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The  $A_\infty$-property of the Kolmogorov measure
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2016 (English)Article in journal (Refereed) Submitted
Abstract [en]

We consider the Kolmogorov-Fokker-Planck operator              \begin{eqnarray*}\label{e-kolm-nd}   \K:=\sum_{i=1}^{m}\partial_{x_i x_i}+\sum_{i=1}^m x_i\partial_{y_{i}}-\partial_t,    \end{eqnarray*}    in unbounded domains of the form         \begin{eqnarray*}\label{dom} \Omega=\{(x,x_{m},y,y_{m},t)\in\mathbb R^{N+1}:\ x_m>\psi(x,y,t)\}.    \end{eqnarray*}    Concerning $\psi$ and $\Omega$ we assume that $\Omega$ is what we call an (unbounded) admissible $\MLip$-domain: $\psi$ 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 $\MLip$-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.

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URN: urn:nbn:se:uu:diva-280644OAI: oai:DiVA.org:uu-280644DiVA: diva2:911517
Available from: 2016-03-13 Created: 2016-03-13 Last updated: 2016-03-13

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Nyström, Kaj
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