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Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

Xilin Zhou , Ziyi He and Dachun Yang EMAIL logo

Abstract

Let (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H*p,q(𝒳) with the optimal range p(ωω+η,) and q ∈ (0, ∞]. When and p(ωω+η,1] q ∈ (0, ∞], the authors establish its real-variable characterizations, respectively, in terms of radial maximal functions, non-tangential maximal functions, atoms, molecules, and various Littlewood–Paley functions. The authors also obtain its finite atomic characterization. As applications, the authors establish a real interpolation theorem on Hardy–Lorentz spaces, and also obtain the boundedness of Calderón–Zygmund operators on them including the critical cases. The novelty of this article lies in getting rid of the reverse doubling assumption of μ by fully using the geometrical properties of 𝒳 expressed via its dyadic reference points and dyadic cubes and, moreover, the results in the case q ∈ (0, 1) of this article are also new even when 𝒳 satisfies the reverse doubling condition.

References

[1] W. Abu-Shammala and A. Torchinsky, The Hardy–Lorentz spaces Hp,q(𝕉n), Studia Math. 182 (2007), 283–294.10.4064/sm182-3-7Search in Google Scholar

[2] A. Almeida and A. M. Caetano, Generalized Hardy spaces, Acta Math. Sin. (Engl. Ser.) 26 (2010), 1673–1692.10.1007/s10114-010-8647-9Search in Google Scholar

[3] J. Alvarez, Hp and weak Hp continuity of Calderón–Zygmund type operators, in: Fourier Analysis (Orono, ME, 1992), 17–34, Lecture Notes in Pure and Appl. Math. 157, Dekker, New York, 1994.10.1201/9781003072133-2Search in Google Scholar

[4] J. Alvarez, Continuity properties for linear commutators of Calderón–Zygmund operators, Collect. Math. 49 (1998), 17–31.Search in Google Scholar

[5] J. Alvarez and M. Milman, Hp continuity properties of Calderón–Zygmund-type operators, J. Math. Anal. Appl. 118 (1986), 63–79.10.1016/0022-247X(86)90290-8Search in Google Scholar

[6] T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588–594.Search in Google Scholar

[7] P. Auscher and T. Hytönen, Orthonormal bases of regular wavelets in spaces of homogeneous type, Appl. Comput. Harmon. Anal. 34 (2013), 266–296.10.1016/j.acha.2012.05.002Search in Google Scholar

[8] P. Auscher and T. Hytönen, Addendum to Orthonormal bases of regular wavelets in spaces of homogeneous type [Appl. Comput. Harmon. Anal. 34(2) (2013) 266–296], Appl. Comput. Harmon. Anal. 39 (2015), 568–569.10.1016/j.acha.2015.03.009Search in Google Scholar

[9] C. Bennett and R. C. Sharpley, Interpolation of Operators, Pure and Applied Mathematics 129, Academic Press, Inc., Boston, MA, 1988.Search in Google Scholar

[10] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin–New York, 1976.10.1007/978-3-642-66451-9Search in Google Scholar

[11] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), no. 781, 1–122.Search in Google Scholar

[12] M. Bownik, B. Li, D. Yang, and Y. Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ. Math. J. 57 (2008), 3065–3100.10.1512/iumj.2008.57.3414Search in Google Scholar

[13] M. Bownik and L. A. D. Wang, A PDE characterization of anisotropic Hardy spaces, Preprint.Search in Google Scholar

[14] H.-Q. Bui, T. A. Bui, and X. T. Duong, Weighted Besov and Triebel–Lizorkin spaces associated to operators and applications, Forum Math. Sigma 8 (2020), e11, 95 pp.10.1017/fms.2020.6Search in Google Scholar

[15] T. A. Bui and X. T. Duong, Sharp weighted estimates for square functions associated to operators on spaces of homogeneous type, J. Geom. Anal. 30 (2020), 874–900.10.1007/s12220-019-00173-8Search in Google Scholar

[16] T. A. Bui, X. T. Duong, and L. D. Ky, Hardy spaces associated to critical functions and applications to T1 theorems, J. Fourier Anal. Appl. 26 (2020), Article number 27, 67 pp.10.1007/s00041-020-09731-zSearch in Google Scholar

[17] T. A. Bui, X. T. Duong, and F. K. Ly, Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type, Trans. Amer. Math. Soc. 370 (2018), 7229–7292.10.1090/tran/7289Search in Google Scholar

[18] T. A. Bui, X. T. Duong, and F. K. Ly, Maximal function characterizations for Hardy spaces on spaces of homogeneous type with finite measure and applications, J. Funct. Anal. 278 (2020), 108423. 55 pp.10.1016/j.jfa.2019.108423Search in Google Scholar

[19] A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190.10.4064/sm-24-2-113-190Search in Google Scholar

[20] A.-P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, Adv. Math. 16 (1975), 1–64.10.1016/0001-8708(75)90099-7Search in Google Scholar

[21] G. Cleanthous, A. G. Georgiadis, and M. Nielsen, Anisotropic mixed-norm Hardy spaces, J. Geom. Anal. 27 (2017), 2758–2787.10.1007/s12220-017-9781-8Search in Google Scholar

[22] R. R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), 247–286.Search in Google Scholar

[23] R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. (French) Étude de Certaines Intégrales Singulières, Lecture Notes in Mathematics 242, Springer-Verlag, Berlin–New York, 1971.10.1007/BFb0058946Search in Google Scholar

[24] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.10.1090/S0002-9904-1977-14325-5Search in Google Scholar

[25] M. Cwikel, The dual of weak Lp, Ann. Inst. Fourier (Grenoble) 25 (1975), 81–126.10.5802/aif.556Search in Google Scholar

[26] M. Cwikel and C. Fefferman, Maximal seminorms on Weak L1, Studia Math. 69 (1980/81), 149–154.10.4064/sm-69-2-149-154Search in Google Scholar

[27] M. Cwikel and C. Fefferman, The canonical seminorm on weak L1, Studia Math. 78 (1984), 275–278.10.4064/sm-78-3-275-278Search in Google Scholar

[28] D. Deng and Y. Han, Harmonic Analysis on Spaces of Homogeneous Type. With a Preface by Yves Meyer. Lecture Notes in Mathematics 1996, Springer-Verlag, Berlin, 2009.10.1007/978-3-540-88745-4Search in Google Scholar

[29] Y. Ding and S. Lu, Hardy spaces estimates for multilinear operators with homogeneous kernels, Nagoya Math. J. 170 (2003), 117–133.10.1017/S0027763000008552Search in Google Scholar

[30] Y. Ding, S. Lu, and S. Shao, Integral operators with variable kernels on weak Hardy spaces, J. Math. Anal. Appl. 317 (2006), 127–135.10.1016/j.jmaa.2005.10.085Search in Google Scholar

[31] Y. Ding, S. Lu, and Q. Xue, Parametrized Littlewood–Paley operators on Hardy and weak Hardy spaces, Math. Nachr. 280 (2007), 351–363.10.1002/mana.200410487Search in Google Scholar

[32] Y. Ding and X. Wu, Weak Hardy space and endpoint estimates for singular integrals on space of homogeneous type, Turkish J. Math. 34 (2010), 235–247.Search in Google Scholar

[33] C. Fefferman, N. M. Rivière, and Y. Sagher, Interpolation between Hp spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75–81.Search in Google Scholar

[34] C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.10.1007/BF02392215Search in Google Scholar

[35] R. Fefferman and F. Soria, The space Weak H1, Studia Math. 85 (1986), 1–16 (1987).10.4064/sm-85-1-1-16Search in Google Scholar

[36] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.10.1515/9780691222455Search in Google Scholar

[37] X. Fu, T. Ma, and D. Yang, Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type, Ann. Acad. Sci. Fenn. Math. 45 (2020), 343–410.10.5186/aasfm.2020.4519Search in Google Scholar

[38] X. Fu and D. Yang, Wavelet characterizations of the atomic Hardy space H1 on spaces of homogeneous type, Appl. Comput. Harmon. Anal. 44 (2018), 1–37.10.1016/j.acha.2016.04.001Search in Google Scholar

[39] X. Fu, D. Yang, and Y. Liang, Products of functions in BMO(𝒳) and Hat1(𝒳) via wavelets over spaces of homogeneous type, J. Fourier Anal. Appl. 23 (2017), 919–990.10.1007/s00041-016-9483-9Search in Google Scholar

[40] L. Grafakos, Hardy space estimates for multilinear operators. II, Rev. Mat. Iberoam. 8 (1992), 69–92.10.4171/RMI/117Search in Google Scholar

[41] L. Grafakos, Classical Fourier Analysis, 3rd edition, Graduate Texts in Mathematics 249, Springer, New York, 2014.10.1007/978-1-4939-1194-3Search in Google Scholar

[42] L. Grafakos, Modern Fourier Analysis, 3rd edition, Graduate Texts in Mathematics 250, Springer, New York, 2014.10.1007/978-1-4939-1230-8Search in Google Scholar

[43] L. Grafakos, L. Liu, D. Maldonado, and D. Yang, Multilinear analysis on metric spaces, Dissertationes Math. 497 (2014), 1–121.10.4064/dm497-0-1Search in Google Scholar

[44] L. Grafakos, L. Liu, and D. Yang, Maximal function characterizations of Hardy spaces on RD-spaces and their applications, Sci. China Ser. A 51 (2008), 2253–2284.10.1007/s11425-008-0057-4Search in Google Scholar

[45] L. Grafakos, L. Liu, and D. Yang, Radial maximal function characterizations for Hardy spaces on RD-spaces, Bull. Soc. Math. France 137 (2009), 225–251.10.24033/bsmf.2574Search in Google Scholar

[46] L. Grafakos, L. Liu, and D. Yang, Vector-valued singular integrals and maximal functions on spaces of homogeneous type, Math. Scand. 104 (2009), 296–310.10.7146/math.scand.a-15099Search in Google Scholar

[47] Ya. Han, Yo. Han, Z. He, J. Li, and C. Pereyra, Geometric characteriztions of embedding theorems — for Sobolev, Besov, and Triebel–Lizorkin spaces on spaces of homogeneous type — via orthonormal wavelets, J. Geom. Anal. (to appear).Search in Google Scholar

[48] Ya. Han, Yo. Han, and J. Li, Criterion of the boundedness of singular integrals on spaces of homogeneous type, J. Funct. Anal. 271 (2016), 3423–3464.10.1016/j.jfa.2016.09.006Search in Google Scholar

[49] Ya. Han, Yo. Han, and J. Li, Geometry and Hardy spaces on spaces of homogeneous type in the sense of Coifman and Weiss, Sci. China Math. 60 (2017), 2199–2218.10.1007/s11425-017-9152-4Search in Google Scholar

[50] Y. Han, J. Li, and L. A. Ward, Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases, Appl. Comput. Harmon. Anal. 45 (2018), 120–169.10.1016/j.acha.2016.09.002Search in Google Scholar

[51] Y. Han, D. Müller, and D. Yang, Littlewood–Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), 1505–1537.10.1002/mana.200610435Search in Google Scholar

[52] Y. Han, D. Müller, and D. Yang, A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot–Carathéodory spaces, Abstr. Appl. Anal. 2008, Art. ID 893409, 1–250.Search in Google Scholar

[53] Y. Han and E. T. Sawyer, Littlewood–Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 110 (1994), no. 530, 1–126.Search in Google Scholar

[54] Z. He, Y. Han, J. Li, L. Liu, D. Yang, and W. Yuan, A complete real-variable theory of Hardy spaces on spaces of homogeneous type, J. Fourier Anal. Appl. 25 (2019), 2197–2267.10.1007/s00041-018-09652-ySearch in Google Scholar

[55] Z. He, L. Liu, D. Yang, and W. Yuan, New Calderón reproducing formulae with exponential decay on spaces of homogeneous type, Sci. China Math. 62 (2019), 283–350.10.1007/s11425-018-9346-4Search in Google Scholar

[56] Z. He, F. Wang, D. Yang, and Wen Yuan, Wavelet characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and its applications, Submitted.Search in Google Scholar

[57] Z. He, D. Yang, and W. Yuan, Real-variable characterizations of local Hardy spaces on spaces of homogeneous type, Math. Nachr. (2019), DOI: 10.1002/mana.201900320.10.1002/mana.201900320Search in Google Scholar

[58] G. Hu, D. Yang, and Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwanese J. Math. 13 (2009), 91–135.Search in Google Scholar

[59] L. Huang, J. Liu, D. Yang, and W. Yuan, Atomic and Littlewood–Paley characterizations of anisotropic mixed-norm Hardy spaces and their applications, J. Geom. Anal. 29 (2019), 1991–2067.10.1007/s12220-018-0070-ySearch in Google Scholar

[60] L. Huang, J. Liu, D. Yang, and W. Yuan, Dual spaces of anisotropic mixed-norm Hardy spaces, Proc. Amer. Math. Soc. 147 (2019), 1201–1215.10.1090/proc/14348Search in Google Scholar

[61] R. A. Hunt, On L(p, q) spaces, Enseign. Math. (2) 12 (1966), 249–276.Search in Google Scholar

[62] T. Hytönen and A. Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126 (2012), 1–33.10.4064/cm126-1-1Search in Google Scholar

[63] T. Hytönen and O. Tapiola, Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, J. Approx. Theory 185 (2014), 12–30.10.1016/j.jat.2014.05.017Search in Google Scholar

[64] N. Ioku, K. Ishige, and E. Yanagida, Sharp decay estimates in Lorentz spaces for nonnegative Schrödinger heat semi-groups, J. Math. Pures Appl. (9) 103 (2015), 900–923.10.1016/j.matpur.2014.09.006Search in Google Scholar

[65] T. Jakab and M. Mitrea, Parabolic initial boundary value problems in nonsmooth cylinders with data in anisotropic Besov spaces, Math. Res. Lett. 13 (2006), 825–831.10.4310/MRL.2006.v13.n5.a12Search in Google Scholar

[66] Y. Jiao, Y. Zuo, D. Zhou, and L. Wu, Variable Hardy–Lorentz spaces Hp(·),q(𝕉n), Math. Nachr. 292 (2019), 309–349.10.1002/mana.201700331Search in Google Scholar

[67] P. Koskela, D. Yang, and Y. Zhou, A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions, J. Funct. Anal. 258 (2010), 2637–2661.10.1016/j.jfa.2009.11.004Search in Google Scholar

[68] P. Koskela, D. Yang, and Y. Zhou, Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), 3579–3621.10.1016/j.aim.2010.10.020Search in Google Scholar

[69] W. Li, A maximal function characterization of Hardy spaces on spaces of homogeneous type, Approx. Theory Appl. (N.S.) 14 (2) (1998), 12–27.Search in Google Scholar

[70] Y. Liang, L. Liu, and D. Yang, An off-diagonal Marcinkiewicz interpolation theorem on Lorentz spaces, Acta Math. Sin. (Engl. Ser.) 27 (2011), 1477–1488.10.1007/s10114-011-0287-1Search in Google Scholar

[71] J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, (French) Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68.10.1007/BF02684796Search in Google Scholar

[72] H. Liu, The weak Hp spaces on homogeneous groups, in: Harmonic Analysis (Tianjin, 1988), 113–118, Lecture Notes in Math. 1494, Springer, Berlin, 1991.Search in Google Scholar

[73] J. Liu, D. Yang, and W. Yuan, Anisotropic Hardy–Lorentz spaces and their applications, Sci. China Math. 59 (2016), 1669–1720.10.1007/s11425-016-5157-ySearch in Google Scholar

[74] J. Liu, D. Yang, and W. Yuan, Anisotropic variable Hardy–Lorentz spaces and their real interpolation, J. Math. Anal. Appl. 456 (2017), 356–393.10.1016/j.jmaa.2017.07.003Search in Google Scholar

[75] J. Liu, D. Yang, and W. Yuan, Littlewood–Paley characterizations of anisotropic Hardy–Lorentz spaces, Acta Math. Sci. Ser. B (Engl. Ed.) 38 (2018), 1–33.Search in Google Scholar

[76] L. Liu, D.-C. Chang, X. Fu, and D. Yang, Endpoint boundedness of commutators on spaces of homogeneous type, Appl. Anal. 96 (2017), 2408–2433.10.1080/00036811.2017.1341628Search in Google Scholar

[77] L. Liu, D.-C. Chang, X. Fu, and D. Yang, Endpoint estimates of linear commutators on Hardy spaces over spaces of homogeneous type, Math. Methods Appl. Sci. 41 (2018), 5951–5984.10.1002/mma.5112Search in Google Scholar

[78] L. Liu, D. Yang, and W. Yuan, Bilinear decompositions for products of Hardy and Lipschitz spaces on spaces of homogeneous type, Dissertationes Math. 533 (2018), 1–93.10.4064/dm774-2-2018Search in Google Scholar

[79] S. Liu and K. Zhao, Various characterizations of product Hardy spaces associated to Schrödinger operators, Sci. China Math. 58 (2015), 2549–2564.10.1007/s11425-015-5071-8Search in Google Scholar

[80] R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257–270.10.1016/0001-8708(79)90012-4Search in Google Scholar

[81] R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979), 271–309.10.1016/0001-8708(79)90013-6Search in Google Scholar

[82] J. Merker and J.-M. Rakotoson, Very weak solutions of Poisson’s equation with singular data under Neumann boundary conditions, Calc. Var. Partial Differential Equations 52 (2015), 705–726.10.1007/s00526-014-0730-0Search in Google Scholar

[83] C. Muscalu, T. Tao, and C. Thiele, A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett. 10 (2003), 237–246.10.4310/MRL.2003.v10.n2.a10Search in Google Scholar

[84] S. Müller, Hardy space methods for nonlinear partial differential equations. Equadiff 8 (Bratislava, 1993), Tatra Mt. Math. Publ. 4 (1994), 159–168.Search in Google Scholar

[85] E. Nakai and Y. Sawano, Orlicz–Hardy spaces and their duals, Sci. China Math. 57 (2014), 903–962.10.1007/s11425-014-4798-ySearch in Google Scholar

[86] E. Nakai and K. Yabuta, Pointwise multipliers for functions of weighted bounded mean oscillation on spaces of homogeneous type, Math. Japon. 46 (1997), 15–28.Search in Google Scholar

[87] R. Oberlin, A. Seeger, T. Tao, C. Thiele, and J. Wright. A variation norm Carleson theorem, J. Eur. Math. Soc. (JEMS) 14 (2012), 421–464.10.4171/JEMS/307Search in Google Scholar

[88] D. V. Parilov, Two theorems on the Hardy–Lorentz classes H1,q, (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 327 (2005), Issled. po Lineĭn. Oper. i Teor. Funkts. 33, 150–167; translation in J. Math. Sci. (N.Y.) 139 (2006), 6447–6456.Search in Google Scholar

[89] N. C. Phuc, The Navier–Stokes equations in nonendpoint borderline Lorentz spaces, J. Math. Fluid Mech. 17 (2015), 741–760.10.1007/s00021-015-0229-2Search in Google Scholar

[90] S. Rolewicz, On a certain class of linear metric spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 471–473.Search in Google Scholar

[91] W. Rudin, Functional Analysis, 2nd edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.Search in Google Scholar

[92] C. Sadosky, Interpolation of Operators and Singular Integrals. An Introduction to Harmonic Analysis, Monographs and Textbooks in Pure and Applied Math. 53, Marcel Dekker, Inc., New York, 1979.Search in Google Scholar

[93] Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), 123–148.10.1007/s00020-013-2073-1Search in Google Scholar

[94] A. Seeger and T. Tao, Sharp Lorentz space estimates for rough operators, Math. Ann. 320 (2001), 381–415.10.1007/PL00004479Search in Google Scholar

[95] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller, Comm. Partial Differential Equations 19 (1994), 277–319.10.1080/03605309408821017Search in Google Scholar

[96] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. With the Assistance of Timothy S. Murphy, Princeton Mathematical Series 43, Monographs in Harmonic Analysis III, Princeton University Press, Princeton, N.J., 1993.10.1515/9781400883929Search in Google Scholar

[97] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series 32, Princeton University Press, Princeton, N.J., 1971.Search in Google Scholar

[98] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Mathematics 1381, Springer-Verlag, Berlin, 1989.10.1007/BFb0091154Search in Google Scholar

[99] T. Tao and J. Wright, Endpoint multiplier theorems of Marcinkiewicz type, Rev. Mat. Iberoam. 17 (2001), 521–558.Search in Google Scholar

[100] H. Triebel, Theory of Function Spaces. III, Monographs in Mathematics 100, Birkhäuser Verlag, Basel, 2006.Search in Google Scholar

[101] F. Wang, Y. Han, Z. He, and D. Yang, Besov spaces and Triebel–Lizorkin spaces on spaces of homogeneous type with their applications to boundedness of Calderón–Zygmund operators, Submitted.Search in Google Scholar

[102] H. Wang, Boundedness of several integral operators with bounded variable kernels on Hardy and weak Hardy spaces, Internat. J. Math. 24 (2013), 1350095, 1–22.10.1142/S0129167X1350095XSearch in Google Scholar

[103] Xin. Wu and Xia. Wu, Weak Hardy spaces Hp, on spaces of homogeneous type and their applications, Taiwanese J. Math. 16 (2012), 2239–2258.Search in Google Scholar

[104] X. Yan, D. Yang, W. Yuan, and C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), 2822–2887.10.1016/j.jfa.2016.07.006Search in Google Scholar

[105] D. Yang, Some new inhomogeneous Triebel–Lizorkin spaces on metric measure spaces and their various characterizations, Studia Math. 167 (2005), 63–98.10.4064/sm167-1-5Search in Google Scholar

[106] D. Yang, Some new Triebel–Lizorkin spaces on spaces of homogeneous type and their frame characterizations, Sci. China Ser. A 48 (2005), 12–39.10.1007/BF02942219Search in Google Scholar

[107] D. Yang and Y. Zhou, Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms, J. Math. Anal. Appl. 339 (2008), 622–635.10.1016/j.jmaa.2007.07.021Search in Google Scholar

[108] D. Yang and Y. Zhou, Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications, Math. Ann. 346 (2010), 307–333.10.1007/s00208-009-0400-2Search in Google Scholar

[109] D. Yang and Y. Zhou, New properties of Besov and Triebel–Lizorkin spaces on RD-spaces, Manuscripta Math. 134 (2011), 59–90.10.1007/s00229-010-0384-ySearch in Google Scholar

[110] Y. Zhang, S. Wang, D. Yang, and W. Yuan, Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón–Zygmund operators, Sci. China Math. (2020), DOI: 10.1007/s11425-019-1645-1.10.1007/s11425-019-1645-1Search in Google Scholar

[111] C. Zhuo, Y. Sawano, and D. Yang, Hardy spaces with variable exponents on RD-spaces and applications, Dissertationes Math. 520 (2016), 1–74.10.4064/dm744-9-2015Search in Google Scholar

Received: 2020-05-13
Revised: 2020-07-19
Accepted: 2020-07-21
Published Online: 2020-08-28
Published in Print: 2020-01-01

© 2018 Xilin Zhou et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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