi t

in unbounded domains of the form

Ω = {(x,xm,y,ym,t) N+1x m > ψ(x,y,t)}.

Concerning ψ and Ω, we assume that Ω is what we call an (unbounded) admissible LipK-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 LipK-domains the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon–Nikodym derivative defines an A weight with respect to this surface measure. Our result is sharp.

Kolmogorov equation, ultraparabolic, hypoelliptic, Lipschitz domain, doubling measure, parabolic measure, Kolmogorov measure, $A_\infty$
Mathematical Subject Classification 2010
Primary: 35K65, 35K70
Secondary: 35H20, 35R03
Received: 1 February 2017
Accepted: 17 June 2017
Published: 1 August 2017
Kaj Nyström
Department of Mathematics
Uppsala University