staple
Avec cedram.org
logo JEP
Table des matières de ce volume | Article précédent | Article suivant
Frédéric Bernicot; Thierry Coulhon; Dorothee Frey
Sobolev algebras through heat kernel estimates
(Algèbres de Sobolev via des estimations du noyau de la chaleur)
Journal de l'École polytechnique — Mathématiques, 3 (2016), p. 99-161, doi: 10.5802/jep.30
Article PDF | TeX source
Class. Math.: 46E35, 22E30, 43A15
Mots clés: Espace de Sobolev, propriété d’algèbre, semi-groupe de la chaleur

Résumé - Abstract

Sur un espace métrique mesuré doublant $(M,d,\mu )$ equipé d’un « carré du champ », soit $\mathcal{L}$ le générateur markovien associé et $\dot{L}^{p}_\alpha (M,\mathcal{L},\mu )$ l’espace de Sobolev homogène correspondant, d’ordre $0<\alpha <1$ dans $L^p$, $1<p<+\infty $, avec la norme $\Vert \mathcal{L}^{\alpha /2}f\Vert _p$. Nous donnons des conditions suffisantes sur le semi-groupe de la chaleur $(e^{-t\mathcal{L}})_{t>0}$ pour garantir que les espaces $\dot{L}^{p}_\alpha (M,\mathcal{L},\mu ) \cap L^\infty (M,\mu )$ sont des algèbres pour le produit ponctuel. Deux approches sont développées, une première utilisant des paraproduits (basée sur l’extrapolation pour obtenir leur bornitude) et une seconde basée sur des fonctionnelles quadratiques géométriques (basée sur la notion d’oscillation). Des règles de composition et de paralinéarisation sont aussi obtenues. En comparaison avec les résultats précédents ([29, 11]), les améliorations principales consistent dans le fait que nous n’avons plus à imposer d’inégalité de Poincaré ou de bornitude $L^p$ des transformées de Riesz, mais seulement des bornitudes $L^p$ du gradient du semi-groupe. Comme conséquence, nous obtenons la propriété d’algèbre de Sobolev pour $p\in (1,2]$, sous la seule hypothèse d’estimations gaussiennes pour le noyau de la chaleur.

Bibliographie

[1] D. Albrecht, X. T. Duong & A. McIntosh, Operator theory and harmonic analysis, Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ. 34, Austral. Nat. Univ., 1996, p. 77–136  MR 1394693 |  Zbl 0903.47010
[2] P. Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\mathbb{R}^n$ and related estimates, Mem. Amer. Math. Soc. 186, no. 871, American Mathematical Society, Providence, R.I., 2007 Article |  MR 2292385 |  Zbl 1221.42022
[3] P. Auscher & T. Coulhon, “Riesz transform on manifolds and Poincaré inequalities”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2005) no. 3, p. 531-555 Numdam |  MR 2185868 |  Zbl 1116.58023
[4] P. Auscher, T. Coulhon, X. T. Duong & S. Hofmann, “Riesz transform on manifolds and heat kernel regularity”, Ann. Sci. École Norm. Sup. (4) 37 (2004) no. 6, p. 911-957 Numdam |  MR 2119242 |  Zbl 1086.58013
[5] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh & Ph. Tchamitchian, “The solution of the Kato square root problem for second order elliptic operators on ${\mathbb{R}}^n$”, Ann. of Math. (2) 156 (2002) no. 2, p. 633-654 Article |  MR 1933726 |  Zbl 1128.35316
[6] P. Auscher, S. Hofmann & J. M. Martell, “Vertical versus conical square functions”, Trans. Amer. Math. Soc. 364 (2012) no. 10, p. 5469-5489 Article |  MR 2931335 |  Zbl 1275.42028
[7] P. Auscher, Ch. Kriegler, S. Monniaux & P. Portal, “Singular integral operators on tent spaces”, J. Evol. Equ. 12 (2012) no. 4, p. 741-765 Article |  MR 3000453 |  Zbl 1279.47072
[8] P. Auscher & J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights”, Adv. Math. 212 (2007) no. 1, p. 225-276 Article |  MR 2319768 |  Zbl 1213.42030
[9] P. Auscher, A. McIntosh & E. Russ, “Hardy spaces of differential forms on Riemannian manifolds”, J. Geom. Anal. 18 (2008) no. 1, p. 192-248 Article |  MR 2365673 |  Zbl 1217.42043
[10] P. Auscher & Ph. Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249, Société Mathématique de France, Paris, 1998  MR 1651262 |  Zbl 0909.35001
[11] N. Badr, F. Bernicot & E. Russ, “Algebra properties for Sobolev spaces—applications to semilinear PDEs on manifolds”, J. Anal. Math. 118 (2012) no. 2, p. 509-544 Article |  MR 3000690 |  Zbl 1286.46033
[12] F. Bernicot, “A $T(1)$-theorem in relation to a semigroup of operators and applications to new paraproducts”, Trans. Amer. Math. Soc. 364 (2012) no. 11, p. 6071-6108 Article |  MR 2946943 |  Zbl 1281.46028
[13] F. Bernicot, T. Coulhon & D. Frey, “Gaussian heat kernel bounds through elliptic Moser iteration”, to appear in J. Math. Pures Appl., arXiv:1407.3906
[14] F. Bernicot & D. Frey, “Pseudodifferential operators associated with a semigroup of operators”, J. Fourier Anal. Appl. 20 (2014) no. 1, p. 91-118 Article |  MR 3180890
[15] F. Bernicot & D. Frey, “Riesz transforms through reverse Hölder and Poincaré inequalities”, arXiv:1503.02508, 2015
[16] F. Bernicot & Y. Sire, “Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure”, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013) no. 5, p. 935-958 Article |  MR 3103176
[17] F. Bernicot & J. Zhao, “New abstract Hardy spaces”, J. Funct. Anal. 255 (2008) no. 7, p. 1761-1796 Article |  MR 2442082 |  Zbl 1171.42012
[18] S. Blunck & P. Ch. Kunstmann, “Calderón-Zygmund theory for non-integral operators and the $H^\infty $ functional calculus”, Rev. Mat. Iberoamericana 19 (2003) no. 3, p. 919-942 Article |  MR 2053568 |  Zbl 1057.42010
[19] G. Bohnke, “Algèbres de Sobolev sur certains groupes nilpotents”, J. Funct. Anal. 63 (1985) no. 3, p. 322-343 Article |  MR 808266 |  Zbl 0574.22008
[20] J.-M. Bony, “Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires”, Ann. Sci. École Norm. Sup. (4) 14 (1981) no. 2, p. 209-246 Numdam |  MR 631751 |  Zbl 0495.35024
[21] G. Bourdaud, “Réalisations des espaces de Besov homogènes”, Ark. Mat. 26 (1988) no. 1, p. 41-54 Article |  MR 948279 |  Zbl 0661.46026
[22] G. Bourdaud, “Le calcul fonctionnel dans les espaces de Sobolev”, Invent. Math. 104 (1991) no. 2, p. 435-446 Article |  MR 1098617 |  Zbl 0699.46019
[23] S. Boutayeb, T. Coulhon & A. Sikora, “A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces”, Adv. Math. 270 (2015), p. 302-374 Article |  MR 3286538 |  Zbl 1304.35314
[24] G. Carron, T. Coulhon & A. Hassell, “Riesz transform and $L^p$-cohomology for manifolds with Euclidean ends”, Duke Math. J. 133 (2006) no. 1, p. 59-93 Article |  MR 2219270 |  Zbl 1106.58021
[25] L. Chen, Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimates, Ph. D. Thesis, Université Paris Sud - Paris XI; Australian national university, tel-01001868, 2014  MR 3415628
[26] R. R. Coifman & Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57, Société Mathématique de France, Paris, 1978  MR 518170 |  Zbl 0483.35082
[27] R. R. Coifman, Y. Meyer & E. M. Stein, “Some new function spaces and their applications to harmonic analysis”, J. Funct. Anal. 62 (1985) no. 2, p. 304-335 Article |  MR 791851 |  Zbl 0569.42016
[28] T. Coulhon & X. T. Duong, “Riesz transforms for $1\le p\le 2$”, Trans. Amer. Math. Soc. 351 (1999) no. 3, p. 1151-1169 Article |  MR 1458299 |  Zbl 0973.58018
[29] T. Coulhon, E. Russ & V. Tardivel-Nachef, “Sobolev algebras on Lie groups and Riemannian manifolds”, Amer. J. Math. 123 (2001) no. 2, p. 283-342  MR 1828225 |  Zbl 0990.43003
[30] T. Coulhon & A. Sikora, “Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem”, Proc. London Math. Soc. (3) 96 (2008) no. 2, p. 507-544 Article |  MR 2396848 |  Zbl 1148.35009
[31] T. Coulhon & A. Sikora, “Riesz meets Sobolev”, Colloq. Math. 118 (2010) no. 2, p. 685-704 Article |  MR 2602174 |  Zbl 1194.58027
[32] M. Cowling, I. Doust, A. McIntosh & A. Yagi, “Banach space operators with a bounded $H^\infty $ functional calculus”, J. Austral. Math. Soc. Ser. A 60 (1996) no. 1, p. 51-89  MR 1364554 |  Zbl 0853.47010
[33] N. Dungey, “Some remarks on gradient estimates for heat kernels”, Abstr. Appl. Anal. (2006) Article |  MR 2211677 |  Zbl 1133.58019
[34] X. T. Duong, E. M. Ouhabaz & A. Sikora, “Plancherel-type estimates and sharp spectral multipliers”, J. Funct. Anal. 196 (2002) no. 2, p. 443-485 Article |  MR 1943098 |  Zbl 1029.43006
[35] X. T. Duong & D. W. Robinson, “Semigroup kernels, Poisson bounds, and holomorphic functional calculus”, J. Funct. Anal. 142 (1996) no. 1, p. 89-128 Article |  MR 1419418 |  Zbl 0932.47013
[36] C. Fefferman & E. M. Stein, “Some maximal inequalities”, Amer. J. Math. 93 (1971), p. 107-115  MR 284802 |  Zbl 0222.26019
[37] J. L. Rubio de Francia, F. J. Ruiz & J. L. Torrea, “Calderón-Zygmund theory for operator-valued kernels”, Adv. Math. 62 (1986) no. 1, p. 7-48 Article |  MR 859252 |  Zbl 0627.42008
[38] D. Frey, “Paraproducts via $H^\infty $-functional calculus”, Rev. Mat. Iberoamericana 29 (2013) no. 2, p. 635-663 Article |  MR 3047431 |  Zbl 1277.42020
[39] D. Frey & P. Ch. Kunstmann, “A $T(1)$-theorem for non-integral operators”, Math. Ann. 357 (2013) no. 1, p. 215-278 Article |  MR 3084347 |  Zbl 1280.42006
[40] D. Frey, A. McIntosh & P. Portal, “Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in $L^p$”, to appear in J. Anal. Math., arXiv:1407.4774
[41] M. Fukushima, Y. Oshima & M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics 19, Walter de Gruyter & Co., Berlin, 2011  MR 2778606 |  Zbl 1227.31001
[42] I. Gallagher & Y. Sire, “Besov algebras on Lie groups of polynomial growth”, Studia Math. 212 (2012) no. 2, p. 119-139 Article |  MR 3008437 |  Zbl 1264.22018
[43] L. Grafakos, Classical Fourier analysis, Graduate Texts in Math. 249, Springer, New York, 2008  MR 2445437 |  Zbl 1220.42001
[44] A. A. Grigorʼyan, “Stochastically complete manifolds”, Dokl. Akad. Nauk SSSR 290 (1986) no. 3, p. 534-537  MR 860324 |  Zbl 0632.58041
[45] A. A. Grigorʼyan, “Gaussian upper bounds for the heat kernel on arbitrary manifolds”, J. Differential Geom. 45 (1997) no. 1, p. 33-52  MR 1443330 |  Zbl 0865.58042
[46] A. Gulisashvili & M. A. Kon, “Exact smoothing properties of Schrödinger semigroups”, Amer. J. Math. 118 (1996) no. 6, p. 1215-1248  MR 1420922 |  Zbl 0864.47019
[47] P. Gyrya & L. Saloff-Coste, Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque 336, Société Mathématique de France, Paris, 2011  MR 2807275 |  Zbl 1222.58001
[48] P. Hajłasz & P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145, no. 688, American Mathematical Society, Providence, R.I., 2000 Article |  MR 1683160 |  Zbl 0954.46022
[49] T. Hytönen & M. Kemppainen, “On the relation of Carleson’s embedding and the maximal theorem in the context of Banach space geometry”, Math. Scand. 109 (2011) no. 2, p. 269-284  MR 2854692 |  Zbl 1251.46017
[50] T. Hytönen, A. McIntosh & P. Portal, “Kato’s square root problem in Banach spaces”, J. Funct. Anal. 254 (2008) no. 3, p. 675-726 Article |  MR 2381159 |  Zbl 1143.47013
[51] T. Kato & G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations”, Comm. Pure Appl. Math. 41 (1988) no. 7, p. 891-907 Article |  MR 951744 |  Zbl 0671.35066
[52] P. Ch. Kunstmann, “On maximal regularity of type $L^p\text{-}L^q$ under minimal assumptions for elliptic non-divergence operators”, J. Funct. Anal. 255 (2008) no. 10, p. 2732-2759 Article |  MR 2464190 |  Zbl 1165.47030
[53] P. Ch. Kunstmann & L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty $-functional calculus, Functional analytic methods for evolution equations, Lect. Notes in Math. 1855, Springer, 2004, p. 65–311 Article |  MR 2108959 |  Zbl 1097.47041
[54] A. McIntosh, Operators which have an $H_\infty $ functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 14, Austral. Nat. Univ., 1986, p. 210–231  MR 912940 |  Zbl 0634.47016
[55] S. Meda, “On the Littlewood-Paley-Stein $g$-function”, Trans. Amer. Math. Soc. 347 (1995) no. 6, p. 2201-2212 Article |  MR 1264824 |  Zbl 0854.42017
[56] Y. Meyer, “Remarques sur un théorème de J.-M. Bony”, Rend. Circ. Mat. Palermo (2) (1981), p. 1-20, suppl. 1  MR 639462 |  Zbl 0473.35021
[57] T. Runst & W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications 3, Walter de Gruyter & Co., Berlin, 1996 Article |  MR 1419319 |  Zbl 0873.35001
[58] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series 289, Cambridge University Press, Cambridge, 2002  MR 1872526 |  Zbl 0991.35002
[59] W. Sickel, “Necessary conditions on composition operators acting on Sobolev spaces of fractional order. The critical case $1<s<n/p$”, Forum Math. 9 (1997) no. 3, p. 267-302 Article |  MR 1441923 |  Zbl 0898.46032
[60] W. Sickel, “Necessary conditions on composition operators acting between Besov spaces. The case $1<s<n/p$. II”, Forum Math. 10 (1998) no. 2, p. 199-231 Article |  MR 1611955 |  Zbl 0914.46030
[61] W. Sickel, “Necessary conditions on composition operators acting between Besov spaces. The case $1<s<n/p$. III”, Forum Math. 10 (1998) no. 3, p. 303-327 Article |  MR 1619719 |  Zbl 0914.46030
[62] A. Sikora & J. Wright, “Imaginary powers of Laplace operators”, Proc. Amer. Math. Soc. 129 (2001) no. 6, p. 1745-1754 (electronic) Article |  MR 1814106 |  Zbl 0969.42007
[63] E. M. Stein, “Interpolation of linear operators”, Trans. Amer. Math. Soc. 83 (1956), p. 482-492  MR 82586 |  Zbl 0072.32402
[64] E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math. Studies 63, Princeton University Press, Princeton, N.J., 1970  MR 252961 |  Zbl 0193.10502
[65] R. S. Strichartz, “Multipliers on fractional Sobolev spaces”, J. Math. Mech. 16 (1967), p. 1031-1060  MR 215084 |  Zbl 0145.38301
[66] K.-T. Sturm, “Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties”, J. reine angew. Math. 456 (1994), p. 173-196 Article |  MR 1301456 |  Zbl 0806.53041
[67] K.-T. Sturm, “Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations”, Osaka J. Math. 32 (1995) no. 2, p. 275-312  MR 1355744 |  Zbl 0854.35015
[68] M. E. Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Math. 100, Birkhäuser Boston, Inc., Boston, MA, 1991 Article |  MR 1121019 |  Zbl 0746.35062