The A-property of the Kolmogorov measure. (English) Zbl 1370.35188

Nyström, Kaj

Anal. PDE 10, No. 7, 1709-1756 (2017).

Summary: We consider the Kolmogorov-Fokker-Planck operator \[ \mathcal{K}:=\sum_{i=1}^{m} \partial_{x_i x_i}+ \sum_{i=1}^m x_i \partial_{y_{i}}-\partial_t \] in unbounded domains of the form \[ \Omega=\{(x,x_{m},y,y_{m},t)\in \mathbb{R}^{N+1}|x_m>\psi(x,y,t)\}\text{.} \] Concerning \(\psi\) and \(\Omega\), we assume that \(\Omega\) is what we call an (unbounded) admissible \(\operatorname{Lip}_K\)-domain: \(\psi\) satisfies a uniform Lipschitz condition, adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator \(\mathcal{K}\), as well as an additional regularity condition formulated in terms of a Carleson measure. We prove that in admissible \(\operatorname{Lip}_K\)-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.

MSC:

35K65	Degenerate parabolic equations
35K70	Ultraparabolic equations, pseudoparabolic equations, etc.
35H20	Subelliptic equations
35R03	PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.

Keywords:

Kolmogorov equation; ultraparabolic; hypoelliptic; Lipschitz domain; doubling measure; parabolic measure; Kolmogorov measure; \(A_\infty\)

References:

[1]	10.1142/S0218202501000945 · Zbl 1034.35166
[2]	10.1007/s00477-008-0276-9 · Zbl 1416.65027
[3]	10.1051/m2an/2010046 · Zbl 1426.76628
[4]	10.1007/s00440-010-0301-z · Zbl 1234.60060
[5]	10.1007/978-3-642-12110-4_123 · Zbl 1308.76218
[6]	; Christ, Colloq. Math., 60/61, 601 (1990)
[7]	10.1007/s11118-010-9172-2 · Zbl 1207.35102
[8]	10.1007/s10231-010-0172-z · Zbl 1241.35127
[9]	103_005
[10]	10.1007/s10851-005-3630-2 · Zbl 1478.92100
[11]	10.1007/978-3-642-34444-2 · Zbl 1287.53001
[12]	; Coifman, Studia Math., 51, 241 (1974)
[13]	10.1007/BFb0058946 · Zbl 0224.43006
[14]	10.1007/BF02656492 · Zbl 0939.35082
[15]	; Fabes, Illinois J. Math., 30, 536 (1986)
[16]	10.1090/S0002-9947-99-02487-3 · Zbl 0976.35031
[17]	10.2307/2001585 · Zbl 0719.35007
[18]	10.1215/S0012-7094-97-09008-6 · Zbl 0941.42006
[19]	10.2307/2118595 · Zbl 0867.35037
[20]	; Hofmann, Illinois J. Math., 43, 752 (1999)
[21]	10.1090/memo/0719 · Zbl 1149.35048
[22]	10.4171/RMI/287 · Zbl 0980.42011
[23]	10.1016/j.jfa.2004.10.016 · Zbl 1065.35125
[24]	; Hofmann, Ann. Acad. Sci. Fenn. Math., 28, 355 (2003)
[25]	10.1215/S0012-7094-04-12222-5 · Zbl 1074.35041
[26]	10.1090/memo/1159 · Zbl 1371.28004
[27]	10.1007/BF02392081 · Zbl 0156.10701
[28]	10.1007/s12220-015-9630-6 · Zbl 1386.35070
[29]	10.2307/1968123 · JFM 60.1159.01
[30]	; Lanconelli, Rend. Sem. Mat. Univ. Politec. Torino, 52, 29 (1994)
[31]	10.1090/memo/0545 · Zbl 0826.35041
[32]	10.1512/iumj.1988.37.37039 · Zbl 0698.35068
[33]	10.1142/3302
[34]	; Manfredini, Adv. Differential Equations, 2, 831 (1997) · Zbl 1023.35518
[35]	10.1512/iumj.1997.46.1277 · Zbl 0878.35010
[36]	10.1016/j.matpur.2016.02.007 · Zbl 1339.35147
[37]	10.1007/978-88-470-1781-8 · Zbl 1214.91002
[38]	10.1017/CBO9780511840531 · Zbl 0966.76002
[39]	10.2307/121104 · Zbl 1157.35391
[40]	10.1007/BF01789473 · Zbl 0477.35049