Radial maximal function characterizations for Hardy spaces on RD-spaces
Bulletin de la Société Mathématique de France, Volume 137 (2009) no. 2, p. 225-251

An RD-space 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type 𝒳 having “dimension” n, there exists a p 0 (n/(n+1),1) such that for certain classes of distributions, the L p (𝒳) quasi-norms of their radial maximal functions and grand maximal functions are equivalent when p(p 0 ,]. This result yields a radial maximal function characterization for Hardy spaces on 𝒳.

Un RD-espace X est un espace de type homogène au sens de Coifman et Weiss, possédant en outre une propriété de doublement inverse. Les auteurs prouvent que pour un espace de type homogène X de « dimension » n, il existe un p 0 (n/(n+1),1) tel que les quasi-normes L p (X) des fonctions radiales maximales et grand-maximales d’une certaine classe de distributions soient équivalentes lorsque p(p 0 ,]. Ce résultat fournit une caractérisation des espaces de Hardy sur X en termes de fonctions radiales maximales.

DOI : https://doi.org/10.24033/bsmf.2574
Classification:  42B25,  42B30,  47B38,  47A30
Keywords: space of homogeneous type, approximation of the identity, space of test function, grand maximal function, radial maximal function, Hardy space
@article{BSMF_2009__137_2_225_0,
     author = {Grafakos, Loukas and Liu, Liguang and Yang, Dachun},
     title = {Radial maximal function characterizations for Hardy spaces on RD-spaces},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {137},
     number = {2},
     year = {2009},
     pages = {225-251},
     doi = {10.24033/bsmf.2574},
     zbl = {1205.42016},
     mrnumber = {2543475},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2009__137_2_225_0}
}
Grafakos, Loukas; Liu, Liguang; Yang, Dachun. Radial maximal function characterizations for Hardy spaces on RD-spaces. Bulletin de la Société Mathématique de France, Volume 137 (2009) no. 2, pp. 225-251. doi : 10.24033/bsmf.2574. http://www.numdam.org/item/BSMF_2009__137_2_225_0/

[1] G. Alexopoulos - « Spectral multipliers on Lie groups of polynomial growth », Proc. Amer. Math. Soc. 120 (1994), p. 973-979. | MR 1172944 | Zbl 0794.43003

[2] M. Christ - « A T(b) theorem with remarks on analytic capacity and the Cauchy integral », Colloq. Math. 60/61 (1990), p. 601-628. | MR 1096400 | Zbl 0758.42009

[3] R. R. Coifman & G. Weiss - Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., Vol. 242, Springer, 1971. | MR 499948 | Zbl 0224.43006

[4] -, « Extensions of Hardy spaces and their use in analysis », Bull. Amer. Math. Soc. 83 (1977), p. 569-645. | MR 447954 | Zbl 0358.30023

[5] D. Danielli, N. Garofalo & D.-M. Nhieu - « Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces », Mem. Amer. Math. Soc. 182 (2006), p. 119. | MR 2229731 | Zbl 1100.43005

[6] X. T. Duong & L. Yan - « Hardy spaces of spaces of homogeneous type », Proc. Amer. Math. Soc. 131 (2003), p. 3181-3189. | MR 1992859 | Zbl 1032.42023

[7] C. Fefferman & E. M. Stein - « H p spaces of several variables », Acta Math. 129 (1972), p. 137-193. | MR 447953 | Zbl 0257.46078

[8] L. Grafakos - Classical Fourier analysis, second éd., Graduate Texts in Math., vol. 249, Springer, 2008. | MR 2445437 | Zbl 1220.42001

[9] L. Grafakos, L. Liu & D. Yang - « Maximal function characterizations of Hardy spaces on RD-spaces and their applications », Sci. China (Ser. A) 51 (2008), p. 2253-2284. | MR 2462027 | Zbl 1176.42017

[10] Y. Han - « Triebel-Lizorkin spaces on spaces of homogeneous type », Studia Math. 108 (1994), p. 247-273. | MR 1259279 | Zbl 0822.46033

[11] Y. Han, D. Müller & D. Yang - « Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type », Math. Nachr. 279 (2006), p. 1505-1537. | MR 2269253 | Zbl 1179.42016

[12] -, « A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces », to appear in Abstr. Appl. Anal, Art. ID 893409, 2008. | MR 2485404 | Zbl 1193.46018

[13] J. Heinonen - Lectures on analysis on metric spaces, Universitext, Springer, 2001. | MR 1800917 | Zbl 0985.46008

[14] R. A. Macías & C. Segovia - « A decomposition into atoms of distributions on spaces of homogeneous type », Adv. in Math. 33 (1979), p. 271-309. | MR 546296 | Zbl 0431.46019

[15] D. Müller & D. Yang - « A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces », to appear in Forum Math. | MR 2503306 | Zbl 1171.42013

[16] A. Nagel & E. M. Stein - « Differentiable control metrics and scaled bump functions », J. Differential Geom. 57 (2001), p. 465-492. | MR 1882665 | Zbl 1041.58006

[17] -, « On the product theory of singular integrals », Rev. Mat. Iberoamericana 20 (2004), p. 531-561. | MR 2073131

[18] A. Nagel, E. M. Stein & S. Wainger - « Balls and metrics defined by vector fields. I. Basic properties », Acta Math. 155 (1985), p. 103-147. | MR 793239 | Zbl 0578.32044

[19] E. M. Stein - Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, 1993. | MR 1232192 | Zbl 0821.42001

[20] -, « Some geometrical concepts arising in harmonic analysis », Geom. Funct. Anal., Special Volume, Part I (2000), p. 434-453, GAFA 2000 (Tel Aviv, 1999). | MR 1826263

[21] E. M. Stein & G. Weiss - « On the theory of harmonic functions of several variables. I. The theory of H p -spaces », Acta Math. 103 (1960), p. 25-62. | MR 121579 | Zbl 0097.28501

[22] A. Uchiyama - « A maximal function characterization of H p on the space of homogeneous type », Trans. Amer. Math. Soc. 262 (1980), p. 579-592. | MR 586737 | Zbl 0503.46020

[23] N. T. Varopoulos - « Analysis on Lie groups », J. Funct. Anal. 76 (1988), p. 346-410. | MR 924464 | Zbl 0634.22008

[24] N. T. Varopoulos, L. Saloff-Coste & T. Coulhon - Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, 1992. | MR 1218884 | Zbl 0813.22003