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Inégalités de Harnack paraboliques et transformées de Riesz sur les groupes de Lie résolubles à croissance polynomiale du volume. (Parabolic Harnack inequalities and Riesz transforms on solvable Lie groups of polynomial growth). (French) Zbl 0699.22014

Let Q be a connected solvable Lie group, not necessarily nilpotent, of polynomial volume growth. Let \(X_ 1,...,X_ n\) be left-invariant vector fields that generate the Lie algebra of Q through successive brackets and put \(L=-(X^ 2_ 1+...+X^ 2_ n)\). The author announces Harnack type inequalities for non-negative solutions of the equation \(((\partial /\partial t)+L)u=0\) and boundedness properties of the Riesz operators \(X_ kL^{-1/2}\), \(L^{-1/2}X_ k\), \(k=1,...,n\).
Reviewer: E.J.Akutowicz

MSC:

22E30 Analysis on real and complex Lie groups
22E25 Nilpotent and solvable Lie groups
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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