On a doubling metric measure space endowed with a “carré du champ”, let be the associated Markov generator and the corresponding homogeneous Sobolev space of order in , , with norm . We give sufficient conditions on the heat semigroup for the spaces to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and a second one through geometrical square functionals (relying on sharp estimates involving oscillations). A chain rule and a paralinearisation result are also given. In comparison with previous results ([29, 11]), the main improvements consist in the fact that we neither require any Poincaré inequalities nor -boundedness of Riesz transforms, but only -boundedness of the gradient of the semigroup. As a consequence, in the range , the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.
Sur un espace métrique mesuré doublant equipé d’un « carré du champ », soit le générateur markovien associé et l’espace de Sobolev homogène correspondant, d’ordre dans , , avec la norme . Nous donnons des conditions suffisantes sur le semi-groupe de la chaleur pour garantir que les espaces 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 des transformées de Riesz, mais seulement des bornitudes du gradient du semi-groupe. Comme conséquence, nous obtenons la propriété d’algèbre de Sobolev pour , sous la seule hypothèse d’estimations gaussiennes pour le noyau de la chaleur.
Accepted:
Published online:
DOI: 10.5802/jep.30
Keywords: Sobolev space, algebra property, heat semigroup
Mot clés : Espace de Sobolev, propriété d’algèbre, semi-groupe de la chaleur
Frédéric Bernicot 1; Thierry Coulhon 2; Dorothee Frey 3
@article{JEP_2016__3__99_0, author = {Fr\'ed\'eric Bernicot and Thierry Coulhon and Dorothee Frey}, title = {Sobolev algebras through heat kernel estimates}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {99--161}, publisher = {ole polytechnique}, volume = {3}, year = {2016}, doi = {10.5802/jep.30}, zbl = {1364.46029}, mrnumber = {3477866}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.30/} }
TY - JOUR AU - Frédéric Bernicot AU - Thierry Coulhon AU - Dorothee Frey TI - Sobolev algebras through heat kernel estimates JO - Journal de l’École polytechnique — Mathématiques PY - 2016 SP - 99 EP - 161 VL - 3 PB - ole polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.30/ DO - 10.5802/jep.30 LA - en ID - JEP_2016__3__99_0 ER -
%0 Journal Article %A Frédéric Bernicot %A Thierry Coulhon %A Dorothee Frey %T Sobolev algebras through heat kernel estimates %J Journal de l’École polytechnique — Mathématiques %D 2016 %P 99-161 %V 3 %I ole polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.30/ %R 10.5802/jep.30 %G en %F JEP_2016__3__99_0
Frédéric Bernicot; Thierry Coulhon; Dorothee Frey. Sobolev algebras through heat kernel estimates. Journal de l’École polytechnique — Mathématiques, Volume 3 (2016), pp. 99-161. doi : 10.5802/jep.30. https://jep.centre-mersenne.org/articles/10.5802/jep.30/
[1] - “Operator theory and harmonic analysis”, in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Austral. Nat. Univ., Canberra, 1996, p. 77-136 | MR
[2] - On necessary and sufficient conditions for -estimates of Riesz transforms associated to elliptic operators on and related estimates, Mem. Amer. Math. Soc., vol. 186, no. 871, American Mathematical Society, Providence, R.I., 2007 | DOI | Zbl
[3] - “Riesz transform on manifolds and Poincaré inequalities”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2005) no. 3, p. 531-555 | Numdam | Zbl
[4] - “Riesz transform on manifolds and heat kernel regularity”, Ann. Sci. École Norm. Sup. (4) 37 (2004) no. 6, p. 911-957 | DOI | Numdam | MR | Zbl
[5] - “The solution of the Kato square root problem for second order elliptic operators on ”, Ann. of Math. (2) 156 (2002) no. 2, p. 633-654 | DOI | MR | Zbl
[6] - “Vertical versus conical square functions”, Trans. Amer. Math. Soc. 364 (2012) no. 10, p. 5469-5489 | DOI | MR | Zbl
[7] - “Singular integral operators on tent spaces”, J. Evol. Equ. 12 (2012) no. 4, p. 741-765 | DOI | MR | Zbl
[8] - “Weighted norm inequalities, off-diagonal estimates and elliptic operators. I. General operator theory and weights”, Adv. Math. 212 (2007) no. 1, p. 225-276 | DOI | MR | Zbl
[9] - “Hardy spaces of differential forms on Riemannian manifolds”, J. Geom. Anal. 18 (2008) no. 1, p. 192-248 | DOI | MR | Zbl
[10] - Square root problem for divergence operators and related topics, Astérisque, vol. 249, Société Mathématique de France, Paris, 1998 | Numdam | Zbl
[11] - “Algebra properties for Sobolev spaces—applications to semilinear PDEs on manifolds”, J. Anal. Math. 118 (2012) no. 2, p. 509-544 | DOI | MR | Zbl
[12] - “A -theorem in relation to a semigroup of operators and applications to new paraproducts”, Trans. Amer. Math. Soc. 364 (2012) no. 11, p. 6071-6108 | DOI | MR | Zbl
[13] - “Gaussian heat kernel bounds through elliptic Moser iteration”, to appear in J. Math. Pures Appl., arXiv:1407.3906 | DOI | Zbl
[14] - “Pseudodifferential operators associated with a semigroup of operators”, J. Fourier Anal. Appl. 20 (2014) no. 1, p. 91-118 | DOI | MR | Zbl
[15] - “Riesz transforms through reverse Hölder and Poincaré inequalities” (2015), arXiv:1503.02508 | Zbl
[16] - “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 | DOI | Numdam | MR | Zbl
[17] - “New abstract Hardy spaces”, J. Funct. Anal. 255 (2008) no. 7, p. 1761-1796 | DOI | MR | Zbl
[18] - “Calderón-Zygmund theory for non-integral operators and the functional calculus”, Rev. Mat. Iberoamericana 19 (2003) no. 3, p. 919-942 | DOI | Zbl
[19] - “Algèbres de Sobolev sur certains groupes nilpotents”, J. Funct. Anal. 63 (1985) no. 3, p. 322-343 | DOI | MR | Zbl
[20] - “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 | DOI | Numdam | Zbl
[21] - “Réalisations des espaces de Besov homogènes”, Ark. Mat. 26 (1988) no. 1, p. 41-54 | DOI | Zbl
[22] - “Le calcul fonctionnel dans les espaces de Sobolev”, Invent. Math. 104 (1991) no. 2, p. 435-446 | DOI | MR | Numdam | Zbl
[23] - “A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces”, Adv. Math. 270 (2015), p. 302-374 | DOI | MR | Zbl
[24] - “Riesz transform and -cohomology for manifolds with Euclidean ends”, Duke Math. J. 133 (2006) no. 1, p. 59-93 | DOI | MR | Zbl
[25] - Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimates, Université Paris Sud - Paris XI; Australian national university, 2014, Ph. D. Thesis
[26] - Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 | Numdam | Zbl
[27] - “Some new function spaces and their applications to harmonic analysis”, J. Funct. Anal. 62 (1985) no. 2, p. 304-335 | DOI | MR | Zbl
[28] - “Riesz transforms for ”, Trans. Amer. Math. Soc. 351 (1999) no. 3, p. 1151-1169 | DOI
[29] - “Sobolev algebras on Lie groups and Riemannian manifolds”, Amer. J. Math. 123 (2001) no. 2, p. 283-342 | DOI | MR | Zbl
[30] - “Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem”, Proc. London Math. Soc. (3) 96 (2008) no. 2, p. 507-544 | DOI | Zbl
[31] - “Riesz meets Sobolev”, Colloq. Math. 118 (2010) no. 2, p. 685-704 | DOI | MR | Zbl
[32] - “Banach space operators with a bounded functional calculus”, J. Austral. Math. Soc. Ser. A 60 (1996) no. 1, p. 51-89 | DOI | MR | Zbl
[33] - “Some remarks on gradient estimates for heat kernels”, Abstr. Appl. Anal. (2006), Art. ID 73020, 10 pages | DOI | MR | Zbl
[34] - “Plancherel-type estimates and sharp spectral multipliers”, J. Funct. Anal. 196 (2002) no. 2, p. 443-485 | DOI | MR | Zbl
[35] - “Semigroup kernels, Poisson bounds, and holomorphic functional calculus”, J. Funct. Anal. 142 (1996) no. 1, p. 89-128 | DOI | MR | Zbl
[36] - “Some maximal inequalities”, Amer. J. Math. 93 (1971), p. 107-115 | DOI | MR | Zbl
[37] - “Calderón-Zygmund theory for operator-valued kernels”, Adv. Math. 62 (1986) no. 1, p. 7-48 | DOI | Zbl
[38] - “Paraproducts via -functional calculus”, Rev. Mat. Iberoamericana 29 (2013) no. 2, p. 635-663 | DOI | MR | Zbl
[39] - “A -theorem for non-integral operators”, Math. Ann. 357 (2013) no. 1, p. 215-278 | DOI | MR
[40] - “Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in ”, to appear in J. Anal. Math., arXiv:1407.4774 | Zbl
[41] - Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011 | MR | Zbl
[42] - “Besov algebras on Lie groups of polynomial growth”, Studia Math. 212 (2012) no. 2, p. 119-139 | DOI | MR | Zbl
[43] - Classical Fourier analysis, Graduate Texts in Math., vol. 249, Springer, New York, 2008 | MR | Zbl
[44] - “Stochastically complete manifolds”, Dokl. Akad. Nauk SSSR 290 (1986) no. 3, p. 534-537 | MR
[45] - “Gaussian upper bounds for the heat kernel on arbitrary manifolds”, J. Differential Geom. 45 (1997) no. 1, p. 33-52 | DOI | MR
[46] - “Exact smoothing properties of Schrödinger semigroups”, Amer. J. Math. 118 (1996) no. 6, p. 1215-1248 | Zbl
[47] - Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque, vol. 336, Société Mathématique de France, Paris, 2011 | Zbl
[48] - Sobolev met Poincaré, Mem. Amer. Math. Soc., vol. 145, no. 688, American Mathematical Society, Providence, R.I., 2000 | DOI | Zbl
[49] - “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 | DOI | MR | Zbl
[50] - “Kato’s square root problem in Banach spaces”, J. Funct. Anal. 254 (2008) no. 3, p. 675-726 | DOI | MR | Zbl
[51] - “Commutator estimates and the Euler and Navier-Stokes equations”, Comm. Pure Appl. Math. 41 (1988) no. 7, p. 891-907 | DOI | MR | Zbl
[52] - “On maximal regularity of type under minimal assumptions for elliptic non-divergence operators”, J. Funct. Anal. 255 (2008) no. 10, p. 2732-2759 | DOI | MR
[53] - “Maximal -regularity for parabolic equations, Fourier multiplier theorems and -functional calculus”, in Functional analytic methods for evolution equations, Lect. Notes in Math., vol. 1855, Springer, Berlin, 2004, p. 65-311 | DOI | MR | Zbl
[54] - “Operators which have an functional calculus”, in Miniconference on operator theory and partial differential equations (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, p. 210-231 | MR | Zbl
[55] - “On the Littlewood-Paley-Stein -function”, Trans. Amer. Math. Soc. 347 (1995) no. 6, p. 2201-2212 | DOI | MR | Zbl
[56] - “Remarques sur un théorème de J.-M. Bony”, Rend. Circ. Mat. Palermo (2) (1981), p. 1-20, suppl. 1 | Zbl
[57] - Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996 | DOI | MR | Zbl
[58] - Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002 | MR | Zbl
[59] - “Necessary conditions on composition operators acting on Sobolev spaces of fractional order. The critical case ”, Forum Math. 9 (1997) no. 3, p. 267-302 | DOI | MR | Zbl
[60] - “Necessary conditions on composition operators acting between Besov spaces. The case . II”, Forum Math. 10 (1998) no. 2, p. 199-231 | DOI | Zbl
[61] - “Necessary conditions on composition operators acting between Besov spaces. The case . III”, Forum Math. 10 (1998) no. 3, p. 303-327 | DOI | MR | Zbl
[62] - “Imaginary powers of Laplace operators”, Proc. Amer. Math. Soc. 129 (2001) no. 6, p. 1745-1754 (electronic) | DOI | MR | Zbl
[63] - “Interpolation of linear operators”, Trans. Amer. Math. Soc. 83 (1956), p. 482-492 | DOI | MR | Zbl
[64] - Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math. Studies, vol. 63, Princeton University Press, Princeton, N.J., 1970 | MR | Zbl
[65] - “Multipliers on fractional Sobolev spaces”, J. Math. Mech. 16 (1967), p. 1031-1060 | MR | Zbl
[66] - “Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and -Liouville properties”, J. reine angew. Math. 456 (1994), p. 173-196 | DOI | MR | Zbl
[67] - “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 | Zbl
[68] - Pseudodifferential operators and nonlinear PDE, Progress in Math., vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991 | DOI | MR | Zbl
Cited by Sources: