Weak Hardy spaces associated with para-accretive functions and their applications

Yan Wang; Xintian Dong; Fanghui Liao; Yan Wang; Xintian Dong; Fanghui Liao

doi:10.3934/math.20241476

AIMS Mathematics

2024, Volume 9, Issue 11: 30572-30596. doi: 10.3934/math.20241476

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Research article

Weak Hardy spaces associated with para-accretive functions and their applications

1.
School of Science, China University of Mining and Technology, Beijing 100083, China
2.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Received: 07 August 2024 Revised: 11 October 2024 Accepted: 16 October 2024 Published: 28 October 2024
MSC : 42B20, 42B25, 42B35

In this paper, we introduced a new class of weak Hardy spaces, denoted by $ H^{p, \infty}_b $, and provided an analysis of their atomic decomposition. As an application, we established the boundedness of Calderón-Zygmund operators (CZOs) from $ H^p $ to $ H^{p, \infty}_b $ including cases at the critical exponent
$ p = \frac{n}{n+\delta}, $
where $ \delta $ represents the regularity index of the distributional kernel. Moreover, the boundedness of CZOs from $ H^{p, \infty} $ to $ H^{p, \infty}_b $ was demonstrated for
$ \frac{n}{n+\delta}<p\leq 1. $
- weak Hardy space,
- para-accretive function,
- atomic decomposition,
- Calderón-Zygmund operator
Citation: Yan Wang, Xintian Dong, Fanghui Liao. Weak Hardy spaces associated with para-accretive functions and their applications[J]. AIMS Mathematics, 2024, 9(11): 30572-30596. doi: 10.3934/math.20241476

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Abstract

In this paper, we introduced a new class of weak Hardy spaces, denoted by $ H^{p, \infty}_b $, and provided an analysis of their atomic decomposition. As an application, we established the boundedness of Calderón-Zygmund operators (CZOs) from $ H^p $ to $ H^{p, \infty}_b $ including cases at the critical exponent

$ p = \frac{n}{n+\delta}, $

where $ \delta $ represents the regularity index of the distributional kernel. Moreover, the boundedness of CZOs from $ H^{p, \infty} $ to $ H^{p, \infty}_b $ was demonstrated for

$ \frac{n}{n+\delta}<p\leq 1. $

References

[1]	R. Ferfferman, F. Soria, The space weak $H^1$, Stud. Math., 85 (1986), 1–16. https://doi.org/10.4064/sm-85-1-1-16 doi: 10.4064/sm-85-1-1-16
[2]	H. Liu, The weak $H^p$ spaces on homogeneous groups, In: M. T. Cheng, D. G. Deng, X. W. Zhou, Harmonic analysis, Springer, 1991. https://doi.org/10.1007/BFb0087762
[3]	D. He, Square function characterization of weak Hardy spaces, J. Fourier Anal. Appl., 20 (2014), 1083–1110. https://doi.org/10.1007/s00041-014-9346-1 doi: 10.1007/s00041-014-9346-1
[4]	X. Yan, D. Yang, W. Yuan, C. Zhuo, Variable weak Hardy spaces and their applications, J. Funct. Anal., 271 (2016), 2822–2887. https://doi.org/10.1016/j.jfa.2016.07.006 doi: 10.1016/j.jfa.2016.07.006
[5]	J. Liu, F. Weisz, D. Yang, W. Yuan, Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl., 25 (2019), 874–922. https://doi.org/10.1007/s00041-018-9609-3 doi: 10.1007/s00041-018-9609-3
[6]	J. Sun, D. Yang, 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), 191. https://doi.org/10.1007/s12220-022-00927-x doi: 10.1007/s12220-022-00927-x
[7]	J. Sun, D. Yang, 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), 1539–1589. https://doi.org/10.1515/forum-2022-0074 doi: 10.1515/forum-2022-0074
[8]	Y. Zuo, K. Saibi, Y. Jiao, Variable Hardy-Lorentz spaces associated to operators satisfying Davies-Gaffney estimates, Banach J. Math. Anal., 13 (2019), 769–797. https://doi.org/10.1215/17358787-2018-0035 doi: 10.1215/17358787-2018-0035
[9]	G. David, J. L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. Math., 120 (1984), 371–397. https://doi.org/10.2307/2006946 doi: 10.2307/2006946
[10]	A. McIntosh, Y. Meyer, Algèbres d'opérateurs définis par intégrales singulières, C. R. Acad. Sci. Paris, Sér. I Math., 301 (1985), 395–397.
[11]	G. David, J. L. Journé, S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoam., 1 (1985), 1–56. https://doi.org/10.4171/RMI/17 doi: 10.4171/RMI/17
[12]	Y. Han, M. Y. Lee, C. C. Lin, Hardy spaces and the $Tb$ theorem, J. Geom. Anal., 14 (2004), 291–318. https://doi.org/10.1007/BF02922074 doi: 10.1007/BF02922074
[13]	D. Deng, D. Yang, Some new Besov and Triebel-Lizorkin spaces associated with para-accretive functions on spaces of homogeneous type, J. Aust. Math. Soc., 80 (2006), 229–262. https://doi.org/10.1017/S1446788700013094 doi: 10.1017/S1446788700013094
[14]	F. Liao, Z. Liu, X. Zhang, The $Tb$ theorem for inhomogeneous Besov and Triebel-Lizorkin spaces and its application, Georgian Math. J., 23 (2016), 253–267. https://doi.org/10.1515/gmj-2015-0062 doi: 10.1515/gmj-2015-0062
[15]	X. Tao, Y. Kang, T. Zheng, The Tb theorem for some inhomogeneous Besov and Triebel-Lizorkin spaces over space of homogeneous type, J. Math. Anal. Appl., 531 (2024), 127879. https://doi.org/10.1016/j.jmaa.2023.127879 doi: 10.1016/j.jmaa.2023.127879
[16]	J. Tan, Weighted variable Hardy spaces associated with Para-accretive functions and boundedness of Calderón-Zygmund operators, J. Geom. Anal., 33 (2023), 61. https://doi.org/10.1007/s12220-022-01121-9 doi: 10.1007/s12220-022-01121-9
[17]	R. R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569–645. https://doi.org/10.1090/S0002-9904-1977-14325-5 doi: 10.1090/S0002-9904-1977-14325-5
[18]	C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), 9–36. https://doi.org/10.1007/BF02394567 doi: 10.1007/BF02394567
[19]	Y. Han, E. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and classical function spaces, Mem. Amer. Math. Soc., 110 (1994), 530. https://doi.org/10.1090/memo/0530 doi: 10.1090/memo/0530
[20]	M. Frazier, B. Jawerth, A discrete transform and decomposition of distridution spaces, J. Funct. Anal., 93 (1990), 34–170. https://doi.org/10.1016/0022-1236(90)90137-A doi: 10.1016/0022-1236(90)90137-A
[21]	Y. Sawano, Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators, Integral Equations Oper. Theory, 77 (2013), 123–148. https://doi.org/10.1007/s00020-013-2073-1 doi: 10.1007/s00020-013-2073-1
[22]	D. Cruz-Uribe, K. Moen, H. V. Nguyen, A new approach to norm inequalities on weighted and variable Hardy spaces, Ann. Fenn. Math., 45 (2020), 175–198. https://doi.org/10.5186/aasfm.2020.4526 doi: 10.5186/aasfm.2020.4526
[23]	J. Tan, Boundedness of multilinear fractional type operators on Hardy spaces with variable exponents, Anal. Math. Phys., 10 (2020), 70. https://doi.org/10.1007/s13324-020-00415-x doi: 10.1007/s13324-020-00415-x
[24]	M. Y. Lee, C. C. Lin, Carleson spaces associated to para-accretive functions, Commun. Contemp. Math., 14 (2012), 1250002. https://doi.org/10.1142/S0219199712500022 doi: 10.1142/S0219199712500022
[25]	R. R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes, Springer-Verlag, 1971. https://doi.org/10.1007/BFb0058946
[26]	N. J. Kalton, Linear operators on $L_p$ for $0 < p < 1$, Trans. Amer. Math. Soc., 259 (1980), 319–355. https://doi.org/10.2307/1998234 doi: 10.2307/1998234
[27]	E. M. Stein, M. H. Taibleson, G. Weiss, Weak type estimates for maximal operators on certain $H^p$ classes, Rend. Circ. Mat. Palermo., 1981, 81–97.
[28]	W. Chen, Y. Han, C. Miao, A note on the boundedness of Calderón-Zygmund operators on Hardy spaces, J. Math. Anal. Appl., 310 (2005), 57–67. https://doi.org/10.1016/j.jmaa.2005.01.021 doi: 10.1016/j.jmaa.2005.01.021

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