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Transformations de Riesz pour les semi-groupes symétriques. II: Étude sous la condition \(\Gamma _ 2\geq 0\). (French) Zbl 0561.42011
Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 145-174 (1985).
[For the entire collection see Zbl 0561.42010.] For part I see the preceding review.
The author considers general symmetric semigroups as envisaged in Part I. The semigroups satisfy \(\Gamma (P_ tf,P_ tf)\leq P_ t\Gamma (f,f)\) (or \(2\Gamma_ 2(f,f)=L\Gamma (f,f)-2\Gamma (f,Lf)\geq 0)\). (Emery shows in the same volume that if L is the Laplacian of a Riemannian variety this amounts to the positivity of the Ricci curvature.) The main results are proved in the form of domination results between \(C=-(L)^{1/2}\) and \(\sqrt{\Gamma}\). In Theorem 2 it is proved that if \(\Gamma_ 2\geq 0\) and \(P_ t\) is a diffusion semigroup, then C is dominated by \(\sqrt{\Gamma}\). The converse is proved for general semigroups and \(p\geq 2\) in Theorem 7. This result allows the extension of Theorem 2 to general semigroups for \(p\leq 2\) by duality. It is also shown that if \(\Gamma_ p(f,g)=(L\Gamma_ p(f,g)-\Gamma_ p(Lf,g)-\Gamma_ p(f,Lg))\) are all positive for \(1\leq p\leq n+1,\) then \(\sqrt{\Gamma_ n}\) is dominated by \(C^ n\) in \(L^ p\) for every \(p\geq 2\). Finally it has been shown that C and \(\sqrt{\Gamma}\) are equivalent for \(1<p<\infty\) for processes for which \(\Gamma (f,f)=\sum_ i(H_ if)^ 2\) where \([L,H_ i]=hH_ i\), with \(h\geq 0\), which includes Brownian motion and Ornstein-Uhlenbeck processes.
Reviewer: R.Johnson

42B25 Maximal functions, Littlewood-Paley theory
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
60J60 Diffusion processes
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