Abstract
As another application of Musielak-Orlicz Hardy space \(H^{\log }(\mathbb{R}^{n})\), we consider the boundedness of commutators in this chapter. It is well known that the linear commutator [b, T], generated by a BMO function b and a Calderón-Zygmund operator T, may not be bounded from \(H^{1}(\mathbb{R}^{n})\) into \(L^{1}(\mathbb{R}^{n})\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J. Alvarez, J. Hounie, Estimates for the kernel and continuity properties of pseudo-differential operators. Ark. Mat. 28, 1–22 (1990)
J. Alvarez, R.J. Bagby, D.S. Kurtz, C. Pérez, Weighted estimates for commutators of linear operators. Stud. Math. 104, 195–209 (1993)
R.R. Coifman, L. Grafakos, Hardy space estimates for multilinear operators, I. Rev. Mat. Iberoam. 8, 45–67 (1992)
R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)
R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 103, 611–635 (1976)
G. Dafni, Local VMO and weak convergence in h 1. Can. Math. Bull. 45, 46–59 (2002)
S. Dobyinsky, La “version ondelettes” du théorème du Jacobien, (French) [The “wavelet version” of the theorem of the Jacobian] Rev. Mat. Iberoam. 11, 309–333 (1995)
C. Fefferman, E.M. Stein, H p spaces of several variables. Acta Math. 129, 137–195 (1972)
M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)
M. Frazier, R.H. Torres, G. Weiss, The boundedness of Calderón-Zygmund operators on the spaces \(\dot{F}_{p}^{\alpha,q}\). Rev. Mat. Iberoam. 4, 41–72 (1988)
D. Goldberg, A local version of real Hardy spaces. Duke Math. J. 46, 27–42 (1979)
E. Harboure, C. Segovia, J.L. Torrea, Boundedness of commutators of fractional and singular integrals for the extreme values of p. Ill. J. Math. 41, 676–700 (1997)
P.W. Jones, J.-L. Journé, On weak convergence in \(H^{1}(\mathbb{R}^{n})\). Proc. Am. Math. Soc. 120, 137–138 (1994)
L.D. Ky, Bilinear decompositions and commutators of singular integral operators. Trans. Am. Math. Soc. 365, 2931–2958 (2013)
M.-Y. Lee, Weighted norm inequalities of Bochner-Riesz means. J. Math. Anal. Appl. 324, 1274–1281 (2006)
Z. Liu, S. Lu, Endpoint estimates for commutators of Calderón-Zygmund type operators. Kodai. Math. J. 25, 79–88 (2002)
S. Lu, Q. Wu, D. Yang, Boundedness of commutators on Hardy type spaces. Sci. China Ser. A. 45, 984–997 (2002)
Y. Meyer, Wavelets and Operators. Advanced Mathematics (Cambridge University Press, Cambridge, 1992)
Y. Meyer, R.R. Coifman, Wavelets, Calderón-Zygmund and Multilinear Operators. Advanced Mathematics (Cambridge University Press, Cambridge, 1997)
C. Pérez, Endpoint estimates for commutators of singular integral operators. J. Funct. Anal. 128, 163–185 (1995)
M.H. Taibleson, G. Weiss, The molecular characterization of certain Hardy spaces, in Representation Theorems for Hardy Spaces Astérisque, vol. 77 (Société Mathématique de France, Paris, 1980), pp. 67–149
D. Yan, G. Hu, J. Lan, Weak-type endpoint estimates for multilinear singular integral operators. Acta Math. Sin. (Engl. Ser.) 21, 209–214 (2005)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Yang, D., Liang, Y., Ky, L.D. (2017). Bilinear Decompositions and Commutators of Calderón-Zygmund Operators. In: Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Mathematics, vol 2182. Springer, Cham. https://doi.org/10.1007/978-3-319-54361-1_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-54361-1_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-54360-4
Online ISBN: 978-3-319-54361-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)