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Riesz transform and related inequalities on non-compact Riemannian manifolds. (English) Zbl 1037.58017
The authors of this interesting paper consider the question when the inequality $\| | \nabla f| \| _p\leq C_p\| \Delta^{1/2}f\| _p\quad f\in C^\infty_0(M)$ holds on a complete Riemannian manifold $$M$$. The inequality is the harder part of the proof of the equivalence of two classical norms in a first order Sobolev space. They conjecture that the inequality holds if $$1<p\leq 2$$ or $$2<p<\infty$$ and a heat semi-group on $$M$$ satisfies an domination condition. The weaker version with $$\sqrt{\| f\| _p\| \Delta f\| _p}$$ instead of $$\| \Delta^{1/2}f\| _p$$ is proved. A generalization with different integral exponent on the left and right sides is given. The original inequality is proved under stronger assumptions. Namely it is proved that if $$M$$ has doubling property, an upper estimate for the heat kernel holds as well as some estimate for the heat kernel on 1-forms then the Riesz transform $$\nabla\Delta^{-1/2}$$ is bounded in $$L^p$$ for $$2<p<\infty$$. This implies the original inequality.

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds 44A15 Special integral transforms (Legendre, Hilbert, etc.) 47B34 Kernel operators
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