Racines carrées d’opérateurs elliptiques et espaces de Hardy
Confluentes Mathematici, Tome 3 (2011) no. 1, pp. 1-119.

Il est bien connu que, pour tout 1 < p < +∞, il existe Cp > 0 tel que

1inifLP(Rn)(Δ)1/2fLP(Rn)     (0.1)

pour toute fonction . Lorsque p = 1, (0.1) est fausse pour L1(ℝn) mais devient

1inifH1(Rn)(Δ)1/2fH1(Rn)  (0.2)

où H1(ℝn) est l'espace de Hardy réel classique. Dans cette vue d'ensemble, nous rassemblons des résultats qui étendent (0.1) et (0.2) dans deux directions. D'une part, nous nous plaçons dans un ouvert fortement lipschitzien de ℝn, ou dans un cadre géométrique non euclidien (variété riemannienne complète ou graphe). D'autre part, nous remplaçons Δ par un opérateur elliptique d'ordre 2 plus général (opérateur sous forme divergence dans un ouvert de ℝn, laplacien de Hodge dans une variété riemannienne, laplacien discret sur un graphe). Dans le cas des domaines fortement lipschitziens de ℝn, ces questions conduisent à introduire des espaces de Hardy–Sobolev et à en donner des propriétés analogues à celles des espaces de Sobolev usuels. Dans le cas des variétés riemanniennes, nous introduisons des espaces de Hardy de formes différentielles exactes. Ces espaces sont adaptés au laplacien de Hodge et possèdent des propriétés analogues à l'espace de Hardy H1(ℝn). Le laplacien de Hodge possède un calcul fonctionnel holomorphe sur ces espaces, et en particulier la transformée de Riesz est continue. Sous des hypothèses géométriques convenables, nous comparons ces espaces de Hardy aux espaces de Hardy usuels (dans le cas d'espaces de fonctions) ou aux espaces Lp. Enfin, sur un graphe vérifiant certaines propriétés géométriques, nous obtenons des versions discrètes de la plupart des résultats correspondants pour les transformées de Riesz et inégalités reliées obtenues dans des variétés riemanniennes. Nous donnons également des résultats concernant les puissances fractionnaires d'opérateurs elliptiques d'ordre 2. Il s'agit de propriétés d'algèbre pour des espaces de Bessel fractionnaires sur des groupes de Lie unimodulaires, généralisant les résultats euclidiens. Nous utilisons également des estimations de la norme L2 de puissances fractionnaires de certains opérateurs pour montrer une inégalité de Poincaré fractionnaire pour certaines mesures de probabilité dans ℝn ou des groupes de Lie à croissance polynomiale.

Publié le :
DOI : 10.1142/S1793744211000278

Emmanuel Russ 1

1
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Emmanuel Russ. Racines carrées d’opérateurs elliptiques et espaces de Hardy. Confluentes Mathematici, Tome 3 (2011) no. 1, pp. 1-119. doi : 10.1142/S1793744211000278. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000278/

[1] R. Adams et J. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Vol. 140 (Elsevier/Academic Press, 2003).

[2] D. R. Adams et L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss., Vol. 314 (Springer, 1996).

[3] V. Adolfsson et D. Jerison, Lp -integrability of the second order derivatives for the Neumann problem in convex domains, Indiana Univ. Math. J. 43 (1993) 1123–1138.

[4] G. Alexopoulos, An application of homogenization theory to harmonic analysis : Harnack inequalities and Riesz transforms on Lie groups with polynomial volume growth, Can. J. Math. 44 (1992) 691–727.

[5] G. Alexopoulos, Random walks on discrete groups of polynomial volume growth, Ann. Probab. 30 (2002) 723–801.

[6] J. M. Angeletti, S. Mazet et P. Tchamitchian, Analysis of second order elliptic operators without boundary conditions and with V MO or Hölderian coefficients, in Multiscale Wavelet Methods for PDE’s, eds. W. Dahmen, A. J. Kurdila et P. Oswald (Academic Press, 1997), pp. 495–539.

[7] D. Aronson, Bounds for fundamental solutions of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967) 890–896.

[8] N. Aronszajn et K. T. Smith, Theory of Bessel potentials I, Ann. Inst. Fourier 11 (1961) 385–475.

[9] G. Auchmuty et J. Alexander, L2 well-posedness of planar div-curl systems, Arch. Rational Mech. Anal. 160 (2001) 91–134.

[10] G. Auchmuty et J. Alexander, L2 -well-posedness of 3d div-curl boundary value problems, Quart. Appl. Math. 63 (2005) 479–508.

[11] P. Auscher, Extrapolation pour les mesures de Carleson et conjecture de Kato, in Séminaire : Equations aux Dérivées Partielles, 2000-2001, Exp. No. I (Sémin. Equ. Dériv. Partielles, Ecole Polytech., Palaiseau, 2001).

[12] P. Auscher, On Lp estimates for square roots of second order elliptic operators on Rn , Publ. Mat. 48 (2004) 159–186.

[13] P. Auscher, On Necessary and Sufficient Conditions for Lp Estimates of Riesz Transforms Associated to Elliptic Operators on Rn and Related Estimates, Mem. Amer. Math. Soc., Vol. 186 (Amer. Math. Soc., 2007).

[14] P. Auscher et T. Coulhon, Gaussian lower bounds for random walks from elliptic regularity, Ann. Inst. H. Poincaré Probab. Statist. 35 (1999) 605–630.

[15] P. Auscher et T. Coulhon, Riesz transforms on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 531–555.

[16] P. Auscher, T. Coulhon, X. T. Duong et S. Hofmann, Riesz transforms on manifolds and heat kernel regularity, Ann. Sci. Ecole Norm. Sup. 37 (2004) 911–957.

[17] P. Auscher, T. Coulhon et P. Tchamitchian, Absence de principe du maximum pour certaines équations paraboliques complexes, Coll. Math. 171 (1996) 87–95.

[18] P. Auscher, X. T. Duong et A. McIntosh, Boundedness of Banach space valued singular integral operators and applications to Hardy spaces, unpublished manuscript.

[19] P. Auscher, S. Hofmann, M. Lacey, J. Lewis, A. McIntosh et P. Tchamitchian, The solution of Kato’s conjectures, C. R. Acad. Sci. Paris, Ser. I Math. 332 (2001) 601–606.

[20] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh et P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn , Ann. Math. 15 (2002) 633–654.

[21] P. Auscher et J.-M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I : General operator theory and weights, Adv. Math. 212 (2007) 225–276.

[22] P. Auscher et J.-M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators : Part IV : Riesz transforms on manifolds and weights, Math. Z. 260 (2008) 527–539.

[23] P. Auscher, A. McIntosh et E. Russ, Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal. 18 (2008) 192–248.

[24] P. Auscher, S. Monniaux et P Portal, The maximal regularity operator on tent spaces, arXiv :1011.1748.

[25] P. Auscher et M. Qafsaoui, Observations on W 1,p estimates for divergence elliptic equations with V MO coefficients, Boll. Unione Mat. Ital. Sez. B, Artic. Ric. Mat. 8 (2002) 487–509.

[26] P. Auscher et E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of Rn , arXiv :math/0201301v1.

[27] P. Auscher et E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of Rn , J. Funct. Anal. 201 (2003) 148–184.

[28] P. Auscher, E. Russ et P. Tchamitchian, Une note sur les lemmes div-curl, C. R. Acad. Sci. Paris Sér I Math. 337 (2003) 511–516.

[29] P. Auscher, E. Russ et P. Tchamitchian, Hardy Sobolev spaces on strongly Lipschitz domains of Rn , J. Funct. Anal. 218 (2005) 54–109.

[30] P. Auscher et P. Tchamitchian, Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Inst. Fourier 45 (1995) 721–778.

[31] P. Auscher et P. Tchamitchian, Square Root Problem for Divergence Operators and Related Topics, Astérisque, Vol. 249 (Soc. Math. France, 1998).

[32] P. Auscher et P. Tchamitchian, Gaussian estimates for second order elliptic divergence operators on Lipschitz and C1 domains, in Evolution Equations and their Applications in Physical and Life Sciences, Lecture Notes in Pure and Applied Mathematics, Vol. 215, eds. G. Lumer et L. Weis (Marcel Dekker, 2001), pp. 15–32.

[33] P. Auscher et P. Tchamitchian, Square roots of elliptic second order divergence operators on strongly Lipschitz domains : Lp theory, Math. Ann. 320 (2001) 577–623.

[34] P. Auscher et P. Tchamitchian, Square roots of second order elliptic divergence operators on strongly Lipschitz domains : L2 theory, J. Anal. Math. 90 (2003) 1–12.

[35] A. Axelsson, S. Keith et A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006) 455–497.

[36] A. Axelsson, S. Keith et A. McIntosh, The Kato square root problem for mixed boundary value problems, J. London Math. Soc. (2) 74 (2006) 113–130.

[37] N. Badr et B. Ben Ali, Lp boundedness of Riesz transform related to Schrödinger operators on a manifold, Ann. Scuola. Norm. Sup. di Pisa 8 (2009) 725–765.

[38] N. Badr et F. Bernicot, Abstract Hardy–Sobolev spaces and interpolation, J. Funct. Anal. 259 (2010) 1169–1208.

[39] N. Badr et F. Bernicot, A new Calder´on–Zygmund decompositions for Sobolev spaces, Coll. Math. 121 (2010) 153–177.

[40] N. Badr et G. Dafni, An atomic decomposition of the Hajlasz–Sobolev space M1 1 on manifolds, J. Funct. Anal. 259 (2010) 1380–1420.

[41] N. Badr et G. Dafni, Maximal characterization of Hardy–Sobolev spaces on mani- folds, to appear in Proc. Int. Workshop at Boca Raton.

[42] N. Badr et J.-M. Martell, Weighted norm inequalities on graphs, preprint.

[43] N. Badr et E. Russ, Interpolation of Sobolev spaces, Littlewood–Paley inequalities and Riesz transforms on graphs, Publ. Mat. 53 (2009) 273–328.

[44] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques, Seconde partie : étude sous la condition Γ2 ≥ 0, in Séminaire de Probabilités XIX, Lect. Notes, Vol. 1123 (Springer, 1985), pp. 145–174.

[45] D. Bakry, Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, in Séminaire de Probabilités, XXI, Lect. Notes in Math., Vol. 1247 (Springer, 1987), pp. 137–172.

[46] D. Bakry, The Riesz transforms associated with second order differential operators, in Seminar on Stochastic Processes, Vol. 88 (Birkhäuser, 1989).

[47] D. Bakry, F. Barthe, P. Cattiaux et A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Electron. Commun. Probab. 13 (2008) 60–66.

[48] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976) 337–403.

[49] R. Banuelos, Martingale transforms and related singular integrals, Trans. Amer. Math. Soc. 293 (1986) 547–564.

[50] L. Baratchart, J. Leblond, S. Rigat et E. Russ, Hardy spaces of the conjugate Beltrami equation, J. Funct. Anal. 259 (2010) 384–427.

[51] M. T. Barlow et R. F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999) 673–744.

[52] M. T. Barlow et E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theor. Relat. Fields 79 (1988) 543–623.

[53] O. Ben-Bassat, R. Strichartz et A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal. 166 (1999) 197–217.

[54] F. Bernicot, Use of abstract Hardy spaces, real interpolation and applications to bilinear operators, Math. Z. 265 (2010) 365–400.

[55] F. Bernicot et J. Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008) 1761–1796.

[56] J. J. Betancor, A. Chicco Ruiz, J. C. Farina et L. Rodriguez-Mesa, Odd BMO(R) functions and Carleson measures in the Bessel setting, Int. Eq. Op. Theory 66 (2010) 463–494.

[57] R. Bishop et R. Crittenden, Geometry of Manifolds (Academic Press, 1964).

[58] G. Bohnke, Algèbres de Sobolev sur certains groupes nilpotents, J. Funct. Anal. 63 (1985) 322–343.

[59] A. Bonami, J. Feuto et S. Grellier, Endpoint for the div-curl lemma in Hardy spaces, Publ. Mat. 54 (2010) 341–358.

[60] A. Bonami, T. Iwaniec, P. Jones et M. Zinsmeister, On the product of functions in BMO and H1 , Ann. Inst. Fourier 57 (2007) 1405–1439.

[61] J.-M. Bouclet, Low frequency estimates for long range perturbations in divergence form, arXiv :0806.3377.

[62] G. Bourdaud, Réalisation des espaces de Besov homogènes, Ark. Mat. 26 (1988) 41–54.

[63] G. Bourdaud, Remarques sur certains sous-espaces de BMO(Rn) et de bmo(Rn), Ann. Inst. Fourier 52 (2002) 1187–1218.

[64] J. Bourgain, Vector-valued singular integrals and the H1 − BMO duality, in Proba- bility Theory and Harmonic Analysis, Monogr. Textbooks Pure Appl. Math., Vol. 98 (Dekker, 1986) (Cleveland, Ohio, 1983), pp. 1–19.

[65] J. Bourgain et H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris 338 (2004) 539–543.

[66] S. C. Brenner et L. R. Scott, The Mathematical Theory of Finite Elements Methods (Springer, 1994).

[67] M. Briane et J. Casado-Diaz, Two-Dimensional Div-Curl Results : Application to the Lack of Nonlocal Effects in Homogenization, Comm. Partial Diff. Eqn. 32 (2007) 935–969.

[68] M. Briane, J. Casado-Diaz et F. Murat, The div-curl lemma “trente ans après”, an extension and an application to the G-convergence of unbounded monotone opera- tors, J. Math. Pures Appl. 91 (2009) 476–494.

[69] A. P. Calder´on, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math. 4 (1961) 33–49.

[70] G. Carbonaro, G. Mauceri et S. Meda, H1 and BMO for certain nondoubling metric measure spaces, Ann. Sc. Norm. Super. Pisa 8 (2009) 543–582.

[71] A. Carbonaro, A. McIntosh et A. Morris, Local Hardy spaces of differential forms on Riemannian manifolds, arXiv :1004.0018.

[72] L. Carleson, An explicit unconditional basis in H1 , Bull. Sci. Math. 104 (1980) 405–416.

[73] G. Carron, Formes harmoniques L2 sur les variétés non-compactes, Rend. Mat. Appl. 7 (2001) 87–119.

[74] G. Carron, Riesz transforms on connected sums, Ann. Inst. Fourier 57 (2007) 2329–2344.

[75] G. Carron, T. Coulhon et A. Hassell, Riesz transform and Lp cohomology for manifolds with Euclidean ends, Duke Math. J. 133 (2006) 59–93.

[76] D. Chafaï, Entropies, convexity, and functional inequalities : on Φ-entropies and Φ-Sobolev inequalities, J. Math. Kyoto Univ. 44 (2004) 325–363.

[77] D.-C. Chang, The dual of Hardy spaces on a bounded domain in Rn , Forum Math. 6 (1994) 65–81.

[78] D.-C. Chang, G. Dafni et C. Sadosky, A div-curl lemma in BMO on a domain, in Harmonic Analysis, Signal Processing and Complexity, Progress in Math., Vol. 238 (Birkhäuser, 2005), pp. 55–65.

[79] D.-C. Chang, G. Dafni et E. M. Stein, Hardy spaces, BMO and boundary value problems for the Laplacian on a smooth domain in RN , Trans. Amer. Math. Soc.

[80] (1999) 1605–1661.

[81] D.-C. Chang, S. G. Krantz et E. M. Stein, Hp theory on a smooth domain in RN and elliptic boundary value problems, J. Funct. Anal. 114 (1993) 286–347.

[82] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428–517.

[83] J.-C. Chen, Heat kernels on positively curved manifolds and applications, Ph.D. Thesis (Hangzhou University, 1987).

[84] Z. Q. Chen et T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stoch. Process. Appl. 108 (2003) 27–62.

[85] Y. K. Cho et J. Kim, Atomic decomposition on Hardy–Sobolev spaces, Studia Math.

[86] (2006) 25–42.

[87] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601–628.

[88] M. Christ et D. Müller, On lp spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (1996) 860–876.

[89] P. G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, 1978).

[90] R. Coifman, A real-variable characterization of Hp , Studia Math. 51 (1974) 269–274.

[91] R. Coifman et L. Grafakos, Hardy space estimates for multilinear operators I, Rev. Mat. Iber. 8 (1992) 45–67.

[92] R. Coifman, P.-L. Lions, Y. Meyer et S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. 9 (1993) 247–286.

[93] R. R. Coifman et Y. Meyer, Au-delà des Opérateurs Pseudo-Différentiels, Astérisque, Vol. 57 (Soc. Math. France, 1978).

[94] R. R. Coifman et Y. Meyer, Nonlinear harmonic analysis, operator theory and PDE, in Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., Vol. 112 (Princeton Univ. Press, 1986), pp. 3–45.

[95] R. Coifman, Y. Meyer et E. M. Stein, Some new function spaces and their applica- tions to harmonic analysis, J. Funct. Anal. 62 (1985) 304–335.

[96] R. Coifman et G. Weiss, Analyse Harmonique Non Commutative sur Certains Espaces Homogènes, Lect. Notes Math., Vol. 242 (Springer-Verlag, 1971).

[97] R. Coifman et G. Weiss, Transference Methods in Analysis (Amer. Math. Soc., 1976).

[98] R. Coifman et G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569–645.

[99] T. Coulhon, Random walks and geometry on infinite graphs, in Lectures Notes on Analysis in Metric Spaces (Appunti Corsi Tenuti Docenti Sc., Scuola Norm. Sup., Pisa, 2000) (Trento, 1999), pp. 5–36.

[100] T. Coulhon et X. T. Duong, Riesz transforms for 1 ≤ p ≤ 2, Trans. Amer. Math. Soc. 351 (1999) 1151–1169.

[101] T. Coulhon et X. T. Duong, Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. Pure Appl. Math. 56 (2003) 1728–1751.

[102] T. Coulhon, X. T. Duong et X. Li, Littlewood–Paley–Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2, Studia Math. 154 (2003) 37–57.

[103] T. Coulhon et A. Grigor’yan, Random walks on graphs with regular volume growth, Geom. Funct. Anal. 8 (1998) 656–701.

[104] T. Coulhon et H. Q. Li, Arch. Math. 83 (2004) 229–242.

[105] T. Coulhon, E. Russ et V. Tardivel-Nachef, Sobolev algebras on Lie groups and Riemannian manifolds, Amer. J. Math. 123 (2001) 283–342.

[106] T. Coulhon et L. Saloff-Coste, Puissances d’un opérateur régularisant, Ann. Inst. H. Poincaré Probab. Statist. 26 (1990) 419–436.

[107] T. Coulhon et L. Saloff-Coste, Semi-groupes d’opérateurs et espaces fonctionnels sur les groupes de Lie, J. Approx. Th. 65 (1991) 176–199.

[108] T. Coulhon et L. Saloff-Coste, Variétés Riemanniennes isométriques à l’infini, Rev. Mat. Iber. 11 (1995) 687–726.

[109] T. Coulhon et Q. S. Zhang, Large time behaviour of heat kernels on forms, J. Differential Geom. 77 (2007) 353–384.

[110] D. Cruz-Uribe et C. Rios, The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds, arXiv :0907.2947.

[111] M. Cwikel et Y. Sagher, Relations between real and complex interpolation spaces, Indiana Univ. Math. J. 36 (1987) 905–912.

[112] G. Dafni, Nonhomogeneous div-curl lemmas and local Hardy spaces, Adv. Diff. Eqns.

[113] (2005) 505–526.

[114] K. Dalrymple, R. Strichartz et J. P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl. 5 (1999) 203–284.

[115] G. David et S. Semmes, Analysis of and on Uniformly Rectifiable Sets, Mathematical Surveys and Monographs, Vol. 38 (Amer. Math. Soc., 1993).

[116] E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992) 99–119.

[117] E. B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141–169.

[118] T. Delmotte, Inégalité de Harnack elliptique sur les graphes, Coll. Math. 72 (1997) 19–37.

[119] T. Delmotte, Parabolic Harnack inequality, Rev. Mat. Iber. 15 (1999) 181–232.

[120] D. Deng, X. T. Duong, A. Sikora et L. Yan, Comparison of the classical BMO with the BMO spaces associated with operators and applications, Rev. Iber. Mat. 24 (2008) 267–296.

[121] D. Deng, L. Song, C. Tan et L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains, J. Geom. Anal. 17 (2007) 455–483.

[122] G. De Rham, Variétés Différentiables, Formes, Courants, Formes Harmoniques (Hermann, 1973).

[123] L. Desvillettes, C. Mouhot et C. Villani, Celebrating Cercignani’s conjecture for the Boltzmann equation, arXiv :1009.4006.

[124] B. Devyver, A Gaussian estimate for the heat Kernel on differential forms and application to the Riesz transform, arXiv :1011.5036.

[125] J.-D. Deuschel et D. Stroock, Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models, J. Funct. Anal. 92 (1990) 30–48.

[126] S. Dobyinsky, Lemme div-curl et renormalisation du produit, J. Math. Pures Appl. 9 (1993) 239–245.

[127] P. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc. 95 (1985) 21–31.

[128] N. Dungey, Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z. 247 (2004) 765–794.

[129] N. Dungey, Riesz transforms on a discrete group of polynomial growth, Bull. London Math. Soc. 36 (2004) 833–840.

[130] N. Dungey, A Littewood–Paley–Stein estimate on graphs and groups, Studia Math.

[131] (2008) 113–129.

[132] J. Duoandikoetxea et J. R. Rubio da Francia, Estimations indépendantes de la dimension pour les transformées de Riesz, C. R. Acad. Sci. Paris 300 (1985) 193–196.

[133] X. T. Duong, S. Hofmann, D. Mitrea, M. Mitrea et L. Yan, Hardy spaces and regu- larity for the inhomogeneous Dirichlet and Neumann problems, preprint.

[134] X. T. Duong et A. McIntosh, The Lp boundedness of Riesz transforms associ- ated with divergence form operators, in Workshop on Analysis and Applications, Brisbane, 1997, Proc. of the CMA, Vol. 37 (Australian National Univ., 1999), pp. 15–25.

[135] X. T. Duong et A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iber. 15 (1999) 233–265.

[136] X. T. Duong et L. X. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005) 943–973.

[137] X. T. Duong et L. X. Yan, New function spaces of BMO type, the John–Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005) 1375–1420.

[138] R. Dur´an, Error estimates for anisotropic finite elements and applications, in Proc. Int. Congress of Mathematicians III (Eur. Maty. Soc. Zürich, 2006), pp. 1181–1200.

[139] R. G. Dur´an, M. A. Muschietti, E. Russ et P. Tchamitchian, Divergence operator and Poincaré inequalities on arbitrary domains of Rn , Complex. Var. Elliptic Eqs.

[140] (2010) 795–816.

[141] P. L. Duren, Theory of Hp Spaces, Pure and Applied Math., Vol. 38 (Academic Press, 1970).

[142] J. Dziub´anski, Atomic decomposition of Hp spaces associated with some Schrödinger operators, Indiana. Univ. Math. J. 47 (1998) 75–98.

[143] J. Dziub´anski, Spectral multipliers for Hardy spaces associated with Schrödinger operators with polynomial potentials, Bull. Lon. Math. Soc. 32 (2000) 571–581.

[144] J. Dziub´anski, G. Garrig´os, T. Martinez, J. L. Torrea et J. Zienkiewicz, BMO spaces related to Schrödinger operator with potential satisfying reverse Hölder inequality, Math. Z. 249 (2005) 329–356.

[145] J. Dziub´anski et J. Zienkiewicz, Hardy spaces associated with some Schrödinger operators, Studia. Math. 126 (1997) 149–160.

[146] J. Dziub´anski et J. Zienkiewicz, Hardy spaces H1 for Schrödinger operators with compactly supported potentials, Ann. Math. 184 (2005) 315–326.

[147] A. F. M. ter Elst, D. Robinson et A. Sikora, Heat kernels and Riesz transforms on nilpotent Lie groups, Coll. Math. 74 (1997) 191–218.

[148] A. F. M. ter Elst, D. Robinson et Y. Zhu, Hardy spaces on Lie groups of polynomial growth, Sci. China Math. 53 (2010) 23–40.

[149] L. C. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991) 101–113.

[150] L. C. Evans et S. Müller, Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity, J. Amer. Math. Soc. 7 (1994) 199–219.

[151] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math.

[152] (1970) 9–36.

[153] C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971) 587–588.

[154] C. Fefferman et E. M. Stein, On some maximal inequalities, Amer. J. Math. 93 (1971) 107–115.

[155] C. Fefferman et E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972) 137–195.

[156] P. J. Fitzsimmons, B. M. Hambly et T. Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Commun. Math. Phys. 165 (1994) 595– 620.

[157] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975) 161–207.

[158] G. B. Folland et E. M. Stein, Hardy Spaces on Homogeneous Groups (Princeton Univ. Press and Univ. of Tokyo Press, 1982).

[159] M. Frazier et B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990) 34–170.

[160] J. Frehse, An irregular complex valued solution to a scalar uniformly elliptic equation, Calc. Var. Partial Differential Equations 33 (2008) 263–266.

[161] K. Fukuya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87 (1987) 517–547.

[162] M. Fukushima et T. Shima, On a spectral analysis for the Sierpinski gasket, Pot. Anal. 1 (1992) 1–35.

[163] M. P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959) 1–11.

[164] I. Gallagher et Y. Sire, Besov algebras on Lie groups of polynomial volume growth and related results, arXiv :1010.0154.

[165] J. Garcia-Cuerva et J. L. Rubio da Francia, Weighted Norm Inequalities and Related Topics, North Holland Mathematics Studies (Elsevier, 1985).

[166] J. B. Garnett, Bounded Analytic Functions, Pure and Applied Math., Vol. 96 (Academic Press, 1981).

[167] G. Gaudry, T. Qian et P. Sjögren, Singular integrals associated to the Laplacian on the affine group ax + b, Ark. Mat. 30 (1992) 259–281.

[168] G. Gaudry et P. Sjögren, Singular integrals on Iwasawa NA groups of rank 1, J. Reine Angew. Math. 479 (1996) 39–66.

[169] G. Gaudry et P. Sjögren, Haar-like expansions and and boundedness of a Riesz operator on a solvable Lie group, Math. Z. 232 (1999) 241–256.

[170] F. W. Gehring, The Lp integrability of the partial derivative of a quasi-conformal mapping, Acta Math. 130 (1973) 265–277.

[171] I. Gentil et C. Imbert, The Lévy–Fokker–Planck equation : Φ-entropies and conver- gence to equilibrium, Asympt. Anal. 59 (2008) 125–138.

[172] G. Geymonat, S. Müller et N. Triantafyllidis, Homogenization of nonlinear elastic materials, microscopic bifurcation and microscopic loss of rank-one convexity, Arch. Rational Mech. Anal. 122 (1993) 231–290.

[173] J. E. Gilbert, J. A. Hogan et J. D. Lakey, Atomic decomposition of divergence-free Hardy spaces, Special Vol., Proc. 5th IWAA, Math. Moravica (1997), pp. 33–52.

[174] J. Gilbert et J. Lakey, On a characterization of the local Hardy space by Gabor frames, in Wavelets, Frames and Operator Theory, Contemp. Math., Vol. 345 (Amer. Math. Soc., 2004), pp. 153–161.

[175] S. Giulini et P. Sjögren, A note on maximal functions on a solvable Lie group, Arch. Mat. (Basel) 55 (1990) 156–160.

[176] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979) 27–42.

[177] V. Gol’dshtein et M. Troyanov, Axiomatic theory of Sobolev spaces, Expo. Math. 19 (2001) 289–336.

[178] R. Gong et J. Li, Hardy–Sobolev spaces on product domains and applications, J. Math. Anal. Appl. 377 (2011) 296–302.

[179] L. Grafakos, Hardy space estimates for multilinear operators II, Rev. Mat. Iber. 8 (1992) 69–92.

[180] R. I. Grigorchuk, On Milnor’s problem on group growth, Sov. Math. Dokl. 28 (1983) 23–26.

[181] R. I. Grigorchuk, The growth degrees of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk. SSSR Ser. Mat. 48 (1984) 939–985.

[182] R. I. Grigorchuk, Degrees of growth of -groups and torsion free groups, Mat. Sb. (N.S.) 126 (1985) 194–214.

[183] A. Grigor’yan, The heat equation on a non-compact Riemannian manifold, Math. USSR Sb. 72 (1992) 47–77.

[184] A. Grigor’yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh 124A (1994) 353–362.

[185] A. Grigor’yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom. 45 (1997) 33–52.

[186] C. Guillarmou et A. Hassell, Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I., Math. Ann. 341 (2008) 859–896.

[187] C. Guillarmou et A. Hassell, Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II., Ann. Inst. Fourier

[188] (2009) 1553–1610.

[189] Y. Guivarc’h, Croissance polynomiale et période des fonctions harmoniques, Bull. Soc. Math. France 101 (1973) 333–379.

[190] A. Gulisashvili et M. A. Kon, Exact smoothing properties of Schrödinger semigroups, Amer. J. Math. 118 (1996) 1215–1248.

[191] R. Gundy et N. Varopoulos, Les transformations de Riesz et les intégrales stochastiques, C. R. Math. Acad. Sci. Paris 289 (1979) 13–16.

[192] P. Hajlasz et J. Kinnunen, Hölder quasicontinuity of Sobolev functions on metric spaces, Rev. Mat. Iber. 14 (1998) 601–622.

[193] P. Hajlasz et P. Koskela, Sobolev Met Poincaré, Mem. Amer. Math. Soc., Vol. 145 (Amer. Math. Soc., 2000).

[194] B. M. Hambly et T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. Lond. Math. Soc. (3) 78 (1999) 431–458.

[195] Y. S. Han, M. Paluszynski et G. Weiss, A new atomic decomposition for the Triebel- Lizorkin spaces, in Harmonic Analysis and Operator Theory : A Conference in Honor of Mischa Cotlar, January 3–8, 1994, Caracas, Venezuela (Amer. Math. Soc., 1995), pp. 235–249.

[196] A. Hassell et A. Sikora, Riesz transforms in one dimension, Indiana Univ. Math. J.

[197] (2009) 823–852.

[198] W. Hebisch, Boundedness of L1 spectral multipliers for an exponential solvable Lie group, Coll. Math. 73 (1997) 155–164.

[199] W. Hebisch et L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993) 673–709.

[200] W. Hebisch et T. Steger, Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (2003) 37–61.

[201] F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991) 591–596.

[202] F. Hélein, Regularity of weakly harmonic maps from a surface into a manifold with symmetries, Manuscripta Math. 70 (1991) 203–218.

[203] I. I. Hirschmann, Fractional integration, Amer. J. Math. 75 (1953) 531–546.

[204] S. Hofmann, G. Lu, D. Mitrea, M. Mitrea et L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies–Gaffney estimates, à paraˆıtre dans Mem. A.M.S.

[205] S. Hofmann et S. Mayboroda, Math. Ann. 344 (2009) 37–116.

[206] S. Hofmann, S. Mayboroda et A. McIntosh, Second order elliptic operators with complex bounded measurable coefficients in Lp , Sobolev and Hardy spaces, à paraˆıtre dans Ann. Sci. E.N.S.

[207] J. Hogan, C. Li, A. McIntosh et K. Zhang, Global higher integrability of Jacobians on bounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 193–217.

[208] G. Hu et D. Yang, h1 , bmo, blo and Littlewood–Paley g-functions with non-doubling measures, Rev. Mat. Iber. 25 (2009) 595–667.

[209] J. Hu et M. Zähle, Potential spaces on fractals, Studia Math. 170 (2005) 259–281.

[210] J. Huang, Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of Rn , Math. Z. 266 (2010) 141–168.

[211] T. Hytönen, Littlewood–Paley–Stein theory for semigroups in UMD spaces, Rev. Matem. Iber. 23 (2007) 973–1009.

[212] T. Hytönen, J. Van Neerven et P. Portal, Conical square function estimates in UMD Banach spaces and applications to H∞ functional calculi, J. Anal. Math. 106 (2008) 317–351.

[213] A. D. Ionescu, Fourier integral operators on noncompact symmetric spaces of real rank one, J. Funct. Anal. 174 (2000) 274–300.

[214] S. Ishiwata, Asymptotic behaviour of a transition probability for a random walk on a nilpotent covering graph, Contemp. Math. 347 (2004) 57–68.

[215] T. Iwaniec et C. Sbordone, Riesz transforms and elliptic PDEs with V MO coeffi- cients, J. Anal. Math. 74 (1998) 183–212.

[216] T. Iwaniec, C. Scott et B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pure Appl. IV, CLXXVII (1999) 37–115.

[217] D. Jerison et C. Kenig, Boundary value problems on Lipschitz domains, in Studies in Partial Differential Equations, MAA Stud. Math., Vol. 23 (Amer. Math. Soc., 1982), pp. 1–68.

[218] D. Jerison et C. Kenig, The functional calculus for the Laplacian on Lipschitz domains, in Journées Equations aux Dérivées Partielles, Exp. 4 (Ecole Polytech., Palaiseau, 1989), pp. 1–10.

[219] L. Ji, P. Kunstmann et A. Weber, Riesz transform on locally symmetric spaces and riemannian manifolds with a spectral gap, Bull. Sci. Math. 134 (2010) 37–43.

[220] R. Jiang et D. Yang, Generalized vanishing mean oscillation spaces associated with divergence form elliptic operators, Int. Eq. Op. Theory 67 (2010) 123–149.

[221] R. Jiang et D. Yang, Orlicz–Hardy spaces associated with operators satisfying Davies–Gaffney estimates, arXiv :0903.4494.

[222] F. John et L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961) 415–426.

[223] P. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980) 41–66.

[224] A. Jonsson, P. Sjögren et H. Wallin, Hardy and Lipschitz subspaces on subsets of Rn , Studia Math. 80 (1984) 141–166.

[225] A. Jonsson et H. Wallin, Function spaces on subsets of Rn (Harwood Academic, 1984).

[226] T. Kato, Integration of the equation of evolution in a Banach space, J. Math. Soc. Jpn. 5 (1953) 208–234.

[227] T. Kato et G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988) 891–907.

[228] S. Keith et X. Zhong, The Poincaré inequality is an open ended condition, Ann. Math. (2) 167 (2008) 575–599.

[229] C. Kenig et J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. Mat. 113 (1993) 447–509.

[230] J. Kigami et M. L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on pcf self-similar fractals, Commun. Math. Phys. 158 (1993) 93–125.

[231] J. Kinnunen et H. Tuominen, Pointwise behaviour of M1,1 Sobolev functions, Math. Z. 257 (2007) 613–630.

[232] H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966) 285–346.

[233] H. Komatsu, Fractional powers of operators, VI : Interpolation of nonnegative oper- ators and imbedding theorems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972) 1–62.

[234] P. Koskela et E. Saksman, Pointwise characterizations of Hardy–Sobolev functions, Math. Res. Lett. 15 (2008) 727–744.

[235] P. Koskela, D. Yang et Y. Zhou, A characterization of Hajlasz–Sobolev and Triebel Lizorkin spaces via grand Littlewood–Paley functions, J. Funct. Anal. 258 (2010) 2637–2661.

[236] T. Kumagai, Function spaces and stochastic processes on fractals, in Fractal Geom- etry and Stochastics III, eds. C. Bandt, U. Mosco et M. Zähle (Birkhäuser, 2004), pp. 221–234.

[237] T. Kumagai et K.-T. Sturm, Construction of diffusion processes on fractals, d-sets, and general metric measure spaces, J. Math. Kyoto Univ. 45 (2005) 307–327.

[238] S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci. 25 (1989) 659–680.

[239] R. H. Latter, A characterization of Hp (Rn ) in terms of atoms, Studia Math. 62 (1978) 93–101.

[240] M. Ledoux, The Concentration of Measure Phenomenon (Amer. Math. Soc., 2001).

[241] H. Q. Li, La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999) 145–238.

[242] P. Li et S. T. Yau, Acta Math. 156 (1986) 153–201.

[243] P. Li et S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Com- mun. Math. Phys. 88 (1983) 309–318.

[244] J. L. Lions et E. Magenes, Problemi ai limiti non omogenei (III), Ann. Scuola Norm. Sup. Pisa 15 (1961) 41–103.

[245] J. L. Lions et E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 1 (Dunod, 1968).

[246] N. Lohoué, Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive, J. Funct. Anal. 61 (1985) 164–201.

[247] N. Lohoué, Estimations des fonctions de Littlewood–Palye–Stein sur les variétés à courbure non-positive, Ann. Sc. ENS 20 (1987) 505–544.

[248] N. Lohoué, Transformées de Riesz et fonctions de Littlewood–Paley sur les groupes non moyennables, C. R. Acad. Sci. Paris 306 (1988) 327–330.

[249] N. Lohoué, Estimation des projecteurs de De Rham Hodge de certaines variétés riemaniennes non compactes, Math. Nachr. 279 (2006) 272–298.

[250] Z. Lou, Div-curl type theorems on Lipschitz domains, Bull. Aust. Math. Soc. 72 (2005) 31–38.

[251] Z. Lou et A. McIntosh, Hardy spaces of exact forms on Lipschitz domains in Rn , Indiana Univ. Math. J. 53 (2004) 583–612.

[252] Z. Lou et A. McIntosh, Hardy spaces of exact forms on Rn , Trans. Amer. Math. Soc.

[253] (2005) 1469–1496.

[254] Z. Lou et S. Yang, An atomic decomposition for the Hardy–Sobolev space, Taiwanese J. Math. 11 (2007) 1167–1176.

[255] R. Macias et C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979) 257–270.

[256] R. Macias et C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979) 271–309.

[257] J. Marcinkiewicz, Sur quelques intégrales du type de Dini, Ann. Soc. Pol. Math. 17 (1938) 42–50.

[258] M. Marias et E. Russ, H1 -boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds, Ark. Mat. 41 (2003) 115–132.

[259] J. M. Martell, Sharp maximal functions associated with approximations of the iden- tity in spaces of homogeneous type and applications, Studia Math. 161 (2004) 113–145.

[260] J. Mateu, P. Mattila, A. Nicolau et J. Orobitg, BMO for nondoubling measures, Duke Math. J. 102 (2000) 533–566.

[261] G. Mauceri, S. Meda et M. Vallarino, Hardy type spaces on certain noncompact manifolds and applications, à paraˆıtre dans, J. London. Math. Soc.

[262] B. Maurey, Isomorphismes entre espaces H1 , Acta Math. 145 (1980) 79–120.

[263] V. G. Maz´ya, S. A. Nazarov et B. A. Plamenevskii, Absence of De Giorgi-type the- orems for strongly elliptic equations with complex coefficients, J. Math. Sci. (N.Y.)

[264] (1985) 726–734.

[265] A. McIntosh, Operators which have an H∞ functional calculus, in Miniconference on Operator Theory and Partial Differential Equations (Canberra), Centre for Math. and Appl., Vol. 14 (Australian National Univ., 1986), pp. 210–231.

[266] A. McIntosh, Square roots of operators and applications to hyperbolic PDE’s, in Miniconference on Operator Theory and Partial Differential Equations (Canberra), Centre for Math. and Appl. (Australian National Univ., 1983).

[267] S. Meda, On the Littlewood–Paley–Stein g-function, Trans. Amer. Math. Soc. 347 (1995) 2201–2212.

[268] P.-A. Meyer, Transformations de Riesz pour les lois gaussiennes, in Séminaire de Probabilités XVIII, Lect. Notes, Vol. 1059 (Springer, 1984), pp. 179–193.

[269] P.-A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley, in Séminaire de Probabilités X, Lect. Notes in Math., Vol. 511 (Springer, 1976), pp. 125–183.

[270] Y. Meyer, Ondelettes et opérateurs, Vol. I (Hermann, 1991).

[271] Y. Meyer, Ondelettes et opérateurs, Vol. II (Hermann, 1991).

[272] N. G. Meyers, An Lp estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa 3 (1963) 189–206.

[273] D. Mitrea, Integral equation methods for div-curl problems for planar vector fields in nonsmooth domains, Diff. Int. Equations 18 (2005) 1039–1054.

[274] A. Miyachi, Hp spaces over open subsets of Rn , Studia Math. 95 (1980) 205–228.

[275] A. Miyachi, Hardy–Sobolev spaces and maximal functions, J. Math. Soc. Jpn. 42 (1990) 73–90.

[276] C. Mouhot, E. Russ et Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures. Appl. 95 (2011) 72–84.

[277] S. Müller, A surprising higher integrability property of mappings with positive deter- minant, Bull. Amer. Math. Soc. 21 (1989) 245–248.

[278] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (1978) 489–507.

[279] F. Murat, Compacité par compensation II, in Proc. Int. Meeting on Recent Methods in Non-Linear Analysis (Rome, May 1978), eds. E. de Giorgi, E. Magenes et U. Mosco (Pitagora Editrice, 1979), pp. 245–256.

[280] R. Nagel, E. M. Stein et S. Wainger, Balls and metrics defined by vector fields I : Basic properties, Acta Math. 155 (1988) 103–147.

[281] F. Nazarov, S. Treil et A. Volberg, The Tb-theorem on non-homogeneous spaces, Acta Math. 190 (2003) 151–239.

[282] J. Neˇcas, Les Méthodes Directes en Théorie des Équations Elliptiques (Masson et Cie, 1967).

[283] E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series, Vol. 31 (Princeton Univ. Press, 2005).

[284] M. I. Ostrovskii, Sobolev spaces on graphs, Quaest. Math. 28 (2005) 501–523.

[285] G. Pisier, Riesz transforms : A simpler analytic proof of P. A. Meyer’s inequality, in Sém. Prob. Strasbourg, Vol. 22 (Springer, 1988), pp. 485–501.

[286] T. Rey, Estimations de De Giorgi-Nash et approximations, Thèse de doctorat de l’Université Paul Cézanne, Marseille (2004).

[287] D. Robinson, Elliptic Operators and Lie Groups (Oxford Univ. Press, 1991).

[288] E. Russ, Riesz transforms on graphs for 1 ≤ p ≤ 2, Math. Scand. 87 (2000) 133–160.

[289] E. Russ, H1 − L1 boundedness of Riesz transforms on Riemannian manifolds and on graphs, Pot. Anal. 14 (2001) 301–330.

[290] E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, in CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis and Related Topics”, Proc. of the Centre for Math. and Appl., Vol. 42 (Australian National Univ., 2007), pp. 125–135.

[291] E. Russ et Y. Sire, Non local Poincaré inequalities on Lie groups with polynomial volume growth, arXiv :1001.4075, à paraˆıtre dans Studia Math.

[292] L. Saloff-Coste, Analyse sur les groupes de Lie à croissance polynomiale, Ark. Mat.

[293] (1990) 315–331.

[294] L. Saloff-Coste, Parabolic Harnack inequality for divergence form second order dif- ferential operators, Pot. Anal. 4 (1995) 429–467.

[295] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975) 391–405.

[296] C. Sbordone, New estimates for div-curl products and very weak solutions of PDEs, Ann. Scuola Norm. Sup. di Pisa 25 (1997) 739–756.

[297] C. Scott, Lp theory of differential forms on manifolds, Trans. A.M.S. 347 (1995) 2075–2096.

[298] Z. Shen, Bounds of Riesz transforms on Lp spaces for second order elliptic operators, Ann. Inst. Fourier 55 (2005) 173–197.

[299] A. Sikora, Riesz transform, Gaussian bounds and the method of wave equation, Math. Z. 247 (2004) 643–662.

[300] P. Sjögren, An estimate for a first-order Riesz operator on the affine group, Trans. Amer. Math. Soc. 351 (1999) 3301–3314.

[301] P. Sjögren et M. Vallarino, Boundedness from H1 to L1 of Riesz transforms on a Lie group of exponential growth, Ann. Inst. Fourier 58 (2008) 1117–1152.

[302] I. Sneiberg, Spectral properties of linear operators in interpolation families of Banach spaces, Mat. Issled. 9 (1974) 214–229.

[303] E. M. Stein, The characterization of functions arising as potentials I, Bull. Amer. Math. Soc. 67 (1961), 102–104, II, Bull. Amer. Math. Soc. 68 (1962) 577–582.

[304] E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, 1970).

[305] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood–Paley Theory (Princeton Univ. Press, 1970).

[306] E. M. Stein, Some results in harmonic analysis in Rn when n → ∞, Bull. Amer. Math. Soc. 9 (1983) 71–73.

[307] E. M. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscil- latory Integrals (Princeton Univ. Press, 1993).

[308] E. M. Stein et G. Weiss, On the theory of harmonic functions of several variables, I : The theory of Hp spaces, Acta Math. 103 (1960) 25–62.

[309] E. M. Stein et G. Weiss, Generalization of the Cauchy–Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968) 163–196.

[310] E. M. Stein et G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, 1971).

[311] R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967) 1031–1060.

[312] R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983) 48–79.

[313] R. Strichartz, Hp Sobolev spaces, Coll. Math. 60–61 (1990) 129–139.

[314] R. Strichartz, Function spaces on fractals, J. Funct. Anal. 198 (2003) 43–83.

[315] R. Strichartz, Solvability for differential equations on fractals, J. Anal. Math. 96 (2005) 247–267.

[316] M. Taibleson et G. Weiss, The Molecular Characterization of Certain Hardy Spaces. Representation Theorems for Hardy Spaces, Astérisque, Vol. 77 (Soc. Math. France, 1980), pp. 67–149.

[317] L. Tartar, Compensated compactness and applications to partial differential equa- tions, in Nonlinear Analysis and Mechanics, Heriot–Watt Symposium IV (Pitman, 1979), pp. 136–212.

[318] L. Tartar, The compensated compactness method applied to systems of conservation laws, in Systems of Nonlinear Partial Differential Equations, ed. J. Ball (D. Reidel, 1983), pp. 263–288.

[319] P. Tchamitchian, The solution of Kato’s conjecture (after Auscher, Hofmann, Lacey, McIntosh and Tchamitchian), in Journées “Equations aux Dérivées Partielles” (Univ. Nantes, 2001), pp. 1–14.

[320] X. Tolsa, BMO, H1 and Calder´on–Zygmund operators for non doubling measures, Math. Ann. 319 (2001) 89–149.

[321] X. Tolsa, A proof of the weak (1,1) inequality for singular integrals with non dou- bling measures based on a Calder´on–Zygmund decomposition, Publ. Mat. 45 (2001) 163–174.

[322] X. Tolsa, The space H1 for nondoubling measures in terms of a grand maximal operator, Trans. Amer. Math. Soc. 355 (2003) 315–348.

[323] H. Triebel, Function spaces on Lie groups, the Riemannian approach, J. London Math. Soc. 35 (1987) 327–338.

[324] H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84 (Birkhäuser, 1992).

[325] H. Triebel, Fractals and Spectra : Related to Fourier Analysis and Function Spaces (Birkhäuser, 1997).

[326] H. Triebel, Theory of Function Spaces III (Birkhäuser, 2006).

[327] H. Triebel et H. Winkelvoss, Intrinsic atomic characterization of function spaces on domains, Math. Z. 221 (1996) 647–673.

[328] M. Vallarino, Spaces H1 and BMO on ax + b groups, Collect. Math. 60 (2009) 277–295.

[329] N. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988) 346–410.

[330] N. Varopoulos, L. Saloff-Coste et T. Coulhon, Analysis and Geometry on Groups (Cambridge Univ. Press, 1992).

[331] J. Verdera, On the T(1)-theorem for the Cauchy integral, Ark. Mat. 38 (2000) 183–199.

[332] C. Villani, Hypocoercivity I, Mem. Amer. Math. Soc., Vol. 202 (Amer. Math. Soc., 2009).

[333] N. F. D. Ward et J. Partington, A construction of rational wavelets and frames in Hardy–Sobolev spaces with applications to system modeling, SIAM J. Control Optim. 36 (1998) 654–679.

[334] M. Weiss et A. Zygmund, A note on smooth functions, Nederl. Akad. Wetensch. Proc. Ser. A 62, Indag. Math. 21 (1959) 52–58.

[335] R. L. Wheeden, On hypersingular integrals and Lebesgue spaces of differentiable functions I, Trans. Amer. Math. Soc. 134 (1968) 421–436.

[336] L. Wu, A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Th. Relat. Fields 118 (2000) 427–438.

[337] D. Yang, Boundedness of linear operators via atoms on Hardy spaces with non- doubling measures, arXiv :0906.1316.

[338] N. Yosida, Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto Univ. 32 (1992) 621–654.

[339] A. Youssfi, Localisation des espaces de Besov homogènes, Indiana Univ. Math. J. 37 (1988) 565–588.

[340] A. Youssfi, Localisation des espaces de Lizorkin–Triebel homogènes, Math. Nachr.

[341] (1990) 107–121.

[342] M. Zähle, Riesz potentials and Liouville operators on fractals, Pot. Anal. 21 (2004) 193–208.

[343] M. Zähle, Local structures and diffusions on fractals, prépublication (2005).

[344] M. Zähle, Harmonic calculus on fractals — A measure geometric approach II, Trans. Amer. Math. Soc. 357 (2005) 3407–3423.

[345] K. Zhang, On the coercivity of elliptic systems in two dimensional spaces, Bull. Austr. Math. Soc. 54 (1996) 423–430.

[346] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944) 170–204.

[347] A. Zygmund, Trigonometric Series (Cambridge Univ. Press, 1959).

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