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**Transformations de Riesz pour les semi-groupes symétriques. I. Étude de la dimension 1.**
*(French)*
Zbl 0561.42010

Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 130-144 (1985).

[For the entire collection see Zbl 0549.00007.]

For symmetric semigroups, Stein has studied Littlewood-Paley inequalities. The author wishes to consider estimates for the Riesz transform. Recall that if \(P_ t\) is a Markovian semigroup with generator L, symmetric with respect to the measure \(\mu\), and if \(V=(- L)^{-1/2}\), on a suitable domain, then the Riesz transform is the operator HV, where H is a linear operator such that \(| Hf|^ 2\leq \Gamma (f,f),\) where \(\Gamma\) is the operator \(\Gamma (f,g)=(L(fg)- fLg-gLf).\) One wants to know about the behavior of HV on \(L^ p\), for \(1<p<\infty\) or on \(H^ 1\) and BMO. For \({\mathbb{R}}^ n\) and spaces of homogeneous type such results are well known, but first by extrinsic methods; i.e., methods that use properties of the underlying space rather than only properties of the semigroup. Later, results were found by intrinsic methods, depending only on the semigroup, its spectral family and the bilinear operator \(\Gamma\).

In this part I the author applies probbilistic methods to situations for which the analytic results are well known. The object is to determine which equalities and inequalities make the proofs work with the idea then to find corresponding conditions in more general cases in part II. In one dimension the generator \(\alpha\geq 0\), \(\beta\geq 0\), \(L_{\alpha,\beta}f(x)=(1-x^ 2)f''-[(\alpha +\beta +2)x+(\alpha - \beta)]f'(x)\) is considered. It is proved that the associated Riesz transform maps a suitable subspace of \(L^ p\) into itself and that the probabilistic \(H^ 1\) and the analytic \(H^ 1\) agree, although it is not shown that HV maps \(H^ 1\) into itself. One thereby obtains the essence of what is outlined in E. Stein [Topics in harmonic analysis. Related to the Littlewood-Paley theory, 138-141 (1970; Zbl 0193.105)].

For symmetric semigroups, Stein has studied Littlewood-Paley inequalities. The author wishes to consider estimates for the Riesz transform. Recall that if \(P_ t\) is a Markovian semigroup with generator L, symmetric with respect to the measure \(\mu\), and if \(V=(- L)^{-1/2}\), on a suitable domain, then the Riesz transform is the operator HV, where H is a linear operator such that \(| Hf|^ 2\leq \Gamma (f,f),\) where \(\Gamma\) is the operator \(\Gamma (f,g)=(L(fg)- fLg-gLf).\) One wants to know about the behavior of HV on \(L^ p\), for \(1<p<\infty\) or on \(H^ 1\) and BMO. For \({\mathbb{R}}^ n\) and spaces of homogeneous type such results are well known, but first by extrinsic methods; i.e., methods that use properties of the underlying space rather than only properties of the semigroup. Later, results were found by intrinsic methods, depending only on the semigroup, its spectral family and the bilinear operator \(\Gamma\).

In this part I the author applies probbilistic methods to situations for which the analytic results are well known. The object is to determine which equalities and inequalities make the proofs work with the idea then to find corresponding conditions in more general cases in part II. In one dimension the generator \(\alpha\geq 0\), \(\beta\geq 0\), \(L_{\alpha,\beta}f(x)=(1-x^ 2)f''-[(\alpha +\beta +2)x+(\alpha - \beta)]f'(x)\) is considered. It is proved that the associated Riesz transform maps a suitable subspace of \(L^ p\) into itself and that the probabilistic \(H^ 1\) and the analytic \(H^ 1\) agree, although it is not shown that HV maps \(H^ 1\) into itself. One thereby obtains the essence of what is outlined in E. Stein [Topics in harmonic analysis. Related to the Littlewood-Paley theory, 138-141 (1970; Zbl 0193.105)].

Reviewer: R.Johnson

### MSC:

42B25 | Maximal functions, Littlewood-Paley theory |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |

60J60 | Diffusion processes |