Abstract
Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H p α (X), H pd (X), and H *,p(X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p(X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ∗,p(X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p(X), the Hardy space H p(X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p(X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.
Similar content being viewed by others
References
Stein E M, Weiss G. On the theory of harmonic functions of several variables I. The theory of H p-spaces. Acta Math, 103: 25–62 (1960)
Fefferman C, Stein E M. H p spaces of several variables. Acta Math, 129: 137–193 (1972)
Stein E M. Harmonic Analysis: Real-VariableMethods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993
Coifman R R, Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math, Vol. 242. Berlin: Springer, 1971
Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 83: 569–645 (1977)
Macías R A, Segovia C. A decomposition into atoms of distributions on spaces of homogeneous type. Adv Math, 33: 271–309 (1979)
Han Y. Triebel-Lizorkin spaces on spaces of homogeneous type. Studia Math, 108: 247–273 (1994)
Duong X T, Yan L. Hardy spaces of spaces of homogeneous type. Proc Amer Math Soc, 131: 3181–3189 (2003)
Li W M. A maximal function characterization of Hardy spaces on spaces of homogeneous type. Anal Theory Appl, 14(2): 12–27 (1998)
Han Y, Sawyer E T. Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Mem Amer Math Soc, 110(530): 1–126 (1994)
Heinonen J. Lectures on Analysis on Metric Spaces. New York: Springer-Verlag, 2001
Triebel H. Theory of Function Spaces, III. Basel: Birkhäuser Verlag, 2006
Alexopoulos G. Spectral multipliers on Lie groups of polynomial growth. Proc Amer Math Soc, 120: 973–979 (1994)
Varopoulos N T. Analysis on Lie groups. J Funct Anal, 76: 346–410 (1988)
Varopoulos N T, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge: Cambridge University Press, 1992
Nagel A, Stein E M, Wainger S. Balls and metrics defined by vector fields. I. Basic properties. Acta Math, 155: 103–147 (1985)
Nagel A, Stein E M. The □b-heat equation on pseudoconvex manifolds of finite type in ℂ2. Math Z, 238: 37–88 (2001)
Nagel A, Stein E M. Differentiable control metrics and scaled bump functions. J Differential Geom, 57: 465–492 (2001)
Nagel A, Stein E M. On the product theory of singular integrals. Rev Mat Iberoamericana, 20: 531–561 (2004)
Nagel A, Stein E M. The \( \bar \partial _b \)-complex on decoupled boundaries in ℂn. Ann of Math (2), 164: 649–713 (2006)
Danielli D, Garofalo N, Nhieu D M. 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(857): 1–119 (2006)
Stein E M. Some geometrical concepts arising in harmonic analysis. Geom Funct Anal, Special Volume: 434–453 (2000)
Han Y, Müller D, Yang D. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr Appl Anal, in press, 2008
Han Y, Müller D, Yang D. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 279: 1505–1537 (2006)
Meda S, Sjögren P, Vallarino M. On the H 1-L 1 boundedness of operators. Proc Amer Math Soc, 136: 2921–2931 (2008)
Meyer Y, Taibleson M, Weiss G. Some functional analytic properties of the spaces B q generated by blocks. Indiana Univ Math J, 34: 493–515 (1985)
García-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics. Amsterdam: North-Holland Publishing Co., 1985
Bownik M. Boundedness of operators on Hardy spaces via atomic decompositions. Proc Amer Math Soc, 133: 3535–3542 (2005)
Meyer Y, Coifman R. Wavelets: Calderón-Zygmund and Multilinear Operators. Cambridge: Cambridge University Press, 1997
Yabuta K. A remark on the (H 1, L 1) boundedness. Bull Fac Sci Ibaraki Univ Ser A, 25: 19–21 (1993)
Yang D, Zhou Y. A bounded criterion via atoms for linear operators in Hardy spaces. Constr Approx, in press
Bownik M, Li B, Yang D, et al. Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators. Indiana Univ Math J, in press
Hu G, Yang D, Zhou Y. Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwanese J Math, in press
Yang D, Zhou Y. Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms. J Math Anal Appl, 339: 622–635 (2008)
Grafakos L, Liu L, Yang D. Radial maximal function characterizations for Hardy spaces on RD-spaces. Bull Soc Math France, in press
Nakai E, Yabuta K. Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type. Math Japon, 46: 15–28 (1997)
Macías R A, Segovia C. Lipschitz functions on spaces of homogeneous type. Adv Math, 3: 257–270 (1979)
Christ M. A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq Math, 60–61: 601–628 (1990)
Wojtaszczyk P. Banach Spaces for Analysts. Cambridge: Cambridge University Press, 1991
Grafakos L, Liu L, Yang D. Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math Scand, in press
Aoki T. Locally bounded linear topological spaces. Proc Imp Acad Tokyo, 18: 588–594 (1942)
Rolewicz S. Metric Linear Spaces, 2nd ed. Mathematics and Its Applications (East European Series), 20. Dordrecht-Warsaw: D. Reidel Publishing Co. and PWN-Polish Scientific Publishers, 1985
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Science Foundation of USA (Grant No. DMS 0400387), the University of Missouri Research Council (Grant No. URC-07-067), the National Science Foundation for Distinguished Young Scholars of China (Grant No. 10425106) and the Program for New Century Excellent Talents in University of the Ministry of Education of China (Grant No. 04-0142)
Rights and permissions
About this article
Cite this article
Grafakos, L., Liu, L. & Yang, D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci. China Ser. A-Math. 51, 2253–2284 (2008). https://doi.org/10.1007/s11425-008-0057-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-008-0057-4
Keywords
- space of homogeneous type
- Calderón reproducing formula
- space of test function
- maximal function
- Hardy space
- atom
- Littlewood-Paley function
- sublinear operator
- quasi-Banach space