Catalogue et commandes en ligne (paiement sécurisé, VISA ou MASTERCARD uniquement)

Revues disponibles par abonnement

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Séries de livres

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Série Chaire Jean Morlet

SMF/AMS Texts and Monographs

La Série T

Fascicules « Journée Annuelle »

Autres livres

Donald E. Knuth - traductions françaises

Rééditions du Séminaire Nicolas Bourbaki

Rééditions des Œuvres de Jean Leray

Revue de l'Institut Elie Cartan

Editions électroniques

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

Plus d'information / Abonnement

Publications grand public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

Pour les auteurs

Soumission des manuscrits

Formats et documentation

Plus d'info

Liste de diffusion électronique (smf.emath.fr)

Information pour les libraires et diffuseurs (smf.emath.fr)

Publications de la SMF
fr en
Votre numéro IP : 54.198.164.83
Accès aux édit. élec. : SémCong

Annales scientifiques de l'ENS

Présentation de la publication

Parutions

Dernières parutions

Comité de rédaction / Secrétariat

Série 4 :
Série 3 :
Série 2 :
Série 1 :

Faire une recherche


Catalogue & commande

Annales scientifiques de l'ENS - Parutions - série 4, 44 (2011)

Parutions < série 4, 44

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 44, fascicule 5 (2011)

Steve Hofmann, Svitlana Mayboroda, Alan McIntosh
Second order elliptic operators with complex bounded measurable coefficients in L^p, Sobolev and Hardy spaces
Annales scientifiques de l'ENS 44, fascicule 5 (2011), 723-800

Télécharger cet article : Fichier PDF
Acheter l'ouvrage

Résumé :
Opérateurs elliptiques du second ordre à coefficients complexes mesurables bornés dans les espaces L^p, de Sobolev et de Hardy
Soit L un opérateur elliptique du second ordre de formes de divergence, à coefficients complexes bornés et mesurables. Les opérateurs associés à L tels que le semi-groupe de la chaleur ou la transformée de Riesz ne sont en général pas de type Calderón-Zygmund et présentent des comportements différents de leurs analogues construits à partir du laplacien. Cet article a pour objectif de décrire de manière exhaustive les propriétés de ces opérateurs dans L^p, dans les espaces de Sobolev ainsi que dans certains nouveaux espaces de Hardy naturellement associés à L. Tout d'abord, nous montrons que les plages de valeurs connues pour lesquelles ces opérateurs sont bornés en norme L^p sont strictes. En particulier, le semi-groupe de la chaleur et la transformée de Riesz ne sont pas obligatoirement bornés si p[2n/(n+2),2n/(n-2)]. Nous fournissons ensuite une description complète de tous les espaces de Sobolev pour lesquels L admet un calcul fonctionnel borné, en particulier, pour lesquels e^-tL est borné. Puis, nous développons une théorie extensive des espaces de Hardy et de Lipschitz associés à L, pour les valeurs de p hors de [2n/(n+2),2n/(n-2)]. Cette théorie comprend, en particulier, des caractérisations par la fonction maximale dièse et par la transformée de Riesz (pour certaines plages de p), ainsi que leur décomposition moléculaire, leur dualité et les théorèmes d'interpolation.

Mots-clefs : Espaces de Hardy et de Lipschitz, opérateurs elliptiques, coefficients complexes, semi-groupe de la chaleur, transformation de Riesz

Abstract:
Let L be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with L, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in L^p, Sobolev, and some new Hardy spaces naturally associated to L. First, we show that the known ranges of boundedness in L^p for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e^-tL need not be bounded in L^p if p[2n/(n+2),2n/(n-2)]. Then we provide a complete description of all Sobolev spaces in which L admits a bounded functional calculus, in particular, where e^-tL is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to L, that serves the range of p beyond [2n/(n+2),2n/(n-2)]. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of p), as well as the molecular decomposition and duality and interpolation theorems.

Keywords: Hardy and Lipschitz spaces, elliptic operators, complex coefficients, heat semigroup, Riesz transform

Class. math. : 42B30, 42B35, 42B25, 35J15


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Albrecht, David and Duong, Xuan and McIntosh, Alan
Operator theory and harmonic analysis
in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995)
Proc. Centre Math. Appl. Austral. Nat. Univ. 34 (1996) 77–136
Math Reviews MR1394696 (97e:47001)
Zentralblatt 903.47010
2
Alvarez, J. and Milman, M.
Spaces of Carleson measures: duality and interpolation
Ark. Mat. 25 (1987) 155–174
Math Reviews MR923404 (89a:46064)
Zentralblatt 638.42020
3
Alvarez, J. and Milman, M.
Interpolation of tent spaces and applications
in Function spaces and applications (Lund, 1986)
Lecture Notes in Math. 1302 (1988) 11–21
Math Reviews MR942254 (89i:46025)
Zentralblatt 662.46076
4
Auscher, Pascal
Some questions on elliptic operators
in Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002)
Contemp. Math. 338 (2003) 1–10
Math Reviews MR2039949 (2004m:35057)
Zentralblatt 1183.35093
5
Auscher, Pascal
On Lp estimates for square roots of second order elliptic operators on Rn
Publ. Mat. 48 (2004) 159–186
Math Reviews MR2044643 (2005m:35065)
6
Auscher, Pascal
On necessary and sufficient conditions for Lp-estimates of Riesz transforms associated to elliptic operators on Rn and related estimates
Mem. Amer. Math. Soc. 186 (2007)
Math Reviews MR2292385 (2007k:42025)
7
Auscher, Pascal and Coulhon, Thierry
Riesz transform on manifolds and Poincaré inequalities
Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005) 531–555
Math Reviews MR2185868 (2007c:58029)
8
Auscher, Pascal and Coulhon, Thierry and Tchamitchian, Philippe
Absence de principe du maximum pour certaines équations paraboliques complexes
Colloq. Math. 71 (1996) 87–95
Math Reviews MR1397370 (97e:35019)
Zentralblatt 960.35011
9
Auscher, Pascal and Duong, Xuan Thinh and McIntosh, Alan
Boundedness of Banach space valued singular integral operators and Hardy spaces
(2005)
10
Auscher, Pascal and Hofmann, Steve and Lacey, Michael and McIntosh, Alan and Tchamitchian, Ph.
The solution of the Kato square root problem for second order elliptic operators on Rn
Ann. of Math. 156 (2002) 633–654
Math Reviews MR1933726 (2004c:47096c)
11
Auscher, Pascal and McIntosh, Alan and Russ, Emmanuel
Hardy spaces of differential forms on Riemannian manifolds
J. Geom. Anal. 18 (2008) 192–248
Math Reviews MR2365673 (2009d:42053)
Zentralblatt pre05319669
12
Auscher, Pascal and Russ, Emmanuel
Hardy spaces and divergence operators on strongly Lipschitz domains of Rn
J. Funct. Anal. 201 (2003) 148–184
Math Reviews MR1986158 (2004c:42049)
13
Auscher, Pascal and Tchamitchian, Philippe
Calcul fontionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux)
Ann. Inst. Fourier (Grenoble) 45 (1995) 721–778
Math Reviews MR1340951 (96f:35036)
14
Auscher, Pascal and Tchamitchian, Philippe
Square root problem for divergence operators and related topics
Astérisque 249 (1998)
Math Reviews MR1651262 (2000c:47092)
Zentralblatt 909.35001
15
Bernal, Antonio
Some results on complex interpolation of Tpq spaces
in Interpolation spaces and related topics (Haifa, 1990)
Israel Math. Conf. Proc. 5 (1992) 1–10
Math Reviews MR1206486 (94b:46101)
16
Bernal, Antonio and Cerdà, J.
Complex interpolation of quasi-Banach spaces with an A-convex containing space
Ark. Mat. 29 (1991) 183–201
Math Reviews MR1150372 (93a:46137)
Zentralblatt 757.41031
19
Calderón, A.-P. and Torchinsky, A.
Parabolic maximal functions associated with a distribution. II
Advances in Math. 24 (1977) 101–171
Math Reviews MR0450888 (56 \#9180)
Zentralblatt 355.46021
20
Cohn, W. S. and Verbitsky, I. E.
Factorization of tent spaces and Hankel operators
J. Funct. Anal. 175 (2000) 308–329
Math Reviews MR1780479 (2001g:42047)
21
Coifman, Ronald R.
A real variable characterization of Hp
Studia Math. 51 (1974) 269–274
Math Reviews MR0358318 (50 \#10784)
22
Coifman, Ronald R. and Meyer, Y. and Stein, Elias M.
Some new function spaces and their applications to harmonic analysis
J. Funct. Anal. 62 (1985) 304–335
Math Reviews MR791851 (86i:46029)
23
Coifman, Ronald R. and Weiss, Guido
Extensions of Hardy spaces and their use in analysis
Bull. Amer. Math. Soc. 83 (1977) 569–645
Math Reviews MR0447954 (56 \#6264)
Zentralblatt 358.30023
24
Cwikel, M. and Milman, M. and Sagher, Y.
Complex interpolation of some quasi-Banach spaces
J. Funct. Anal. 65 (1986) 339–347
Math Reviews MR826431 (87h:46152)
Zentralblatt 586.46054
25
Davies, E. B.
Limits on Lp regularity of self-adjoint elliptic operators
J. Differential Equations 135 (1997) 83–102
Math Reviews MR1434916 (98i:47044)
Zentralblatt 871.35020
26
Duong, Xuan Thinh and Yan, Lixin
Duality of Hardy and BMO spaces associated with operators with heat kernel bounds
J. Amer. Math. Soc. 18 (2005) 943–973
Math Reviews MR2163867 (2006d:42037)
Zentralblatt 1078.42013
27
Duong, Xuan Thinh and Yan, Lixin
New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications
Comm. Pure Appl. Math. 58 (2005) 1375–1420
Math Reviews MR2162784 (2006i:26012)
Zentralblatt 1153.26305
28
Duong, Xuan Thinh and Xiao, Jie and Yan, Lixin
Old and new Morrey spaces with heat kernel bounds
J. Fourier Anal. Appl. 13 (2007) 87–111
Math Reviews MR2296729 (2007k:42027)
Zentralblatt 1133.42017
29
Duren, P. L. and Romberg, B. W. and Shields, A. L.
Linear functionals on Hp spaces with 0<p<1
J. reine angew. Math. 238 (1969) 32–60
Math Reviews MR0259579 (41 \#4217)
32
Fefferman, C. and Stein, Elias M.
Hp spaces of several variables
Acta Math. 129 (1972) 137–193
Math Reviews MR0447953 (56 \#6263)
34
Frehse, Jens
An irregular complex valued solution to a scalar uniformly elliptic equation
Calc. Var. Partial Differential Equations 33 (2008) 263–266
Math Reviews MR2429531 (2009h:35084)
Zentralblatt 1157.35026
35
García-Cuerva, José and Rubio de Francia, José L.
Weighted norm inequalities and related topics
North-Holland Publishing Co., 1985
Math Reviews MR807149 (87d:42023)
Zentralblatt 578.46046
36
Giaquinta, Mariano
Multiple integrals in the calculus of variations and nonlinear elliptic systems
Princeton Univ. Press, 1983
Math Reviews MR717034 (86b:49003)
Zentralblatt 516.49003
37
Gomez, M. E. and Milman, M.
Complex interpolation of Hp spaces on product domains
Ann. Mat. Pura Appl. 155 (1989) 103–115
Math Reviews MR1042830 (91g:46030)
Zentralblatt 712.46040
38
39
Hofmann, Steve and Martell, José María
Lp bounds for Riesz transforms and square roots associated to second order elliptic operators
Publ. Mat. 47 (2003) 497–515
Math Reviews MR2006497 (2004i:35067)
40
Hofmann, Steve and Mayboroda, Svitlana
Hardy and BMO spaces associated to divergence form elliptic operators
Math. Ann. 344 (2009) 37–116
Math Reviews MR2481054 (2009m:42038)
Zentralblatt 1162.42012
41
43
Janson, Svante and Jones, Peter W.
Interpolation between Hp spaces: the complex method
J. Funct. Anal. 48 (1982) 58–80
Math Reviews MR671315 (84c:46021)
44
Jiang, Renjin and Yang, Dachun
New Orlicz-Hardy spaces associated with divergence form elliptic operators
J. Funct. Anal. 258 (2010) 1167–1224
Math Reviews MR2565837 (2011e:42047)
Zentralblatt 1205.46014
45
Kalton, Nigel and Mayboroda, Svitlana and Mitrea, Marius
Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations
in Interpolation theory and applications
Contemp. Math. 445 (2007) 121–177
Math Reviews MR2381891 (2009g:46031)
Zentralblatt 1158.46013
46
Kalton, Nigel and Mitrea, Marius
Stability of Fredholm properties on interpolation scales of quasi-Banach spaces and applications
Trans. Amer. Math. Soc. 350 (1998) 3837–3901
Math Reviews MR1443193 (98m:46094)
Zentralblatt 902.46002
47
Latter, Robert H.
A characterization of Hp(<b>R</b>n) in terms of atoms
Studia Math. 62 (1978) 93–101
Math Reviews MR0482111 (58 \#2198)
48
Martell, José María
Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications
Studia Math. 161 (2004) 113–145
Math Reviews MR2033231 (2005b:42016)
Zentralblatt 1044.42019
49
Mayboroda, Svitlana
The connections between Dirichlet, regularity and Neumann problems for second order elliptic operators with complex bounded measurable coefficients
Adv. Math. 225 (2010) 1786–1819
Math Reviews MR2680190
Zentralblatt 1203.35087
50
Mazya, V. G. and Nazarov, S. A. and Plamenevski, B. A.
Absence of a De Giorgi-type theorem for strongly elliptic equations with complex coefficients
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982) 156–168
Math Reviews MR660079 (84j:35059)
51
McIntosh, Alan
Operators which have an H functional calculus
in Miniconference on operator theory and partial differential equations (North Ryde, 1986)
Proc. Centre Math. Anal. Austral. Nat. Univ. 14 (1986) 210–231
Math Reviews MR912940 (88k:47019)
Zentralblatt 634.47016
52
Mendez, Osvaldo and Mitrea, Marius
The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations
J. Fourier Anal. Appl. 6 (2000) 503–531
Math Reviews MR1781091 (2001k:46056)
Zentralblatt 972.46017
54
Šneberg, I. Ja.
Spectral properties of linear operators in interpolation families of Banach spaces
Mat. Issled. 9 (1974) 214–229, 254–255
Math Reviews MR0634681 (58 \#30362)
55
Stein, Elias M.
Singular integrals and differentiability properties of functions
Princeton Univ. Press, 1970
Math Reviews MR0290095 (44 \#7280)
Zentralblatt 207.13501
56
Stein, Elias M.
Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals
Princeton Univ. Press, 1993
Math Reviews MR1232192 (95c:42002)
Zentralblatt 821.42001
57
Stein, Elias M. and Weiss, Guido
On the theory of harmonic functions of several variables. I. The theory of Hp-spaces
Acta Math. 103 (1960) 25–62
Math Reviews MR0121579 (22 \#12315)
58
Taibleson, Mitchell H. and Weiss, Guido
The molecular characterization of certain Hardy spaces
Astérisque 77 (1980) 67–149
Math Reviews MR604370 (83g:42012)
Zentralblatt 472.46041
60
Wolff, Thomas H.
A note on interpolation spaces
in Harmonic analysis (Minneapolis, Minn., 1981)
Lecture Notes in Math. 908 (1982) 199–204
Math Reviews MR654187 (83f:46082)
Zentralblatt 517.46054
61
Yan, Lixin
Classes of Hardy spaces associated with operators, duality theorem and applications
Trans. Amer. Math. Soc. 360 (2008) 4383–4408
Math Reviews MR2395177 (2010f:42045)
Zentralblatt pre05308769