Abstract
We present the details of an extension of known results about the compactness of the commutators of bilinear Calderón–Zygmund operators with multiplication by appropriate functions in the John–Nirenberg space, to the case when the target of the operators is a quasi-Banach Lebesgue space.
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Notes
See the comments in [1, p. 3612] and the references therein for the difference between CMO and other similar spaces in the literature and the not uniformly used notation such as, for example, VMO.
References
Bényi, Á., and R.H. Torres. 2013. Compact bilinear operators and commutators. Proceedings of the American Mathematical Society 141 (10): 3609–3621.
Bényi, Á., W. Damián, K. Moen, and R.H. Torres. 2015. Compact bilinear commutators: The weighted case. Michigan Mathematical Journal 64 (1): 39–51.
Chaffee, L., P. Chen, Y. Han, R.H. Torres, and L.A. Ward. 2018. Characterization of compactness of commutators of bilinear singular integral operators. Proceedings of the American Mathematical Society 146 (9): 3943–3953.
Coifman, R., R. Rochberg, and G. Weiss. 1976. Factorization theorems for Hardy spaces in several variables. Annals of Mathematics 103 (3): 611–635.
Grafakos, L., and R.H. Torres. 2002. Multilinear Calderón–Zygmund theory. Advances in Mathematics 165 (1): 124–164.
Grafakos, L., and R.H. Torres. 2002. Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana University Mathematics Journal 51 (5): 1261–1276.
Lerner, A., S. Ombrosi, C. Pérez, R. Torres, and R. Trujillo-González. 2009. New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Advances in Mathematics 220 (4): 1222–1264.
Pérez, C., and R.H. Torres. 2003. Sharp maximal function estimates for multilinear singular integrals. Contemporary Mathematics 320: 323–331.
Tang, L. 2008. Weighted estimates for vector-valued commutators of multilinear operators. Proceedings of the Royal Society of Edinburgh Section A 138: 897–922.
Tsuji, M. 1951. On the compactness of space \(L^p\) (\(p > 0\)) and its application to integral operators. Kodai Mathematical Seminar Reports 3: 33–36.
Uchiyama, A. 1978. On the compactness of operators of Hankel type. Tôhoku Mathematical Journal 30(2) (1): 163–171.
Yan, J. 2015. Multilinear vector-valued singular integral operators and square operators (in Chinese). PhD thesis, Beijing Normal University, Beijing, People’s Republic of China.
Yosida, K. 1995. Functional analysis. Berlin: Springer.
Acknowledgements
R. H. Torres would like to thank B. Wick for pointing out the reference [10]. He would also like to thank G. P. Youvaraj for his warm hospitality during the International Conference on Harmonic Analysis and Wavelets at the Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India, March 2017, where he presented several results related to the content of this article.
Funding
Q. Xue was partly supported by NSFC (nos. 11471041, 11671039).
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Torres, R.H., Xue, Q. & Yan, J. Compact bilinear commutators: the quasi-Banach space case. J Anal 26, 227–234 (2018). https://doi.org/10.1007/s41478-018-0144-z
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DOI: https://doi.org/10.1007/s41478-018-0144-z