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Spherical analysis on harmonic AN groups. (English) Zbl 0881.22008
The paper under review considers harmonic \(AN\) groups and analysis thereon. Those groups are solvable Lie groups with a left-invariant metric and are defined as a semi-direct product \(S=A \times N\) of \(A\cong \mathbb{R}\) and a Heisenberg type (\(H\)-type) Lie group \(N\). The \(H\)-type Lie groups \(N\) [A. Kaplan, Trans. Am. Math. Soc. 258, 147-153 (1980; Zbl 0416.35022)] are nilpotent step two Lie groups with a linear map \(J_Z: n \to n\) \[ \langle J_Z X, Y \rangle = \langle Z , [X, Y] \rangle, \qquad (X,Y \in n,\;Z \in z), \] where \(z\) is the center of the Lie algebra of \(N\) and \(n=z^\perp\) its orthogonal completion. As Riemannian manifolds \(AN\) groups include all symmetric spaces of noncompact type and rank one. Algebraic properties of \(AN\) groups are described in Section 1 of the paper and many basic facts on analysis thereon are given in Section 2. The paper under review establishes a series of analytical results like a sharp criterion for the \( L_p \to L_p\) and the weak \(L_1 \to L_1\) boundedness of positive convolution kernels, a Kunze-Stein phenomenon, the \(L_p\) behavior of (functions of) the Laplace-Beltrami operator, analysis of the heat kernel and semigroup, the weak \(L_1 \to L_1\) boundedness of both the heat maximal operator and the Riesz transform.
Reviewer: V.V.Kisil (Gent)

MSC:
22E30 Analysis on real and complex Lie groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A80 Analysis on other specific Lie groups
43A85 Harmonic analysis on homogeneous spaces
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