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Riesz transforms through reverse Hölder and Poincaré inequalities. (English) Zbl 1368.58013
The paper improves previous results from [P. Auscher et al., Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 911–957 (2004; Zbl 1086.58013)] and [P. Auscher and T. Coulhon, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 3, 531–555 (2005; Zbl 1116.58023)] concerning the relationship between the $$L^p$$-boundedness of the Riesz transform and the $$L^p$$-gradient estimates for the semigroup. The Riesz transform $$\mathcal{R}=\Gamma L^{-1/2}$$ is considered on $$L^p$$ modelled on a metric measure space $$(X,d,\mu)$$ with a doubling measure $$\mu$$, $$\Gamma$$ is an abstract gradient operator defined on a dense subset $$\mathcal{F}$$ of $$L^p$$ which satisfies a relative Faber-Krahn inequality and $$L$$ is an injective, $$\omega$$-accreative ($$\omega\in[0,\pi/2]$$) operator, defined on $$\mathcal{D}\subset\mathcal{F}$$, satisfying the $$L^2$$-Davies-Gaffney estimates.
The authors show that if $$L$$ is non-negative self-adjoint, then, under the assumption that a reverse Hölder-type inequality holds, the Riesz transform is always bounded on $$L^p$$ for $$p\in[2,2+\varepsilon)$$ (with some $$\varepsilon>0$$) and $$L^p$$-gradient estimates for the semigroup (with $$p>2$$) imply boundedness of the Riesz transform in $$L^q$$ for $$q\in[2,p)$$.
In the last section, the authors study elliptic perturbations of Riesz transforms.
Reviewer: Petr Gurka (Praha)

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