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) no. 1, pp. 137-156.

Nous obtenons de nouveaux résultats sur les fonctions carrées

x F = 0 F ( t A ) x 2 dt t 1/2 p
associées à un opérateur sectoriel A sur L p pour 1<p<. Quand A est en fait R-sectoriel, on montre des équivalences de la forme K -1 x G x F Kx G pour des fonctions F,G appropriées. On démontre également que A possède un calcul fonctionnel H borné par rapport à . F . Puis nous appliquons nos résultats à l’étude de conditions impliquant une inégalité du type ( 0 |Ce -tA (x)| 2 dt) 1/2 q Mx p , où -A engendre un semigroupe borné e -tA sur L p et C:D(A)L q est une application linéaire.

We give new results on square functions

x F = 0 F ( t A ) x 2 dt t 1/2 p
associated to a sectorial operator A on L p for 1<p<. Under the assumption that A is actually R-sectorial, we prove equivalences of the form K -1 x G x F Kx G for suitable functions F,G. We also show that A has a bounded H functional calculus with respect to . F . Then we apply our results to the study of conditions under which we have an estimate ( 0 |Ce -tA (x)| 2 dt) 1/2 q Mx p , when -A generates a bounded semigroup e -tA on L p and C:D(A)L q is a linear mapping.

DOI : 10.24033/bsmf.2462
Classification : 47A60, 47D06
Keywords: sectorial operators, $H^{\infty }$ functional calculus, square functions, $R$-boundedness, admissibility
Mot 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},
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     volume = {132},
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     url = {http://archive.numdam.org/articles/10.24033/bsmf.2462/}
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Le Merdy, Christian. On square functions associated to sectorial operators. Bulletin de la Société Mathématique de France, Tome 132 (2004) no. 1, pp. 137-156. doi : 10.24033/bsmf.2462. http://archive.numdam.org/articles/10.24033/bsmf.2462/

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