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An application of
homogenization theory to harmonic analysis on solvable Lie
groups of polynomial growth
G. Alexopoulos |
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Vol. 159 (1993), No. 1, 19–45
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Abstract |
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Let Q be a connected solvable Lie group of polynomial growth. Let also E1,…,Ep be left invariant vector fields on G that satisfy Hörmander’s condition and denote by L = −(E12 + ⋯ + Ep2) the associated sub-Laplacian and by S(x,t) the ball which is centered at x ∈ Q and it is of radius t > 0 with respect to the control distance associated to those vector fields. The goal of this article is to prove the following Harnack inequality: there is a constant c > 0 such that |Eiu(x)|≤ ct−1u(x), x ∈ Q, t ≥ 1, 1 ≤ i ≤ p, for all u ≥ 0 such that Lu = 0 in S(x,t). This inequality is proved by adapting some ideas from the theory of homogenization. |
Mathematical Subject Classification 2000
Primary: 22E30
Secondary: 35B27, 43A80, 58G11
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Milestones
Received: 27 February 1990
Revised: 24 February 1992
Published: 1 May 1993
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Authors |
| G. Alexopoulos | |
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