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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 \(L\).
43A85 Harmonic analysis on homogeneous spaces
53C30 Differential geometry of homogeneous manifolds
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