Abstract
In the paper, we prove that the maximal commutator of conditional expectations \(\sup _{n\ge 0}|[{\mathbb {E}}_n,g]|\) is bounded from \(H_1^S\) to \(L_{1,\infty }\) provided \(g\in \textrm{BMO}_2\). If \(g\in \textrm{bmo}_2\), we also show that a modified maximal operator \(\sup _{n\ge 0}{\mathbb {E}}_n(|[{\mathbb {E}}_n,g]|)\) is bounded from \(\textrm{h}_1\) to \(L_{1,\infty }\).
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Agcayazi, M., Gogatishvili, A., Koca, K., Mustafayev, R.: A note on maximal commutators and commutators of maximal functions. J. Math. Soc. Jpn. 67(2), 581–593 (2015)
Alphonse, A.: An end point estimate for maximal commutators. J. Fourier Anal. Appl. 6(4), 449–456 (2000)
Burkholder, D.: Martingale transforms. Ann. Math. Stat. 37, 1494–1504 (1966)
Chao, J., Daly, J., Ombe, H.: Factorizations of Hardy spaces of simple martingales. Tamkang J. Math. 19(4), 57–65 (1988)
Chao, J., Ombe, H.: Commutators on dyadic martingales. Proc. Jpn. Acad. Ser. A Math. Sci. 61(2), 35–38 (1985)
Chen, W., Sawyer, E.: Endpoint estimates for commutators of singular integrals on spaces of homogeneous type. J. Math. Anal. Appl. 282(2), 553–566 (2003)
Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)
Davis, B.: On the integrability of the martingale square function. Israel J. Math. 8, 187–190 (1970)
Fu, Z., Pozzi, E., Wu, Q.: Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions, arXiv preprint arXiv:1907.09953 (2019)
García-Cuerva, J., Harboure, E., Segovia, C., Torrea, J.: Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ. Math. J. 40(4), 1397–1420 (1991)
Garsia, A.: Martingale Inequalities: Seminar Notes on Recent Progress. Mathematics Lecture Notes Series. W. A. Benjamin, Inc., Reading (1973)
Gundy, R.: A decomposition for \(L^{1}\)-bounded martingales. Ann. Math. Stat. 39, 134–138 (1968)
Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Mat. 16(2), 263–270 (1978)
Janson, S.: BMO and commutators of martingale transforms. Ann. Inst. Fourier (Grenoble) 31(1, viii), 265–270 (1981)
Li, D., Hu, G., Shi, X.: Weighted norm inequalities for the maximal commutators of singular integral operators. J. Math. Anal. Appl. 319(2), 509–521 (2006)
Long, R.: Martingale Spaces and Inequalities. Peking University Press and Friedr. Vieweg & Sohn, Beijing and Braunschweig (1993)
Meyer, P.: Notes sur les intégrales stochastiques. IV. Caractérisation de BMO par un opérateur maximal, Sèminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976), Lecture Notes in Mathematics, vol. 581, pp. 470–475 (1977)
Nakai, E., Sadasue, G.: Commutators of fractional integrals on martingale Morrey spaces. Math. Inequal. Appl. 22(2), 631–655 (2019)
Pérez, C.: Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128(1), 163–185 (1995)
Segovia, C., Torrea, J.: Higher order commutators for vector-valued Calderón–Zygmund operators. Trans. Am. Math. Soc. 336(2), 537–556 (1993)
Treil, S.: Commutators, paraproducts and BMO in non-homogeneous martingale settings. Rev. Mat. Iberoam. 29(4), 1325–1372 (2013)
Weisz, F.: Martingale Hardy Spaces and Their Applications in Fourier Analysis. Lecture Notes in Mathematics, vol. 1568. Springer, Berlin (1994)
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Communicated by See Keong Lee.
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Li, Y. Maximal Commutator of Conditional Expectations. Bull. Malays. Math. Sci. Soc. 46, 29 (2023). https://doi.org/10.1007/s40840-022-01434-6
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DOI: https://doi.org/10.1007/s40840-022-01434-6