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An Application of Homogenization Theory to Harmonic Analysis: Harnack Inequalities And Riesz Transforms on Lie Groups of Polynomial Growth
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- Journal:
- Canadian Journal of Mathematics / Volume 44 / Issue 4 / 01 August 1992
- Published online by Cambridge University Press:
- 20 November 2018, pp. 691-727
- Print publication:
- 01 August 1992
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- Article
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We prove a homogenization formula for a sub-Laplacian
are left invariant Hörmander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms 
, are bounded on Lq,1 < q <+∞ and from L1to weak-L1.
are left invariant Hörmander vector fields) on a connected Lie group 
, are bounded on 