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Riesz transform on manifolds and Poincaré inequalities. (English) Zbl 1116.58023
The authors study the validity of the $$L^p$$ inequality for the Riesz transform when $$p>2$$ and of its reverse inequality when $$1<p<2$$ on complete Riemannian manifolds under the doubling property and some Poincaré inequality.
Let $$M$$ be a non-compact complete Riemannian manifold. Consider the $$L^p$$ inequality $||\;|\nabla \Delta^{-1/2}f|\;||_p \leq C_p ||f||_p, \tag{1}$ and the “reverse inequality” $||\Delta^{1/2}f||_p \leq C_p ||\;|\nabla f|\;||_p. \tag{2}$ A manifold is said to satisfy the volume doubling property if there exists a constant $$C$$ such that for all $$x\in M$$ and $$r>0$$, $V(x,2r)\leq C V(x,r).$ It satisfies the scaled Poincaré inequalities $$P_p$$ if there exists $$C>0$$ such that for every ball $$B(x,r)$$, and every $$f$$ with $$f$$, $$\nabla f$$ locally $$p$$-integrable, then $\int_B |f-f_B|^p d\mu \leq Cr^p \int_B |\nabla f|^p\, d\mu, \tag{$$P_p$$}$ where $$f_B$$ is the mean of $$f$$ on $$B$$.
The main theorem is: Let $$M$$ be a complete non-compact Riemannian manifold satisfying the volume doubling property, and $$(P_2)$$. Then there exists $$\epsilon>0$$ such that (1) holds for $$2<p<2+\epsilon$$.
An important step in the proof is the following theorem: Let $$M$$ be a complete non-compact Riemannian manifold satisfying the volume doubling property and $$P_q$$ for some $$q\in [1,2]$$. Then (2) holds for $$q<p<2$$. If $$q=1$$, there is a weak-type $$(1,1)$$ estimate.

##### MSC:
 58J35 Heat and other parabolic equation methods for PDEs on manifolds 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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