Abstract
In this paper, we prove that the weighted BMO space
is independent of the scale p ∈ (0, ∞) in sense of norm when ω ∈ A1. Moreover, we can replace Lp(ω) by Lp,∞(ω). As an application, we characterize this space by the boundedness of the bilinear commutators [b, T]j(j = 1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol b, from \({L^{{p_1}}}(\omega ) \times {L^{{p_2}}}(\omega )\) to Lp(ω1−p) with 1 < p1,p2 < ∞ and 1/p =1/p1 + 1/p2. Thus we answer the open problem proposed by Chaffee affirmatively.
Similar content being viewed by others
References
Bloom, S.: A commutator theorem and weighted BMO. Trans. Amer. Math. Soc., 292, 103–122 (1985)
Chaffffee, L.: Characterizations of BMO through commutators of bilinear singular integral operators. Proc. Royal Soc. Edinburgh A., 146, 1159–1166 (2016)
Chanillo, S.: A note on commutators. Indiana Univ. Math. J., 31, 7–16 (1982)
Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. of Math., 103, 611–635 (1976)
Fefferman, C., Stein, E. M.: Hp spaces of several variables. Acta Math., 129, 137–193 (1972)
García-Cuerva, J.: Hardy spaces and Beurling algebras. J. Lond. Math. Soc., 39, 499–513 (1989)
Janson, S.: Mean oscillation and commutators of singular integral operators. Ark. Math., 16, 263–270 (1978)
John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm Pure Appl. Math., 2, 415–426 (1961)
Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Differential Equations, 19, 959–1014 (1994)
Krantz, S., Li, S. Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications, I. J. Math. Anal. Appl., 258, 629–641 (2001)
Krantz, S., Li, S. Y.: Boundedness and compactness of integral operators on spaces of homogeneous type and applications, II. J. Math. Anal. Appl., 258, 642–657 (2001)
Lerner, A. K., Ombrosi, S., Pérez, C., et al.: New maximal functions and multiple weights for the multilinear Calderon-Zygmund theory. Adv. Math., 220, 1222–1264 (2009)
Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165, 207–226 (1972)
Stein, E. M.: Singular integral and differentiability properties of functions. Princeton University Press, Princeton, 1971
Strömberg, J. O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana U. Math. J., 23, 511–544 (1979)
Uchiyama, A.: On the compactness of operators of Hankel type. Tôhoku Math. J., 30, 163–171 (1978)
Wang, D. H., Zhou, J., Chen, W. Y.: Another characterizations of Muckenhoupt Ap class. Acta Math. Sci. Ser. B, 37, 1761–1774 (2017)
Wang, S. B, Jiang, Y. S., Pan, J. B.: Necessary and sufficient conditions for boundedness of commutators of multilinear fractional integral operators. Acta Math. Sci. Ser. A, 35, 1106–1114 (2015)
Acknowledgements
We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by National Natural Science Foundation of China (Nos. 11971237, 12071223), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 19KJA320001) and Doctoral Scientific Research Foundation (Grant No. 903/752041)
Rights and permissions
About this article
Cite this article
Wang, D.H., Zhou, J. & Teng, Z.D. Characterizations of Weighted BMO Space and Its Application. Acta. Math. Sin.-English Ser. 37, 1278–1292 (2021). https://doi.org/10.1007/s10114-021-9567-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-021-9567-6