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Variational characterizations of weighted Hardy spaces and weighted $BMO$ spaces

Part of: Harmonic analysis in several variables Special classes of linear operators

Published online by Cambridge University Press:  01 December 2021

Weichao Guo ,
Yongming Wen ,
Huoxiong Wu  and
Dongyong Yang
Weichao Guo
Affiliation:
School of Science, Jimei University, Xiamen 361021, China (weichaoguomath@gmail.com)
Yongming Wen*
Affiliation:
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China (wenyongmingxmu@163.com)
Huoxiong Wu
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China (huoxwu@xmu.edu.cn; dyyang@xmu.edu.cn)
Dongyong Yang
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China (huoxwu@xmu.edu.cn; dyyang@xmu.edu.cn)
*
*Corresponding author.
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Abstract

This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.

Keywords

Variation operatorsapproximate identitiesweighted Hardy spacesweighted BMO(ℝn) spaces

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis 47B47: Commutators, derivations, elementary operators, etc.
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 6 , December 2022 , pp. 1613 - 1632
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Accomazzo, N.. A characterization of BMO in terms of endpoint bounds for commutators of singular integrals. Isr. J. Math. 228 (2018), 787–800.CrossRefGoogle Scholar
Accomazzo, N., Martínez-Perales, J. C. and Rivera-Ríos, I. P.. On Bloom type estimates for iterated commutators of fractional integrals. Indiana Univ. Math. J. 69 (2020), 1207–1230.CrossRefGoogle Scholar
Betancor, J. J., Fariña, J. C., Harbour, E. and Rodríguez-Mesa, L.. $L^{p}$-boundedness properties of variation operators in the Schrödinger setting. Rev. Mat. Complut. 26 (2013), 485–534.CrossRefGoogle Scholar
Bloom, S.. A commutator theorem and weighted BMO. Trans. Am. Math. Soc. 292 (1985), 103–122.CrossRefGoogle Scholar
Bourgain, J.. Pointwise ergodic theorems for arithmetric sets. Publ. Math. Inst. Hautes Études Sci. 69 (1989), 5–45.CrossRefGoogle Scholar
Bownik, M., Li, B., Yang, D. and Zhou, Y.. Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators. Indiana Univ. Math. J. 57 (2008), 3065–3100.Google Scholar
Campbell, J. T., Jones, R. L., Reinhold, K. and Wierdl, M.. Oscillations and variation for the Hilbert transform. Duke Math. J. 105 (2000), 59–83.CrossRefGoogle Scholar
Campbell, J. T., Jones, R. L., Reinhold, K. and Wierdl, M.. Oscillations and variation for singular integrals in higher dimensions. Trans. Am. Math. Soc. 355 (2003), 2115–2137.CrossRefGoogle Scholar
Chen, Y., Ding, Y., Hong, G. and Liu, H.. Weighted jump and variational inequalities for rough operators. J. Funct. Anal. 275 (2018), 2446–2475.CrossRefGoogle Scholar
Chen, Y., Ding, Y., Hong, G. and Liu, H.. Variational inequalities for the commutators of rough operators with BMO functions. Sci. China Math. In press.Google Scholar
Coifman, R. R., Rochberg, R. and Weiss, G.. Factorization theorems for Hardy spaces in several variables. Ann. Math. 103 (1976), 611–635.CrossRefGoogle Scholar
Crescimbeni, R., Macías, R. A., Menárguez, T., Torrea, J. L. and Viviani, B.. The $\rho$-variation as an operator between maximal operators and singular integrals. J. Evol. Equ. 9 (2009), 81–102.CrossRefGoogle Scholar
Cruz-Uribe, D., Martell, J. M. and Pérez, C.. Sharp weighted estimates for approximating dyadic operators. Electron. Res. Announce. Math. Sci. 17 (2010), 12–19.Google Scholar
Ding, Y., Hong, G. and Liu, H.. Jump and variational inequalities for rough operators. J. Fourier Anal. Appl. 23 (2017), 679–711.CrossRefGoogle Scholar
Domingo-Salazar, C., Lacey, M. and Rey, G.. Boderline weak-type estimates for singular integrals and square functions. Bull. London Math. Soc. 48 (2016), 63–73.CrossRefGoogle Scholar
García-Cuerva, J.. Weighted $H^{p}$ spaces. Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63 pp.Google Scholar
Gillespie, T. A. and Torrea, J. L.. Dimension free estimates for the oscillation of Riesz transform. Israel J. Math. 141 (2004), 125–144.CrossRefGoogle Scholar
Guo, W., Lian, J. and Wu, H.. The unified necessity of bounded commutators and applications. J. Geom. Anal. 30 (2020), 3995–4035.CrossRefGoogle Scholar
Holmes, I., Lacey, M. T. and Wick, B. D.. Commutators in the two-weight setting. Math. Ann. 367 (2016), 51–80.CrossRefGoogle Scholar
Holmes, I., Rahm, R. and Spencer, S.. Commutators with fractional operators. Stud. Math. 233 (2016), 279–291.Google Scholar
Hytönen, T. P.. The $L^{p}$-to-$L^{q}$ boundedness of commutators with applications to the Jacobian operator. J. Math. Pures Appl. In press.Google Scholar
Hytönen, T. P., Lacey, M. and Pérez, C.. Sharp weighted bounds for the q-variation of singular integrals. Bull. London Math. Soc. 45 (2013), 529–540.CrossRefGoogle Scholar
Hytönen, T. P. and Pérez, C.. The $L(\log L)^{\epsilon }$ endpoint estimate for maximal singular integral operators. J. Math. Anal. Appl. 428 (2015), 605–626.CrossRefGoogle Scholar
Jones, R. L.. Variation inequalities for singular integrals and related operators. In Harmonic Analysis. Calderón-Zygmund and Beyond (Chicago 2002). Contemporary Mathematics, vol. 411, pp. 89–121 (Providence: American Mathematical Society, 2006).CrossRefGoogle Scholar
Jones, R. L. and Reinhold, K.. Oscillation and variation inequalities for convolution powers. Ergod. Theory Dyn. Syst. 21 (2001), 1809–1829.CrossRefGoogle Scholar
Lépingle, D.. La variation d'order $p$ des semi-martingales. Z. Wahrsch. Verw. Geb. 36 (1976), 295–316.CrossRefGoogle Scholar
Lerner, A. K.. On pointwise estimates involving sparse operators. N. Y. J. Math. 22 (2017), 341–349.Google Scholar
Lerner, A. K., Ombrosi, S. and Rivera-Ríos, I. P.. On pointwise and weighted estimates for commutators of Calderón-Zygmund operators. Adv. Math. 319 (2017), 153–181.CrossRefGoogle Scholar
Lerner, A. K., Ombrosi, S. and Rivera-Ríos, I. P.. Commutators of singular integrals revisited. Bull. London Math. Soc. 51 (2019), 107–119.CrossRefGoogle Scholar
Liang, Y., Ky, L. D. and Yang, D.. Weighted endpoint estimates for commutators of Calderón-Zygmund operators. Proc. Am. Math. Soc. 144 (2016), 5171–5181.CrossRefGoogle Scholar
Liu, F. and Wu, H.. A criterion on oscillation and variation for the commutators of singular integrals. Forum Math. 27 (2015), 77–97.CrossRefGoogle Scholar
Liu, H.. Variational characterization of $H^{p}$. Proc. R. Soc. Edinburgh Sect. A 149 (2019), 1123–1134.CrossRefGoogle Scholar
Ma, T., Torrea, J. L. and Xu, Q.. Weighted variation inequalities for differential operators and singular integrals. J. Funct. Anal. 268 (2015), 376–416.CrossRefGoogle Scholar
Ma, T., Torrea, J. L. and Xu, Q.. Weighted variation inequalities for differential operators and singular integrals in higher dimensions. Sci. China Math. 60 (2017), 1419–1442.CrossRefGoogle Scholar
Moen, K.. Sharp weighted bounds without testing or extrapolation. Arch. Math. (Basel) 99 (2012), 457–466.CrossRefGoogle Scholar
Muckenhoupt, B. and Wheeden, R. L.. On the dual of weighted $H^{1}$ of the half-space. Stud. Math. 63 (1978), 53–79.CrossRefGoogle Scholar
Nazarov, F., Treil, S. and Volberg, S. A.. Weak type estimates and Cotlar inequalities for Calderón–Zygmund operators on nonhomogeneous spaces. Int. Math. Res. Not. 9 (1998), 463–487.CrossRefGoogle Scholar
Paluszyński, M.. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J. 44 (1995), 1–17.CrossRefGoogle Scholar
Pereyra, M. C.. Dyadic harmonic analysis and weighted inequalities: the sparse revolution. New Trends in Applied Harmonic Analysis, vol. 2, pp. 159–239 (Cham: Birkhäuser, 2019).Google Scholar
Strömberg, J.-O. and Torchinsky, A.. Weighted hardy spaces. Lecture Notes in Mathematics, vol. 1381 (Berlin: Springer-Verlag, 1989).CrossRefGoogle Scholar
Wen, Y., Guo, W. and Wu, H.. Variation and oscillation inequalities for commutators in two-weight setting. Forum Math. 32 (2020), 1459–1475.CrossRefGoogle Scholar
Wen, Y., Wu, H. and Zhang, J.. Weighted variation inequalities for singular integrals and commutators. J. Math. Anal. Appl. 485 (2020), 123825.CrossRefGoogle Scholar