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

 ${\parallel x\parallel }_{F}=\parallel {\left({\int }_{0}^{\infty }{\left|F\left(tA\right)x\right|}^{2}\frac{\phantom{\rule{0.55542pt}{0ex}}\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_{p}$
associées à un opérateur sectoriel $A$ sur ${L}^{p}$ pour $1. 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 ${\parallel \phantom{\rule{0.166667em}{0ex}}.\phantom{\rule{0.166667em}{0ex}}\parallel }_{F}$. Puis nous appliquons nos résultats à l’étude de conditions impliquant une inégalité du type $\parallel {\left({\int }_{0}^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^{2}\mathrm{d}t{\right)}^{1/2}\parallel }_{q}\le M{\parallel x\parallel }_{p}$, 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.

We give new results on square functions

 ${\parallel x\parallel }_{F}=\parallel {\left({\int }_{0}^{\infty }{\left|F\left(tA\right)x\right|}^{2}\frac{\phantom{\rule{0.55542pt}{0ex}}\mathrm{d}t}{t}\right)}^{1/2}{\parallel }_{p}$
associated to a sectorial operator $A$ on ${L}^{p}$ for $1. 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 ${\parallel \phantom{\rule{0.166667em}{0ex}}.\phantom{\rule{0.166667em}{0ex}}\parallel }_{F}$. Then we apply our results to the study of conditions under which we have an estimate $\parallel {\left({\int }_{0}^{\infty }|C{\mathrm{e}}^{-tA}{\left(x\right)|}^{2}\mathrm{d}t{\right)}^{1/2}\parallel }_{q}\le M{\parallel x\parallel }_{p}$, 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.

DOI : https://doi.org/10.24033/bsmf.2462
Classification : 47A60,  47D06
Mots clés : opérateurs sectoriels, calcul fonctionnel ${H}^{\infty }$, fonctions carrées, $R$-bornitude, admissibilité
@article{BSMF_2004__132_1_137_0,
author = {Le Merdy, Christian},
title = {On square functions associated to sectorial operators},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {137--156},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {132},
number = {1},
year = {2004},
doi = {10.24033/bsmf.2462},
zbl = {1066.47013},
language = {en},
url = {http://archive.numdam.org/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) no. 1, pp. 137-156. doi : 10.24033/bsmf.2462. http://archive.numdam.org/item/BSMF_2004__132_1_137_0/

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