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Fundamental solutions of differential operators on homogeneous manifolds of negative curvature and related Riesz transforms. (English) Zbl 0878.22007

Let \(S\) be a connected, simply connected homogeneous manifold of negative curvature, then \(S\) is a semidirect product of its maximal nilpotent normal subgroup \(N\) and the Abel subgroup \(A= \mathbb{R}^+\). Let \(\pi:S\to A\), \(\pi(xa) =a\) be the canonical homomorphism of \(S\) onto \(A\). One considers a left-invariant second order subelliptic operator \(L=Y^2_1+ \cdots+ Y^2_p+Y +\gamma\), where \(Y_1, \dots, Y_p\) generate the Lie algebra Lie \((S)\), \(\pi(L)= (a\partial_a)^2 -\alpha a\partial_a +\gamma\) and \(\gamma< \alpha^2/4\). There is no need for further assumptions on \(L\) like symmetry or \(L\) being the Laplace-Beltrami operator; only using Ancona’s approach to the Martin boundary on negatively curved manifolds, the author formulates certain Harnack inequalities, which give sharp pointwise estimates from above and below for the global Green function \(G\) of the operator \(L+\lambda I\), where \(\lambda\leq \alpha^2/4-\gamma\).
For \(L\) an elliptic with \(\gamma =0\) and \(\alpha=Q\), i.e. in the case where \(L\) has a spectral gap on \(L^2(m)\), consider the first and second Riesz transforms \(f\to\nabla^j (-L)^{j/2}f\), \(j=1,2\). In the case of the first order Riesz transform, one needs the additional assumption \(Y=- Qa\partial_a\) and so \(L\) is selfadjoint on \(L^2(m)\). By dividing the Green function \(G\) into the local part \(G_1\) and the global part \(G_2\), \(G=G_1+G_2\), \(G_1 =\varphi G\), \(G_2=(1-\varphi)G\) with \(\varphi\in C^\infty_0(S)\), \(0\leq \varphi\leq 1\) and \(\varphi \equiv 1\) in a neighborhood of \(e\), we know that the corresponding local parts of the Riesz transforms are standard, while the global parts of these transforms can be dominated by the Green function \(G\). Using the pointwise estimate for \(G\), the author proves that the first and the second order Riesz transforms are weak type (1,1).
Reviewer: Zhu Fuliu (Hubei)

MSC:

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
31B25 Boundary behavior of harmonic functions in higher dimensions
53C30 Differential geometry of homogeneous manifolds
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