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2017 The $A_\infty$-property of the Kolmogorov measure
Kaj Nyström
Anal. PDE 10(7): 1709-1756 (2017). DOI: 10.2140/apde.2017.10.1709


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.


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Kaj Nyström. "The $A_\infty$-property of the Kolmogorov measure." Anal. PDE 10 (7) 1709 - 1756, 2017.


Received: 1 February 2017; Accepted: 17 June 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1370.35188
MathSciNet: MR3683926
Digital Object Identifier: 10.2140/apde.2017.10.1709

Primary: 35K65 , 35K70
Secondary: 35H20 , 35R03

Keywords: $A_\infty$ , doubling measure , hypoelliptic , Kolmogorov equation , Kolmogorov measure , Lipschitz domain , Parabolic measure , ultraparabolic

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.10 • No. 7 • 2017
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