On square functions associated to sectorial operators
Bulletin de la Société Mathématique de France, Volume 132 (2004) no. 1, pp. 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: 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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     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, Volume 132 (2004) no. 1, pp. 137-156. doi : 10.24033/bsmf.2462. http://archive.numdam.org/articles/10.24033/bsmf.2462/

[1] W. Arendt & S. Bu - « The operator valued Marcinkiewicz multiplier theorem and maximal regularity », Math. Z. 240 (2002), p. 311-343. | MR | Zbl

[2] P. Auscher, X. Duong & A. McIntosh - in preparation.

[3] P. Auscher, A. Mcintosh & A. Nahmod - « Holomorphic functional calculi of operators, quadratic estimates and interpolation », Indiana Univ. Math. J. 46 (1997), p. 375-403. | MR | Zbl

[4] P. Clément, B. De Pagter, F. Sukochev & H. Witvliet - « Schauder decompositions and multiplier theorems », Studia Math. 138 (2000), p. 135-163. | MR | Zbl

[5] P. Clément & J. Pruss - « An operator valued transference principle and maximal regularity on vector valued L p -spaces », Proc. of the Sixth International Conference on Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998) (G. Lumer & L. Weis, éds.), Marcel Dekker, New-York, 2001, p. 67-87. | MR | Zbl

[6] M. Cowling, I. Doust, A. Mcintosh & A. Yagi - « Banach space operators with a bounded H functional calculus », J. Austr. Math. Soc. 60 (1996), p. 51-89. | MR | Zbl

[7] J. Garnett - Bounded analytic functions, Pure and applied Mathematics, vol. 96, Academic Press, 1981. | MR | Zbl

[8] P. Grabowsky & F. Callier - « Admissible observation operators. Semigroup criteria of admissibility », Int. Equ. Oper. Theory 25 (1996), p. 182-198. | MR | Zbl

[9] B. Jacob & J. Partington - « The Weiss conjecture on admissibility of observation operators for contraction semigroups », Int. Equ. Oper. Theory 40 (2001), p. 231-243. | MR | Zbl

[10] B. Jacob, J. Partington & S. Pott - « Admissible and weakly admissible observation operators for the right shift semigroup », Proc. Edinburgh Math. Soc. 45 (2002), p. 353-362. | MR | Zbl

[11] B. Jacob, O. Staffans & H. Zwart - « Weak admissibility does not imply admissibility for analytic semigroups », 2003. | MR | Zbl

[12] B. Jacob & H. Zwart - « Disproof of two conjectures of George Weiss », Preprint, 2000.

[13] N. Kalton & G. Lancien - « A solution to the problem of L p -maximal regularity », Math. Z. 235 (2000), p. 559-568. | MR | Zbl

[14] N. Kalton & L. Weis - « The H calculus and sums of closed operators », Math. Ann. 321 (2001), p. 319-345. | MR | Zbl

[15] F. Lancien, G. Lancien & C. Le Merdy - « A joint functional calculus for sectorial operators with commuting resolvents », Proc. London Math. Soc. 77 (1998), p. 387-414. | MR | Zbl

[16] C. Le Merdy - « The Weiss conjecture for bounded analytic semigroups », J. London Math. Soc. (2) 67 (2003), p. 715-738. | MR | Zbl

[17] J. Lindenstrauss & L. Tzafriri - Classical Banach spaces II, Springer Verlag, Berlin, 1979. | MR | Zbl

[18] A. Mcintosh - « Operators which have an H functional calculus », Miniconference on operator theory and partial differential equations, Proc. of CMA, Canberra, vol. 14, 1986, p. 210-231. | MR | Zbl

[19] A. Mcintosh & A. Yagi - « Operators of type ω without a bounded H functional calculus », Miniconference on operators in analysis, Proc. of CMA, Canberra, vol. 24, 1989, p. 159-172. | MR | Zbl

[20] J. Partington & G. Weiss - « Admissible observation operators for the right shift semigroup », Math. Cont. Signals Systems 13 (2000), p. 179-192. | MR | Zbl

[21] L. Weis - « A new approach to maximal L p -regularity », Proc. of the Sixth International Conference on Evolution Equations and their Applications in Physical and Life Sciences (Bad Herrenalb, 1998) (G. Lumer & L. Weis, éds.), Lecture Notes in Pure and Appl. Math., vol. 215, Marcel Dekker, New-York, 2001, p. 195-214. | MR | Zbl

[22] -, « Operator valued Fourier multiplier theorems and maximal regularity », Math. Ann. 319 (2001), p. 735-758. | MR | Zbl

[23] G. Weiss - « Admissibility of unbounded control operators », SIAM J. Control Optim. 27 (1989), p. 527-545. | MR | Zbl

[24] -, « Admissible observation operators for linear semigroups », Israel J. Math. 65 (1989), p. 17-43. | MR

[25] -, « Two conjectures on the admissibility of control operators », Estimation and control of distributed parameter systems, Birkäuser Verlag, 1991, p. 367-378. | MR | Zbl

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