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Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations

  • Jingsong Sun , Dachun Yang ORCID logo EMAIL logo and Wen Yuan
Published/Copyright: September 30, 2022

Abstract

Let ( 𝕏 , d , μ ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ( 𝕏 ) be a ball quasi-Banach function space on 𝕏 . In this article, the authors introduce the weak Hardy space W H ~ X ( 𝕏 ) associated with X ( 𝕏 ) via the Lusin area function. Then the authors characterize W H ~ X ( 𝕏 ) by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 𝕏 expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ( 𝕏 ) and the reverse doubling condition of μ, and that the authors use the tent space on 𝕏 × to characterize W H ~ X ( 𝕏 ) by the Littlewood–Paley g λ * -function, where the range of λ might be best possible in some cases.


Communicated by Jan Frahm


Award Identifier / Grant number: 2020YFA0712900

Award Identifier / Grant number: 11971058

Award Identifier / Grant number: 12071197

Award Identifier / Grant number: 12122102

Award Identifier / Grant number: 11871100

Funding statement: This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900) and the National Natural Science Foundation of China (Grant Nos. 11971058, 12071197, 12122102 and 11871100).

Acknowledgements

The authors would like to thank the referee for her/his carefully reading and several motivating and useful comments which indeed improved the presentation of this article.

References

[1] A. Amenta, Tent spaces over metric measure spaces under doubling and related assumptions, Operator Theory in Harmonic and Non-Commutative Analysis, Oper. Theory Adv. Appl. 240, Birkhäuser/Springer, Cham (2014), 1–29. 10.1007/978-3-319-06266-2_1Search in Google Scholar

[2] T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588–594. 10.3792/pia/1195573733Search in Google Scholar

[3] P. Auscher, Change of angle in tent spaces, C. R. Math. Acad. Sci. Paris 349 (2011), no. 5–6, 297–301. 10.1016/j.crma.2011.01.023Search in Google Scholar

[4] P. Auscher and T. Hytönen, Orthonormal bases of regular wavelets in spaces of homogeneous type, Appl. Comput. Harmon. Anal. 34 (2013), no. 2, 266–296. 10.1016/j.acha.2012.05.002Search in Google Scholar

[5] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, 1988. Search in Google Scholar

[6] M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), no. 781, 1–122. 10.1090/memo/0781Search in Google Scholar

[7] M. Bownik and K.-P. Ho, Atomic and molecular decompositions of anisotropic Triebel–Lizorkin spaces, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1469–1510. 10.1090/S0002-9947-05-03660-3Search in Google Scholar

[8] 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), no. 7, 3065–3100. 10.1512/iumj.2008.57.3414Search in Google Scholar

[9] M. Bownik and L.-A. D. Wang, Fourier transform of anisotropic Hardy spaces, Proc. Amer. Math. Soc. 141 (2013), no. 7, 2299–2308. 10.1090/S0002-9939-2013-11623-0Search in Google Scholar

[10] T. A. Bui and X. T. Duong, Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces, J. Math. Soc. Japan 73 (2021), no. 2, 597–631. 10.2969/jmsj/83938393Search in Google Scholar

[11] T. A. Bui, X. T. Duong and L. D. Ky, Hardy spaces associated to critical functions and applications to T 1 theorems, J. Fourier Anal. Appl. 26 (2020), no. 2, Paper No. 27. 10.1007/s00041-020-09731-zSearch in Google Scholar

[12] 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), no. 10, 7229–7292. 10.1090/tran/7289Search in Google Scholar

[13] 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), no. 8, Article ID 108423. 10.1016/j.jfa.2019.108423Search in Google Scholar

[14] D.-C. Chang, S. Wang, D. Yang and Y. Zhang, Littlewood–Paley characterizations of Hardy-type spaces associated with ball quasi-Banach function spaces, Complex Anal. Oper. Theory 14 (2020), no. 3, Paper No. 40. 10.1007/s11785-020-00998-0Search in Google Scholar

[15] G. Cleanthous, A. G. Georgiadis, G. Kerkyacharian, P. Petrushev and D. Picard, Kernel and wavelet density estimators on manifolds and more general metric spaces, Bernoulli 26 (2020), no. 3, 1832–1862. 10.3150/19-BEJ1171Search in Google Scholar

[16] G. Cleanthous, A. G. Georgiadis and E. Porcu, Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces, J. Nonparametr. Stat. (2022), 10.1080/10485252.2022.2070162. 10.1080/10485252.2022.2070162Search in Google Scholar

[17] R. R. Coifman, Y. Meyer and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. 10.1016/0022-1236(85)90007-2Search in Google Scholar

[18] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes. Étude de certaines intégrales singulières, Lecture Notes in Math. 242, Springer, Berlin, 1971. 10.1007/BFb0058946Search in Google Scholar

[19] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. 10.1515/9781400827268.295Search in Google Scholar

[20] D. V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Heidelberg, 2013. 10.1007/978-3-0348-0548-3Search in Google Scholar

[21] R. del Campo, A. Fernández, F. Mayoral and F. Naranjo, Orlicz spaces associated to a quasi-Banach function space: Applications to vector measures and interpolation, Collect. Math. 72 (2021), no. 3, 481–499. 10.1007/s13348-020-00295-1Search in Google Scholar

[22] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011. 10.1007/978-3-642-18363-8Search in Google Scholar

[23] Y. Ding and X. Wu, Weak Hardy space and endpoint estimates for singular integrals on space of homogeneous type, Turkish J. Math. 34 (2010), no. 2, 235–247. 10.3906/mat-0809-37Search in Google Scholar

[24] X. T. Duong, R. Gong, M.-J. S. Kuffner, J. Li, B. D. Wick and D. Yang, Two weight commutators on spaces of homogeneous type and applications, J. Geom. Anal. 31 (2021), no. 1, 980–1038. 10.1007/s12220-019-00308-xSearch in Google Scholar

[25] X. T. Duong, G. Hu and J. Li, Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators, J. Math. Soc. Japan 71 (2019), no. 1, 91–115. 10.2969/jmsj/78287828Search in Google Scholar

[26] X. T. Duong and L. Yan, Hardy spaces of spaces of homogeneous type, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3181–3189. 10.1090/S0002-9939-03-06868-0Search in Google Scholar

[27] C. Fefferman, N. M. Rivière and Y. Sagher, Interpolation between H p spaces: The real method, Trans. Amer. Math. Soc. 191 (1974), 75–81. 10.2307/1996982Search in Google Scholar

[28] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), no. 3–4, 137–193. 10.1007/BF02392215Search in Google Scholar

[29] R. Fefferman and F. Soria, The space Weak H 1 , Studia Math. 85 (1986), no. 1, 1–16. 10.4064/sm-85-1-1-16Search in Google Scholar

[30] 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), no. 1, 343–410. 10.5186/aasfm.2020.4519Search in Google Scholar

[31] A. G. Georgiadis, G. Kerkyacharian, G. Kyriazis and P. Petrushev, Homogeneous Besov and Triebel–Lizorkin spaces associated to non-negative self-adjoint operators, J. Math. Anal. Appl. 449 (2017), no. 2, 1382–1412. 10.1016/j.jmaa.2016.12.049Search in Google Scholar

[32] A. G. Georgiadis, G. Kerkyacharian, G. Kyriazis and P. Petrushev, Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators, J. Fourier Anal. Appl. 25 (2019), no. 6, 3259–3309. 10.1007/s00041-019-09702-zSearch in Google Scholar

[33] A. G. Georgiadis and G. Kyriazis, Embeddings between Triebel–Lizorkin spaces on metric spaces associated with operators, Anal. Geom. Metr. Spaces 8 (2020), no. 1, 418–429. 10.1515/agms-2020-0120Search in Google Scholar

[34] L. Grafakos, Classical Fourier Analysis, 3rd ed., Grad. Texts in Math. 249, Springer, New York, 2014. 10.1007/978-1-4939-1194-3Search in Google Scholar

[35] 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), no. 12, 2253–2284. 10.1007/s11425-008-0057-4Search in Google Scholar

[36] L. Grafakos, L. Liu and D. Yang, Vector-valued singular integrals and maximal functions on spaces of homogeneous type, Math. Scand. 104 (2009), no. 2, 296–310. 10.7146/math.scand.a-15099Search in Google Scholar

[37] 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), no. 1, 120–169. 10.1016/j.acha.2016.09.002Search in Google Scholar

[38] Y. Han, D. Müller and D. Yang, Littlewood–Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), no. 13–14, 1505–1537. 10.1002/mana.200610435Search in Google Scholar

[39] 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 (2008), Article ID 893409. 10.1155/2008/893409Search in Google Scholar

[40] 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), no. 5, 2197–2267. 10.1007/s00041-018-09652-ySearch in Google Scholar

[41] 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), no. 2, 283–350. 10.1007/s11425-018-9346-4Search in Google Scholar

[42] Z. He, F. Wang, D. Yang and W. Yuan, Wavelet characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and its applications, Appl. Comput. Harmon. Anal. 54 (2021), 176–226. 10.1016/j.acha.2021.03.007Search in Google Scholar

[43] Z. He, D. Yang and W. Yuan, Real-variable characterizations of local Hardy spaces on spaces of homogeneous type, Math. Nachr. 294 (2021), no. 5, 900–955. 10.1002/mana.201900320Search in Google Scholar

[44] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001. 10.1007/978-1-4613-0131-8Search in Google Scholar

[45] K.-P. Ho, Atomic decompositions of weighted Hardy–Morrey spaces, Hokkaido Math. J. 42 (2013), no. 1, 131–157. 10.14492/hokmj/1362406643Search in Google Scholar

[46] K.-P. Ho, Atomic decomposition of Hardy–Morrey spaces with variable exponents, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 31–62. 10.5186/aasfm.2015.4002Search in Google Scholar

[47] K.-P. Ho, Atomic decompositions and Hardy’s inequality on weak Hardy–Morrey spaces, Sci. China Math. 60 (2017), no. 3, 449–468. 10.1007/s11425-016-0229-1Search in Google Scholar

[48] K.-P. Ho, Atomic decompositions of weighted Hardy spaces with variable exponents, Tohoku Math. J. (2) 69 (2017), no. 3, 383–413. 10.2748/tmj/1505181623Search in Google Scholar

[49] G. Hu, D. Yang and Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type, Taiwanese J. Math. 13 (2009), no. 1, 91–135. 10.11650/twjm/1500405274Search in Google Scholar

[50] T. Hytönen and A. Kairema, Systems of dyadic cubes in a doubling metric space, Colloq. Math. 126 (2012), no. 1, 1–33. 10.4064/cm126-1-1Search in Google Scholar

[51] 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

[52] S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), no. 4, 959–982. 10.1215/S0012-7094-80-04755-9Search in Google Scholar

[53] H. Jia and H. Wang, Decomposition of Hardy–Morrey spaces, J. Math. Anal. Appl. 354 (2009), no. 1, 99–110. 10.1016/j.jmaa.2008.12.051Search in Google Scholar

[54] R. Jiang and D. Yang, New Orlicz–Hardy spaces associated with divergence form elliptic operators, J. Funct. Anal. 258 (2010), no. 4, 1167–1224. 10.1016/j.jfa.2009.10.018Search in Google Scholar

[55] 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), no. 8, 2637–2661. 10.1016/j.jfa.2009.11.004Search in Google Scholar

[56] P. Koskela, D. Yang and Y. Zhou, Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings, Adv. Math. 226 (2011), no. 4, 3579–3621. 10.1016/j.aim.2010.10.020Search in Google Scholar

[57] L. D. Ky, New Hardy spaces of Musielak–Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), no. 1, 115–150. 10.1007/s00020-013-2111-zSearch in Google Scholar

[58] M.-Y. Lee, J. Li and L. A. Ward, On weak-star convergence in product Hardy spaces on spaces of homogeneous type, Studia Math. 235 (2016), no. 3, 251–267. 10.4064/sm8574-8-2016Search in Google Scholar

[59] J. Li, Atomic decomposition of weighted Triebel–Lizorkin spaces on spaces of homogeneous type, J. Aust. Math. Soc. 89 (2010), no. 2, 255–275. 10.1017/S144678871000159XSearch in Google Scholar

[60] J. Li and L. A. Ward, Singular integrals on Carleson measure spaces CMO p on product spaces of homogeneous type, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2767–2782. 10.1090/S0002-9939-2013-11604-7Search in Google Scholar

[61] Y. Liang, J. Huang and D. Yang, New real-variable characterizations of Musielak–Orlicz Hardy spaces, J. Math. Anal. Appl. 395 (2012), no. 1, 413–428. 10.1016/j.jmaa.2012.05.049Search in Google Scholar

[62] Y. Liang, D. Yang and R. Jiang, Weak Musielak–Orlicz Hardy spaces and applications, Math. Nachr. 289 (2016), no. 5–6, 634–677. 10.1002/mana.201500152Search in Google Scholar

[63] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II: Function Spaces, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1979. 10.1007/978-3-662-35347-9Search in Google Scholar

[64] H. P. Liu, The weak H p spaces on homogeneous groups, Harmonic Analysis (Tianjin 1988), Lecture Notes in Math. 1494, Springer, Berlin (1991), 113–118. 10.1007/BFb0087762Search in Google Scholar

[65] R. A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. Math. 33 (1979), no. 3, 271–309. 10.1016/0001-8708(79)90013-6Search in Google Scholar

[66] R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), no. 3, 257–270. 10.1016/0001-8708(79)90012-4Search in Google Scholar

[67] D. Müller and D. Yang, A difference characterization of Besov and Triebel–Lizorkin spaces on RD-spaces, Forum Math. 21 (2009), no. 2, 259–298. 10.1515/FORUM.2009.013Search in Google Scholar

[68] E. Nakai, Singular and fractional integral operators on preduals of Campanato spaces with variable growth condition, Sci. China Math. 60 (2017), no. 11, 2219–2240. 10.1007/s11425-017-9154-ySearch in Google Scholar

[69] E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665–3748. 10.1016/j.jfa.2012.01.004Search in Google Scholar

[70] E. Nakai and Y. Sawano, Orlicz–Hardy spaces and their duals, Sci. China Math. 57 (2014), no. 5, 903–962. 10.1007/s11425-014-4798-ySearch in Google Scholar

[71] T. Quek and D. Yang, Calderón–Zygmund-type operators on weighted weak Hardy spaces over 𝐑 n , Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 1, 141–160. 10.1007/s101149900022Search in Google Scholar

[72] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 146, Marcel Dekker, New York, 1991. Search in Google Scholar

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

[74] E. Russ, The atomic decomposition for tent spaces on spaces of homogeneous type, CMA/AMSI Research Symposium “Asymptotic Geometric Analysis, Harmonic Analysis, and Related Topics”, Proc. Centre Math. Appl. Austral. Nat. Univ. 42, Australian National University, Canberra (2007), 125–135. Search in Google Scholar

[75] Y. Sawano, A vector-valued sharp maximal inequality on Morrey spaces with non-doubling measures, Georgian Math. J. 13 (2006), no. 1, 153–172. 10.1515/GMJ.2006.153Search in Google Scholar

[76] Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Operator Theory 77 (2013), no. 1, 123–148. 10.1007/s00020-013-2073-1Search in Google Scholar

[77] Y. Sawano, K.-P. Ho, D. Yang and S. Yang, Hardy spaces for ball quasi-Banach function spaces, Dissertationes Math. 525 (2017), 102. 10.4064/dm750-9-2016Search in Google Scholar

[78] L. Song and L. Wu, A q-atomic decomposition of weighted tent spaces on spaces of homogeneous type and its application, J. Geom. Anal. 31 (2021), no. 3, 3029–3059. 10.1007/s12220-020-00382-6Search in Google Scholar

[79] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables. I. The theory of H p -spaces, Acta Math. 103 (1960), 25–62. 10.1007/BF02546524Search in Google Scholar

[80] J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer, Berlin, 1989. 10.1007/BFb0091154Search in Google Scholar

[81] J. Sun, D. Yang and W. Yuan, Weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Decompositions, real interpolation, and Calderón–Zygmund operators, J. Geom. Anal. 32 (2022), no. 7, Paper No. 191. 10.1007/s12220-022-00927-xSearch in Google Scholar

[82] F. Wang, Y. Han, Z. He and D. Yang, Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators, Dissertationes Math. 565 (2021), 1–113. 10.4064/dm821-4-2021Search in Google Scholar

[83] F. Wang, Z. He, D. Yang and W. Yuan, Difference characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Commun. Math. Stat. (2021), 10.1007/s40304-021-00243-w. 10.1007/s40304-021-00243-wSearch in Google Scholar

[84] F. Wang, D. Yang and S. Yang, Applications of Hardy spaces associated with ball quasi-Banach function spaces, Results Math. 75 (2020), no. 1, Paper No. 26. 10.1007/s00025-019-1149-xSearch in Google Scholar

[85] S. Wang, D. Yang, W. Yuan and Y. Zhang, Weak Hardy-type spaces associated with ball quasi-Banach function spaces II: Littlewood–Paley characterizations and real interpolation, J. Geom. Anal. 31 (2021), no. 1, 631–696. 10.1007/s12220-019-00293-1Search in Google Scholar

[86] X. Wu and X. Wu, Weak Hardy space H p , on spaces of homogeneous type and their applications, Taiwanese J. Math. 16 (2012), no. 6, 2239–2258. 10.11650/twjm/1500406849Search in Google Scholar

[87] X. Yan, Z. He, D. Yang and W. Yuan, Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces, Math. Nachr. (2022), 10.1002/mana.202100432. 10.1002/mana.202100432Search in Google Scholar

[88] X. Yan, Z. He, D. Yang and W. Yuan, Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type: Littlewood–Paley characterizations with applications to boundedness of Calderón–Zygmund operators, Acta Math. Sin. (Engl. Ser.) 38 (2022), no. 7, 1133–1184. 10.1007/s10114-022-1573-9Search in Google Scholar

[89] X. Yan, D. Yang and W. Yuan, Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces, Front. Math. China 15 (2020), no. 4, 769–806. 10.1007/s11464-020-0849-6Search in Google Scholar

[90] X. Yan, D. Yang, W. Yuan and C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal. 271 (2016), no. 10, 2822–2887. 10.1016/j.jfa.2016.07.006Search in Google Scholar

[91] D. Yang, Y. Liang and L. D. Ky, Real-Variable Theory of Musielak–Orlicz Hardy Spaces, Lecture Notes in Math. 2182, Springer, Cham, 2017. 10.1007/978-3-319-54361-1Search in Google Scholar

[92] D. Yang and Y. Zhou, New properties of Besov and Triebel–Lizorkin spaces on RD-spaces, Manuscripta Math. 134 (2011), no. 1–2, 59–90. 10.1007/s00229-010-0384-ySearch in Google Scholar

[93] Y. Zhang, D. Yang, W. Yuan and S. Wang, Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón–Zygmund operators, Sci. China Math. 64 (2021), no. 9, 2007–2064. 10.1007/s11425-019-1645-1Search in Google Scholar

[94] X. Zhou, Z. He and D. Yang, Real-variable characterizations of Hardy–Lorentz spaces on spaces of homogeneous type with applications to real interpolation and boundedness of Calderón–Zygmund operators, Anal. Geom. Metr. Spaces 8 (2020), no. 1, 182–260. 10.1515/agms-2020-0109Search in Google Scholar

[95] 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: 2022-03-04
Revised: 2022-07-02
Published Online: 2022-09-30
Published in Print: 2022-11-01

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