On square functions associated to sectorial operators

Le Merdy, Christian

Bulletin de la Société Mathématique de France, Tome 132 (2004) no. 1, pp. 137-156

We give new results on square functions

$${\parallel x\parallel }_F=\parallel {\left({\int }_0^{\infty }{\left|F\left(tA\right)x\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_p$$

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 $K^{-1}{\parallel x\parallel }_G\le {\parallel x\parallel }_F\le K{\parallel x\parallel }_G$ for suitable functions $F,G$. We also show that $A$ has a bounded $H^{\infty }$ functional calculus with respect to ${\parallel .\parallel }_F$. Then we apply our results to the study of conditions under which we have an estimate $\parallel {({\int }_0^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^2\mathrm{d}t{)}^{1/2}\parallel }_q\le M{\parallel x\parallel }_p$, when $-A$ generates a bounded semigroup ${\mathrm{e}}^{-tA}$ on $L^p$ and $C:D\left(A\right)\to L^q$ is a linear mapping.

Nous obtenons de nouveaux résultats sur les fonctions carrées

$${\parallel x\parallel }_F=\parallel {\left({\int }_0^{\infty }{\left|F\left(tA\right)x\right|}^2\frac{\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_p$$

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 $K^{-1}{\parallel x\parallel }_G\le {\parallel x\parallel }_F\le K{\parallel x\parallel }_G$ pour des fonctions $F,G$ appropriées. On démontre également que $A$ possède un calcul fonctionnel $H^{\infty }$ borné par rapport à ${\parallel .\parallel }_F$. Puis nous appliquons nos résultats à l’étude de conditions impliquant une inégalité du type $\parallel {({\int }_0^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^2\mathrm{d}t{)}^{1/2}\parallel }_q\le M{\parallel x\parallel }_p$, où $-A$ engendre un semigroupe borné ${\mathrm{e}}^{-tA}$ sur $L^p$ et $C:D\left(A\right)\to L^q$ est une application linéaire.

Classification : 47A60, 47D06
Keywords: sectorial operators, $H^{\infty }$ functional calculus, square functions, $R$-boundedness, admissibility
Mots-clés : opérateurs sectoriels, calcul fonctionnel $H^{\infty }$, fonctions carrées, $R$-bornitude, admissibilité

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