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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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