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Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group. (English) Zbl 0936.43006
The present paper concerns boundedness properties of the Riesz operator $$\Delta^{-1/2}X$$ on Lebesgue spaces defined with respect to the left Haar measure. It is proved that the operator $$\Delta^{-1/2}X$$ is bounded on $$L^p(G)$$, $$1<p< \infty$$, and of weak type (1,1). This theorem is a generalization of a well-known theorem in the case $$p=2$$. It also provides answers, in a particular case, to the open problem of the boundedness of Riesz operators on Lie groups of exponential growth. The main part of the proof of the theorem is a weak type (1,1) estimation which implies $$L^p$$ boundedness for $$1<p<2$$ using interpolation. In previous works other authors investigated the case of Lie groups of polynomial growth and on nonamenable Lie groups. The results of these works also imply the boundedness of the Riesz transforms on compact Lie groups.

##### MSC:
 43A80 Analysis on other specific Lie groups 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 22E30 Analysis on real and complex Lie groups
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