On square functions associated to sectorial operators
Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 1, p. 137-156

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.

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.

DOI : https://doi.org/10.24033/bsmf.2462
Classification:  47A60,  47D06
Keywords: sectorial operators, H functional calculus, square functions, R-boundedness, admissibility
@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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {132},
     number = {1},
     year = {2004},
     pages = {137-156},
     doi = {10.24033/bsmf.2462},
     zbl = {1066.47013},
     language = {en},
     url = {http://www.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, Volume 132 (2004) no. 1, pp. 137-156. doi : 10.24033/bsmf.2462. http://www.numdam.org/item/BSMF_2004__132_1_137_0/

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