Home Weighted estimates for product singular integral operators in Journé’s class on RD-spaces
Article
Licensed
Unlicensed Requires Authentication

Weighted estimates for product singular integral operators in Journé’s class on RD-spaces

  • Taotao Zheng ORCID logo EMAIL logo , Yanmei Xiao and Xiangxing Tao
Published/Copyright: April 18, 2024

Abstract

An RD-space 𝑀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝑀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé’s class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces.

MSC 2020: 42B20; 42B25; 46E35

Award Identifier / Grant number: 12326307

Award Identifier / Grant number: 12326308

Award Identifier / Grant number: 12271483

Award Identifier / Grant number: LQ17A010002

Funding statement: This research was funded by National Natural Science Foundation of China (grant numbers 12326307, 12326308, 12271483) and Natural Science Foundation of Zhejiang Province (grant number LQ17A010002).

  1. Communicated by: Christopher D. Sogge

References

[1] R. Alvarado, D. Yang and W. Yuan, A measure characterization of embedding and extension domains for Sobolev, Triebel–Lizorkin, and Besov spaces on spaces of homogeneous type, J. Funct. Anal. 283 (2022), no. 12, Article ID 109687. 10.1016/j.jfa.2022.109687Search in Google Scholar

[2] S.-Y. A. Chang and R. Fefferman, The Calderón–Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), no. 3, 455–468. 10.2307/2374150Search in Google Scholar

[3] L.-K. Chen and D. Fan, The multiplier operators on the weighted product spaces, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3755–3765. 10.1090/S0002-9939-96-03616-7Search in Google Scholar

[4] M. Christ, A T ( b ) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. 10.4064/cm-60-61-2-601-628Search in Google Scholar

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

[6] G. David and J.-L. Journé, A boundedness criterion for generalized Calderón–Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. 10.2307/2006946Search in Google Scholar

[7] D. Deng and Y. Han, T 1 theorem for Besov and Triebel–Lizorkin spaces, Sci. China Ser. A 48 (2005), no. 5, 657–665. 10.1360/04sys0114Search in Google Scholar

[8] W. Ding and G. Lu, Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators, Forum Math. 34 (2022), no. 6, 1679–1705. 10.1515/forum-2022-0192Search in Google Scholar

[9] Y. Ding, Y. Han, G. Lu and X. Wu, Boundedness of singular integrals on multiparameter weighted Hardy spaces H w p ( R n × R m ) , Potential Anal. 37 (2012), no. 1, 31–56. 10.1007/s11118-011-9244-ySearch in Google Scholar

[10] D. Fan, K. Guo and Y. Pan, Singular integrals with rough kernels on product spaces, Hokkaido Math. J. 28 (1999), no. 3, 435–460. 10.14492/hokmj/1351001230Search in Google Scholar

[11] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. (2) 126 (1987), no. 1, 109–130. 10.2307/1971346Search in Google Scholar

[12] R. Fefferman, A p weights and singular integrals, Amer. J. Math. 110 (1988), no. 5, 975–987. 10.2307/2374700Search in Google Scholar

[13] R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. Math. 45 (1982), no. 2, 117–143. 10.1016/S0001-8708(82)80001-7Search in Google Scholar

[14] 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

[15] 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

[16] Y. Han and S. Hofmann, T1 theorems for Besov and Triebel–Lizorkin spaces, Trans. Amer. Math. Soc. 337 (1993), no. 2, 839–853. 10.1090/S0002-9947-1993-1097168-4Search in Google Scholar

[17] Y. Han, M.-Y. Lee, C.-C. Lin and Y.-C. Lin, Calderón–Zygmund operators on product Hardy spaces, J. Funct. Anal. 258 (2010), no. 8, 2834–2861. 10.1016/j.jfa.2009.10.022Search in Google Scholar

[18] Y. Han, J. Li and C.-C. Lin, Criterion of the L 2 boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 3, 845–907. 10.2422/2036-2145.201411_002Search in Google Scholar

[19] Y. Han, J. Li and G. Lu, Duality of multiparameter Hardy spaces H p on spaces of homogeneous type, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 4, 645–685. 10.2422/2036-2145.2010.4.01Search in Google Scholar

[20] Y. Han, J. Li and G. Lu, Multiparameter Hardy space theory on Carnot–Carathéodory spaces and product spaces of homogeneous type, Trans. Amer. Math. Soc. 365 (2013), no. 1, 319–360. 10.1090/S0002-9947-2012-05638-8Search in Google Scholar

[21] Y. Han, G. Lu and Z. Ruan, Boundedness of singular integrals in Journé’s class on weighted multiparameter Hardy spaces, J. Geom. Anal. 24 (2014), no. 4, 2186–2228. 10.1007/s12220-013-9421-xSearch in Google Scholar

[22] 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

[23] 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. 10.1090/memo/0530Search in Google Scholar

[24] Y. Han and D. Yang, H p boundedness of Calderón–Zygmund operators on product spaces, Math. Z. 249 (2005), no. 4, 869–881. 10.1007/s00209-004-0741-1Search in Google Scholar

[25] Y. Han and D. Yang, Boundedness of Calderón–Zygmund operators in product Hardy spaces, Appl. Math. J. Chinese Univ. Ser. B 24 (2009), no. 3, 321–335. 10.1007/s11766-009-1030-xSearch in Google Scholar

[26] S. He and J. Chen, Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type, Forum Math. 34 (2022), no. 1, 175–196. 10.1515/forum-2021-0204Search in Google Scholar

[27] 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

[28] 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

[29] 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

[30] 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

[31] J.-L. Journé, Calderón–Zygmund operators on product spaces, Rev. Mat. Iberoam. 1 (1985), no. 3, 55–91. 10.4171/rmi/15Search in Google Scholar

[32] M.-Y. Lee, Boundedness of Calderón–Zygmund operators on weighted product Hardy spaces, J. Operator Theory 72 (2014), no. 1, 115–133. 10.7900/jot.2012nov06.1993Search in Google Scholar

[33] H. Li and T. Zheng, Calderón–Zygmund operators on Lipschitz spaces over RD spaces, Quaest. Math. 44 (2021), no. 4, 473–494. 10.2989/16073606.2019.1709579Search in Google Scholar

[34] P. I. Lizorkin, Properties of functions in the spaces Λ p , θ r . Studies in the theory of differentiable functions of several variables and its applications, Trudy Mat. Inst. Steklov. 131 (1974), 158–181. Search in Google Scholar

[35] G. Lu and Y. Zhu, Singular integrals and weighted Triebel–Lizorkin and Besov spaces of arbitrary number of parameters, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 1, 39–52. 10.1007/s10114-012-1402-7Search in Google Scholar

[36] 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

[37] 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

[38] J. Peetre, On spaces of Triebel–Lizorkin type, Ark. Mat. 13 (1975), 123–130. 10.1007/BF02386201Search in Google Scholar

[39] J. Peetre, New Thoughts on Besov Spaces, Duke University, Durham, 1976. Search in Google Scholar

[40] J. Sun, D. Yang and W. Yuan, 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, Forum Math. 34 (2022), no. 6, 1539–1589. 10.1515/forum-2022-0074Search in Google Scholar

[41] H. Triebel, Theory of Function Spaces, Monogr. Math. 78, Birkhäuser, Basel, 1983. 10.1007/978-3-0346-0416-1Search in Google Scholar

[42] 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

[43] X. Yan, Littlewood–Paley g λ -function characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type, Probl. Anal. Issues Anal. 13(31) (2024), no. 1, 100–123. 10.15393/j3.art.2024.15310Search in Google Scholar

[44] 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

[45] 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. 296 (2023), no. 7, 3056–3116. 10.1002/mana.202100432Search in Google Scholar

[46] D. Yang, New frames of Besov and Triebel–Lizorkin spaces, J. Funct. Spaces Appl. 3 (2005), no. 1, 1–16. 10.1155/2005/139260Search in Google Scholar

[47] 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

[48] 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

[49] T. Zheng, J. Chen, J. Dai, S. He and X. Tao, Calderón–Zygmund operators on homogeneous product Lipschitz spaces, J. Geom. Anal. 31 (2021), no. 2, 2033–2057. 10.1007/s12220-019-00331-ySearch in Google Scholar

[50] T. Zheng, Y. Xiao, S. He and X. Tao, T 1 theorem on homogeneous product Besov spaces and product Triebel–Lizorkin spaces, Banach J. Math. Anal. 16 (2022), no. 3, Paper No. 50. 10.1007/s43037-022-00202-9Search in Google Scholar

[51] T. Zheng, Y. Xiao and X. Tao, The T 1 theorem for the generalized product Calderón–Zygmund operator on product endpoint function spaces over RD spaces (in Chinese), Sci. Sin. Math. 53 (2023), no. 3, 441–472. 10.1360/SSM-2021-0189Search in Google Scholar

Received: 2023-07-29
Revised: 2024-03-16
Published Online: 2024-04-18
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 19.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2023-0273/html?lang=en
Scroll to top button