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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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