## 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

### MSC:

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

### Keywords:

symmetric semigroups; diffusion semigroup

Zbl 0561.42010
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