Home Variation and oscillation inequalities for commutators in two-weight setting
Article
Licensed
Unlicensed Requires Authentication

Variation and oscillation inequalities for commutators in two-weight setting

  • Yongming Wen , Weichao Guo and Huoxiong Wu EMAIL logo
Published/Copyright: July 16, 2020

Abstract

This paper studies the two-weight estimates of variation and oscillation operators for commutators of singular integrals with weighted BMO functions. A new characterization of weighted BMO spaces via the boundedness of variation and oscillation operators for the iterated commutators of Calderón–Zygmund singular integrals in the two-weight setting is given.

MSC 2010: 42B20; 42B25

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 11771358

Award Identifier / Grant number: 11871101

Award Identifier / Grant number: 11701112

Award Identifier / Grant number: 11671414

Funding statement: Supported by the NNSF of China (nos. 11771358, 11871101, 11701112, 11671414).

References

[1] M. Akcoglu, R. L. Jones and P. Schwartz, Variation in probability, ergodic theory and analysis, Illinois J. Math. 42 (1998), 154–177. 10.1215/ijm/1255985619Search in Google Scholar

[2] J. J. Betancor, J. C. Farina, E. Harbour and L. Rodriguez-Mesa, Lp-boundedness properities of variation operators in the Schrödinger setting, Rev. Mat. Complut. 26 (2013), no. 2, 485–534. 10.1007/s13163-012-0094-ySearch in Google Scholar

[3] S. Bloom, A commutator theorem and weighted BMO, Trans. Amer. Math. Soc. 292 (1985), 103–122. 10.1090/S0002-9947-1985-0805955-5Search in Google Scholar

[4] J. Bourgain, Pointwise ergodic theorems for arithmetric sets, Publ. Math. Inst. Hautes Études Sci. 69 (1989), 5–45. 10.1007/BF02698838Search in Google Scholar

[5] J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl, Oscillations and variation for the Hilbert transform, Duke Math. J. 105 (2000), 59–83. 10.1215/S0012-7094-00-10513-3Search in Google Scholar

[6] J. T. Campbell, R. L. Jones, K. Reinhold and M. Wierdl, Oscillations and variation for singular integrals in higher dimensions, Trans. Amer. Math. Soc. 355 (2003), 2115–2137. 10.1090/S0002-9947-02-03189-6Search in Google Scholar

[7] Y. Chen, Y. Ding, G. Hong and H. Liu, Variational inequalities for the commutators of rough operators with BMO functions, J. Funct. Anal. 275 (2018), no. 8, 2446–2475. 10.1007/s11425-019-1713-xSearch in Google Scholar

[8] R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. 10.2307/1970954Search in Google Scholar

[9] R. Crescimbeni, F. J. Martin-reyes, A. L. Torre and J. L. Torrea, The ρ-variation of the Hermitian Riesz transform, Acta Math. Sin. (Engl. Ser.) 26 (2010), 1827–1838. 10.1007/s10114-010-9122-3Search in Google Scholar

[10] Y. Ding and S. Lu, Higher order commutators for a class of rough operators, Ark. Mat. 37 (1999), 33–44. 10.1007/BF02384827Search in Google Scholar

[11] Y. Ding, G. Hong and H. Liu, Jump and variational inequalities for rough operators, J. Fourier. Anal. Appl. 23 (2017), no. 3, 679–711. 10.1007/s00041-016-9484-8Search in Google Scholar

[12] J. García-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), no. 4, 1397–1420. 10.1512/iumj.1991.40.40063Search in Google Scholar

[13] T. A. Gillespie and J. L. Torrea, Dimension free estimates for the oscillation of Riesz transform, Israel J. Math. 141 (2014), 125–144. 10.1007/BF02772215Search in Google Scholar

[14] W. Guo, H. Wu and D. Yang, A revisit on the compactness of commutators, preprint (2017), https://arxiv.org/abs/1712.08292v1. 10.4153/S0008414X20000644Search in Google Scholar

[15] I. Holmes, M. T. Lacey and B. D. Wick, Commutators in the two-weight setting, Math. Ann. 367 (2016), no. 1–2, 51–80. 10.1007/s00208-016-1378-1Search in Google Scholar

[16] I. Holmes and B. D. Wick, Two weight inequalities for iterated commutators with Calderón–Zygmund operators, J. Operator Theory 79 (2018), 33–54. Search in Google Scholar

[17] R. L. Jones, Ergodic theory and connections with analysis and probability, New York J. Math. 3A (1997), 31–67. Search in Google Scholar

[18] R. L. Jones, Variation inequalities for singular integrals and related operators, Contemp. Math. 411 (2006), 89–122. 10.1090/conm/411/07750Search in Google Scholar

[19] R. L. Jones and K. Reinhold, Oscillation and variation inequalities for convolution powers, Ergodic Theory Dynam. Systems. 21 (2001), 1809–1829. 10.1017/S0143385701001869Search in Google Scholar

[20] A. K. Lerner, S. Ombrosi and I. P. Rivera-Ríos, On pointwise and weighted estimates for commutators of Calderón–Zygmund operators, Adv. Math. 319 (2017), 153–181. 10.1016/j.aim.2017.08.022Search in Google Scholar

[21] A. K. Lerner, S. Ombrosi and I. P. Rivera-Ríos, Commutators of singular integrals revisited, Bull. Lond. Math. Soc. 51 (2019), 107–119. 10.1112/blms.12216Search in Google Scholar

[22] F. Liu and H. Wu, A criterion on oscillation and variation for the commutators of singular integrals, Forum Math. 27 (2015), 77–97. 10.1515/forum-2012-0019Search in Google Scholar

[23] T. Ma, J. L. Torrea and Q. Xu, Weighted variation inequalities for differential operators and singular integrals, J. Funct. Anal. 268 (2015), no. 2, 376–416. 10.1016/j.jfa.2014.10.008Search in Google Scholar

[24] T. Ma, J. L. Torrea and Q. Xu, Weighted variation inequalities for differential operators and singular integrals in higher dimensions, Sci. China Math. 60 (2017), no. 8, 1419–1442. 10.1007/s11425-016-9012-7Search in Google Scholar

[25] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. 10.1090/S0002-9947-1972-0293384-6Search in Google Scholar

[26] C. Segovia and J. L. Torrea, Higher order commutators for vector-valued Calderón–Zygmund operators, Trans. Amer. Math. Soc. 336 (1993), no. 2, 537–556. 10.1090/S0002-9947-1993-1074151-6Search in Google Scholar

[27] J. Zhang and H. Wu, Weighted oscillation and variation inequalities for singular integrals and commutators satisfying Hörmander conditions, Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 10, 1397–1420. 10.1007/s10114-017-6379-9Search in Google Scholar

Received: 2019-07-29
Published Online: 2020-07-16
Published in Print: 2020-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 18.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0217/html
Scroll to top button