Littlewood–Paley–Stein functionals: an -boundedness approach
Annales de l'Institut Fourier, Online first, 46 p.

Let L=Δ+V be a Schrödinger operator with a non-negative potential V on a complete Riemannian manifold M. We prove that the vertical Littlewood-Paley-Stein functional associated with L is bounded on L p (M) if and only if the set {te -tL ,t>0} is -bounded on L p (M). We also introduce and study more general functionals. For a sequence of functions m k :[0,), we define

H((f k )):= k 0 |m k (tL)f k | 2 dt 1/2 + k 0 |Vm k (tL)f k | 2 dt 1/2 .

We prove boundedness of H on L p (M) in the sense

H((f k )) p C k |f k | 2 1/2 p

for some constant C independent of (f k ) k . A lower estimate is also proved on the dual space L p . We introduce and study boundedness of other Littlewood-Paley-Stein type functionals and discuss their relationships to the Riesz transform. Several examples are given in the paper.

Soit L=Δ+V un opérateur de Schrödinger avec un potentiel positif V sur une variété Riemannianne complète M. Nous montrons que les fonctionnelles verticales de Littlewood–Paley–Stein associées à L sont bornées sur L p (M) si et seulement si l’ensemble {te -tL ,t>0} est -borné sur L p (M). Nous introduisons et étudions d’autres fonctionnelles plus générales. Pour une suite de fonctions m k :[0,) données, on définit

H((f k )):= k 0 |m k (tL)f k | 2 dt 1/2 + k 0 |Vm k (tL)f k | 2 dt 1/2 .

Nous montrons que H est bornée sur L p (M) au sens

H((f k )) p C k |f k | 2 1/2 p

avec une constante C indépendante de (f k ) k . Une estimation inférieure est aussi démontrée sur l’espace dual L p . Nous discuterons le lien entre ces fonctionnelles et la transformée de Riesz. Plusieurs exemples et contre-exemples sont donnés dans le papier.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3634
Classification: 42B25, 58J35, 47D08, 60G50
Keywords: Littlewood–Paley–Stein functionals, Riesz transforms, Kahane–Khintchine inequality, spectral multipliers, Schrödinger operators, elliptic operators.
Mot clés : Fonctionnelles de Littlewood–Paley–Stein, transformée de Riesz, inégalités de Kahane–Khintchine, multiplicateurs spectraux, opérateurs de Schrödinger, opérateurs elliptiques.
Cometx, Thomas 1; Ouhabaz, El Maati 2

1 IMB, CNRS and Univ. Bordeaux, 351 Cours de la Libération, 33405 Talence (France)
2 IMB, CNRS, Univ. Bordeaux, 351 Cours de la Libération, 33405 Talence (France)
@unpublished{AIF_0__0_0_A63_0,
     author = {Cometx, Thomas and Ouhabaz, El Maati},
     title = {Littlewood{\textendash}Paley{\textendash}Stein functionals: an ${\mathcal{R}}$-boundedness approach},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3634},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Cometx, Thomas
AU  - Ouhabaz, El Maati
TI  - Littlewood–Paley–Stein functionals: an ${\mathcal{R}}$-boundedness approach
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3634
LA  - en
ID  - AIF_0__0_0_A63_0
ER  - 
%0 Unpublished Work
%A Cometx, Thomas
%A Ouhabaz, El Maati
%T Littlewood–Paley–Stein functionals: an ${\mathcal{R}}$-boundedness approach
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3634
%G en
%F AIF_0__0_0_A63_0
Cometx, Thomas; Ouhabaz, El Maati. Littlewood–Paley–Stein functionals: an ${\mathcal{R}}$-boundedness approach. Annales de l'Institut Fourier, Online first, 46 p.

[1] Assaad, Joyce; Ouhabaz, El Maati Riesz transforms of Schrödinger operators on manifolds, J. Geom. Anal., Volume 22 (2012) no. 4, pp. 1108-1136 | DOI | MR | Zbl

[2] Auscher, Pascal; Ben Ali, Besma Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Volume 57 (2007) no. 6, pp. 1975-2013 | DOI | Numdam | MR | Zbl

[3] Auscher, Pascal; Coulhon, Thierry; Duong, Xuan Thinh; Hofmann, Steve Riesz transform on manifolds and heat kernel regularity, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 6, pp. 911-957 | DOI | Numdam | MR | Zbl

[4] Auscher, Pascal; Hofmann, Steve; Martell, José-María Vertical versus conical square functions, Trans. Am. Math. Soc., Volume 364 (2012) no. 10, pp. 5469-5489 | DOI | MR | Zbl

[5] Auscher, Pascal; Tchamitchian, Philippe Square root problem for divergence operators and related topics, Astérisque, Société Mathématique de France, 1998 no. 249, viii+172 pages | Numdam | MR | Zbl

[6] Bakry, Dominique étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités, XXI (Lecture Notes in Mathematics), Volume 1247, Springer, 1987, pp. 137-172 | DOI | Numdam | MR | Zbl

[7] Bernicot, Frédéric; Frey, Dorothee Riesz transforms through reverse Hölder and Poincaré inequalities, Math. Z., Volume 284 (2016) no. 3-4, pp. 791-826 | DOI | MR | Zbl

[8] Carbonaro, Andrea; Dragičević, Oliver Functional calculus for generators of symmetric contraction semigroups, Duke Math. J., Volume 166 (2017) no. 5, pp. 937-974 | DOI | MR | Zbl

[9] Carron, Gilles; Coulhon, Thierry; Hassell, Andrew Riesz transform and L p -cohomology for manifolds with Euclidean ends, Duke Math. J., Volume 133 (2006) no. 1, pp. 59-93 | DOI | MR | Zbl

[10] Chen, Ji-Cheng Heat kernels on positively curved manifolds and applications., Ph. D. Thesis, Hangzhu Univ. (1987)

[11] Chen, Li Quasi transformées de Riesz, espaces de Hardy et estimations sous-gaussiennes du noyau de la chaleur, Thèse de doctorat Mathématiques, Université Paris 11 (2014) (http://www.theses.fr/2014PA112068)

[12] Chen, Peng; Magniez, Jocelyn; Ouhabaz, El Maati The Hodge–de Rham Laplacian and L p -boundedness of Riesz transforms on non-compact manifolds, Nonlinear Anal., Theory Methods Appl., Volume 125 (2015), pp. 78-98 | DOI | MR | Zbl

[13] Chen, Peng; Ouhabaz, El Maati; Sikora, Adam; Yan, Lixin Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means, J. Anal. Math., Volume 129 (2016), pp. 219-283 | DOI | MR | Zbl

[14] Clément, Philippe; de Pagter, Bernardus; Sukochev, Fedor A.; Witvliet, H. Schauder decomposition and multiplier theorems, Stud. Math., Volume 138 (2000) no. 2, pp. 135-163 | MR | Zbl

[15] Coulhon, Thierry; Duong, Xuan Thinh Riesz transforms for 1p2, Trans. Am. Math. Soc., Volume 351 (1999) no. 3, pp. 1151-1169 | DOI | MR | Zbl

[16] Coulhon, Thierry; Duong, Xuan Thinh Riesz transform and related inequalities on noncompact Riemannian manifolds, Commun. Pure Appl. Math., Volume 56 (2003) no. 12, pp. 1728-1751 | DOI | MR | Zbl

[17] Coulhon, Thierry; Duong, Xuan Thinh; Li, Xiang Dong Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1p2, Stud. Math., Volume 154 (2003) no. 1, pp. 37-57 | DOI | MR | Zbl

[18] Coulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam Gradient estimates for heat kernels and harmonic functions, J. Funct. Anal., Volume 278 (2020) no. 8, 108398, 67 pages | DOI | MR | Zbl

[19] Cowling, Michael; Doust, Ian; McIntosh, Alan; Yagi, Atsushi Banach space operators with a bounded H functional calculus, J. Aust. Math. Soc., Ser. A, Volume 60 (1996) no. 1, pp. 51-89 | DOI | MR | Zbl

[20] Deleaval, Luc; Kriegler, Christoph Maximal Hörmander functional calculus on L p spaces and UMD lattices, Int. Math. Res. Not. (2023) no. 6, pp. 4643-4694 | DOI | MR | Zbl

[21] Devyver, Baptiste Heat kernel and Riesz transform of Schrödinger operators, Ann. Inst. Fourier, Volume 69 (2019) no. 2, pp. 457-513 | DOI | Numdam | MR | Zbl

[22] Driver, Bruce K.; Thalmaier, Anton Heat equation derivative formulas for vector bundles, J. Funct. Anal., Volume 183 (2001) no. 1, pp. 42-108 | DOI | MR | Zbl

[23] Duong, Think Xuan; Ouhabaz, El Maati; Sikora, Adam Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal., Volume 196 (2002) no. 2, pp. 443-485 | DOI | MR | Zbl

[24] Duong, Xuan Thinh; Ouhabaz, El Maati; Yan, Lixin Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Ark. Mat., Volume 44 (2006) no. 2, pp. 261-275 | DOI | MR | Zbl

[25] Guillarmou, Colin; Hassell, Andrew Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II, Ann. Inst. Fourier, Volume 59 (2009) no. 4, pp. 1553-1610 | DOI | Numdam | MR | Zbl

[26] Hytönen, Tuomas; van Neerven, Jan; Veraar, Mark; Weis, Lutz Analysis in Banach spaces. Vol. II. Probabilistic methods and operator theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 67, Springer, 2017, xxi+616 pages | DOI | MR | Zbl

[27] Ivanovici, Oana; Planchon, Fabrice Square function and heat flow estimates on domains, Commun. Partial Differ. Equations, Volume 42 (2017) no. 9, pp. 1447-1466 | DOI | MR | Zbl

[28] Kalton, Nigel J.; Weis, Lutz The H -calculus and sums of closed operators, Math. Ann., Volume 321 (2001) no. 2, pp. 319-345 | DOI | MR | Zbl

[29] Komatsu, Hikosaburo Fractional powers of operators, Pac. J. Math., Volume 19 (1966), pp. 285-346 | DOI | MR | Zbl

[30] Le Merdy, Christian On square functions associated to sectorial operators, Bull. Soc. Math. Fr., Volume 132 (2004) no. 1, pp. 137-156 | DOI | Numdam | MR | Zbl

[31] Lohoué, Noël Estimation des fonctions de Littlewood-Paley-Stein sur les variétés riemanniennes à courbure non positive, Ann. Sci. Éc. Norm. Supér., Volume 20 (1987) no. 4, pp. 505-544 | DOI | Numdam | MR | Zbl

[32] Ouhabaz, El Maati Analysis of heat equations on domains, London Mathematical Society Monographs, 31, Princeton University Press, 2005, xiv+284 pages | MR | Zbl

[33] Ouhabaz, El Maati Littlewood–Paley–Stein functions for Schrödinger operators, Front. Sci. Eng., Volume 6 (2016) no. 1, pp. 99-109 (See also https://arxiv.org/abs/1705.06794)

[34] Shen, Zhong Wei L p estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, Volume 45 (1995) no. 2, pp. 513-546 | DOI | Numdam | MR | Zbl

[35] Stein, Elias M. Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, 1970, xiv+290 pages | MR | Zbl

[36] Stein, Elias M. Topics in harmonic analysis related to the Littlewood–Paley theory, Annals of Mathematics Studies, 63, Princeton University Press; University of Tokyo Press, 1970, viii+146 pages | DOI | MR | Zbl

[37] Weis, Lutz A new approach to maximal L p -regularity, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998) (Lecture Notes in Pure and Applied Mathematics), Volume 215, Marcel Dekker, 2001, pp. 195-214 | MR | Zbl

Cited by Sources: