Sur la théorie de Littlewood-Paley-Stein (d’après Coifman-Rochberg-Weiss et Cowling).

*(French)*Zbl 0574.47028
Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 113-129 (1985).

[For the entire collection see Zbl 0549.00021.]

The author presents recent proofs of inequalities for semigroups of operators proved using the transference method which allows some improvement of the results of Stein proved using Littlewood-Paley theory and of the results obtained by use of martingales. The new method (due to Cowling) allows the semigroup to be submarkovian rather than Markovian. The author’s emphasis is on filling in details sketched hastily in the papers he covers.

The results can be expressed in terms of the semigroup \(P_ t\) or its generator defined by \(P_ t=e^{-tL}\). Let \(\mu\) be a tempered distribution on \(R_+\) and \[ m(z)=\int^{\infty}_{0}e^{-zs}\mu (ds) \] be its Laplace transform. The problem is to give \(L^ p\) estimates for the operator \[ P_{\mu}=\int^{\infty}_{0}P_ s\mu (ds) \] which corresponds to m(L) defined by spectral theory.

The author first sketches the approach of E. M. Stein [Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. Math. Studies 63 (1970; Zbl 0193.105)]. He then covers in detail the results of M. G. Cowling using the transference method [Harmonic analysis on semigroups, Ann. Math., II. Ser. 117, 267-283 (1983; Zbl 0528.42006)]. Cowling shows that if m(z) is bounded in every angle of opening less than \(\theta\), then \(P_{\mu}\) is bounded on \(L^ p\) for \(1<p<\infty\) if \(\theta =\pi /2\) recovering the result of Stein. He also shows that this holds for \(\theta =\pi /2\) with \(P_ t\) submarkovian and for \(\theta >\pi /2\), the inequality holds even if \(P_ t\) is not symmetric. For \(\theta <\pi /2\), the method of Cowling gives a range of p for which the operator is bounded on \(L^ p\). The remainder of the paper is a detailed look at the proof of Cowling.

The author presents recent proofs of inequalities for semigroups of operators proved using the transference method which allows some improvement of the results of Stein proved using Littlewood-Paley theory and of the results obtained by use of martingales. The new method (due to Cowling) allows the semigroup to be submarkovian rather than Markovian. The author’s emphasis is on filling in details sketched hastily in the papers he covers.

The results can be expressed in terms of the semigroup \(P_ t\) or its generator defined by \(P_ t=e^{-tL}\). Let \(\mu\) be a tempered distribution on \(R_+\) and \[ m(z)=\int^{\infty}_{0}e^{-zs}\mu (ds) \] be its Laplace transform. The problem is to give \(L^ p\) estimates for the operator \[ P_{\mu}=\int^{\infty}_{0}P_ s\mu (ds) \] which corresponds to m(L) defined by spectral theory.

The author first sketches the approach of E. M. Stein [Topics in harmonic analysis related to the Littlewood-Paley theory, Ann. Math. Studies 63 (1970; Zbl 0193.105)]. He then covers in detail the results of M. G. Cowling using the transference method [Harmonic analysis on semigroups, Ann. Math., II. Ser. 117, 267-283 (1983; Zbl 0528.42006)]. Cowling shows that if m(z) is bounded in every angle of opening less than \(\theta\), then \(P_{\mu}\) is bounded on \(L^ p\) for \(1<p<\infty\) if \(\theta =\pi /2\) recovering the result of Stein. He also shows that this holds for \(\theta =\pi /2\) with \(P_ t\) submarkovian and for \(\theta >\pi /2\), the inequality holds even if \(P_ t\) is not symmetric. For \(\theta <\pi /2\), the method of Cowling gives a range of p for which the operator is bounded on \(L^ p\). The remainder of the paper is a detailed look at the proof of Cowling.

Reviewer: R.Johnson

##### MSC:

47D03 | Groups and semigroups of linear operators |

47D07 | Markov semigroups and applications to diffusion processes |

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