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Estimates for derivatives of the Green functions for the noncoercive differential operators on homogeneous manifolds of negative curvature. II. (English) Zbl 1041.43004
[Part I by the same author in Potential Anal. 19, 317–339 (2003; Zbl 1028.43010).]
The author studies noncoercive operators \(L\) of the following form \[ L= \sum^m_{j=1} \phi_a(X_j)^2+ \phi_a(X)+ a^2\partial^2_a+ \partial_a, \] where \(X,X_1,\dots, X_m\) are left-invariant vector fields on \(N\); moreover, the vector fields \(X_1,\dots, X_m\) are linearly independent and generate \(n\), \[ \phi_a= \text{Ad}_{\exp(\log a)Y_0}= e^{(\log a)\text{ad\,}Y_0}= e^{(\log a)D}, \] where \(D= \text{ad}_{Y_0}\) is a derivation of the Lie algebra \(n\) of the Lie group \(N\) such that the real parts \(d_j\) of the eigenvalues \(\lambda_j\) of \(D\) are positive. The main goal of the paper is to prove estimates for the derivatives of the Green function for \(L\).
MSC:
43A85 Harmonic analysis on homogeneous spaces
53C30 Differential geometry of homogeneous manifolds
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