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The Riesz transformation on conical varieties. (La transformation de Riesz sur les variétés coniques.) (French) Zbl 0937.43004
Let $$N$$ be a connected Riemannian manifold of dimension $$n -1\geq 1$$. This paper is concerned with the investigation of the Riesz transform $$\nabla(-\Delta)^{-1/2}$$ on the cone $$C(N)= R^+\times N$$. When $$N$$ is compact, let $$\lambda_1$$ be the smallest non-zero eigenvalue of the Laplace-Beltrami operator on $$N$$ and $p_0 =\text{sup}\bigl\{p>1\mid p.\bigl(\tfrac n2-\sqrt{(\tfrac{n-2}{2})^2+\lambda_1}\bigr)<n\bigr\}.$ The author shows that $$\nabla(-\Delta)^{-1/2}$$ is of weak type (1,1) and type $$(p,p)$$ for all $$1 <p <p_0$$ and that it is not of type $$(p,p)$$ for any $$p >p_0$$.
When $$N$$ is a complete noncompact Riemannian manifold, the author studies the behavior of the Riesz transform on the cone $$C(N)$$ in several cases. For example, if $$N =G/K$$ is a Riemannian symmetric space of noncompact type then $$\nabla(-\Delta)^{-1/2}$$ is bounded on $$L^p(C(N))$$ for all $$1< p < n$$, while if $$N= R^{n-1}$$ with $$n \geq 3$$, then $$\nabla(- \Delta)^{-1/2}$$ is not bounded on $$L^p(C(N))$$ for any $$p>n$$.
Reviewer: Zhu Fulin (Hubei)

##### MSC:
 43A85 Harmonic analysis on homogeneous spaces 22E30 Analysis on real and complex Lie groups
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