On square functions associated to sectorial operators

Le Merdy, Christian

On square functions associated to sectorial operators
[Sur les fonctions carrées associées aux opérateurs sectoriels]

Bulletin de la Société Mathématique de France, Tome 132 (2004), p. 137-156 / Harvested from Numdam

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Résumé
Abstract

Nous obtenons de nouveaux résultats sur les fonctions carrées ${∥ x ∥}_{F} = ∥ {(\int_{0}^{\infty} {|F (t A) x|}^{2} \frac{d t}{t})}^{1 / 2} ∥_{p}$ associées à un opérateur sectoriel $A$ sur $L^{p}$ pour $1 < p < \infty$ . Quand $A$ est en fait $R$ -sectoriel, on montre des équivalences de la forme $K^{- 1} {∥ x ∥}_{G} \leq {∥ x ∥}_{F} \leq K {∥ x ∥}_{G}$ pour des fonctions $F, G$ appropriées. On démontre également que $A$ possède un calcul fonctionnel $H^{\infty}$ borné par rapport à ${∥ . ∥}_{F}$ . Puis nous appliquons nos résultats à l’étude de conditions impliquant une inégalité du type $∥ {(\int_{0}^{\infty} | C e^{- t A} {(x) |}^{2} d t)^{1 / 2} ∥}_{q} \leq M {∥ x ∥}_{p}$ , où $- A$ engendre un semigroupe borné $e^{- t A}$ sur $L^{p}$ et $C : D (A) \to L^{q}$ est une application linéaire.

We give new results on square functions ${∥ x ∥}_{F} = ∥ {(\int_{0}^{\infty} {|F (t A) x|}^{2} \frac{d t}{t})}^{1 / 2} ∥_{p}$ associated to a sectorial operator $A$ on $L^{p}$ for $1 < p < \infty$ . Under the assumption that $A$ is actually $R$ -sectorial, we prove equivalences of the form $K^{- 1} {∥ x ∥}_{G} \leq {∥ x ∥}_{F} \leq K {∥ x ∥}_{G}$ for suitable functions $F, G$ . We also show that $A$ has a bounded $H^{\infty}$ functional calculus with respect to ${∥ . ∥}_{F}$ . Then we apply our results to the study of conditions under which we have an estimate $∥ {(\int_{0}^{\infty} | C e^{- t A} {(x) |}^{2} d t)^{1 / 2} ∥}_{q} \leq M {∥ x ∥}_{p}$ , when $- A$ generates a bounded semigroup $e^{- t A}$ on $L^{p}$ and $C : D (A) \to L^{q}$ is a linear mapping.

Publié le : 2004-01-01

Zbl 1066.47013 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/bsmf.2462
Classification: 47A60, 47D06
Mots clés: opérateurs sectoriels, calcul fonctionnel

H^{\infty}

, fonctions carrées,

R

-bornitude, admissibilité

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     author = {Le Merdy, Christian},
     title = {On square functions associated to sectorial operators},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     volume = {132},
     year = {2004},
     pages = {137-156},
     doi = {10.24033/bsmf.2462},
     zbl = {1066.47013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BSMF_2004__132_1_137_0}
}

Le Merdy, Christian. On square functions associated to sectorial operators. Bulletin de la Société Mathématique de France, Tome 132 (2004) pp. 137-156. doi : 10.24033/bsmf.2462. http://gdmltest.u-ga.fr/item/BSMF_2004__132_1_137_0/

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