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A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

Malte Litsgård and Kaj Nyström

Vol. 16 (2023), No. 7, 1547–1588
Abstract

We consider the operators

X (A(X)X),X (A(X)X) t,X (A(X)X) + X Y t,

where X Ω, (X,t) Ω × and (X,Y,t) Ω × m × , respectively, and where Ω m is an (unbounded) Lipschitz domain with defining function ψ : m1 being Lipschitz with constant bounded by M. Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous with respect to the surface measure d σ(X) and that the corresponding Radon–Nikodym derivative or Poisson kernel satisfies a scale-invariant reverse Hölder inequality in Lp , for some fixed p, 1 < p < , with constants depending only on the constants of A, m and the Lipschitz constant of ψ, M. Under this assumption we prove that the same conclusions are also true for the parabolic measures associated to the second and third operators with d σ(X) replaced by the surface measures d σ(X)d t and d σ(X)d Y d t, respectively. This structural theorem allows us to reprove several results previously established in the literature, as well as to deduce new results in, for example, the context of homogenization for operators of Kolmogorov type. Our proof of the structural theorem is based on recent results established by the authors concerning boundary Harnack inequalities for operators of Kolmogorov type in divergence form with bounded, measurable and uniformly elliptic coefficients.

Keywords
Kolmogorov equation, elliptic, parabolic, ultraparabolic, hypoelliptic, operators in divergence form, Dirichlet problem, Lipschitz domain, doubling measure, elliptic measure, parabolic measure, Kolmogorov measure, $A_{\infty}$, Lie group, homogenization
Mathematical Subject Classification
Primary: 35K65, 35K70, 35H20, 35R03
Milestones
Received: 14 December 2020
Revised: 15 December 2021
Accepted: 14 February 2022
Published: 21 September 2023
Authors
Malte Litsgård
Department of Mathematics
Uppsala University
Uppsala
Sweden
Kaj Nyström
Department of Mathematics
Uppsala University
Uppsala
Sweden

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