Abstract
We prove that the weak Morrey space WM pq is contained in the Morrey space \(M_{{q_1}}^p\) for 1 ≤ q1 < q ≤ p < ∞. As applications, we show that if the commutator [b, T] is bounded from Lp to Lp,∞ for some p ∈ (1, ∞), then b ∈ BMO, where T is a Calderón-Zygmund operator. Also, for 1 < p ≤ q < ∞, b ∈ BMO if and only if [6, T] is bounded from M pq to WM pq . For b belonging to Lipschitz class, we obtain similar results.
Similar content being viewed by others
References
Adams, D.R. A note on Riesz potentials. Duke Math. J., 42: 765–778 (1975)
Coifman, R., Rochberg, R., Weiss, G. Factorization theorems for Hardy spaces in several variables. Ann. of Math., 103: 611–635 (1976)
Devore, R.A., Sharpley, R.C. Maximal functions measuring smoothness. Mem. Amer. Math. Soc., 47: 293 (1984)
Ding, Y. A characterization of BMO via commutators for some operators. Northeast. Math. J., 13: 422–432 (1997)
Janson, S. Mean oscillation and commutators of singular integral operators. Ark. Math., 16: 263–270 (1978)
Janson, S., Taibleson, M., Weiss, G. Elementary characterization of the Morrey-Campanato spaces. Lect. Notes in Math., 992: 101–114 (1983)
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)
Paluszyński, M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ. Math. J., 44: 1–17 (1995)
Peetre, J. On the theory \({{\cal L}_{p,\lambda}}\) spaces. J. Funct. Anal., 4: 71–87 (1969)
Shi, S. G., Lu, S. Z. Some characterizations of Campanato spaces via commutators on Morrey spaces. Pacific J. Math., 264: 221–234 (2013)
Tang, L. Endpoint estimates for multilinear fractional integrals. J. Aust. Math. Soc., 84: 419–429 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Conflict of Interest
The authors declare no conflict of interest.
The project is supported by the National Natural Science Foundation of China (No. 12101010) and the Natural Science Foundation of Anhui Province (No.2108085QA19).
Rights and permissions
About this article
Cite this article
Wang, Dh., Zhou, J. Characterizations of the BMO and Lipschitz Spaces via Commutators on Weak Lebesgue and Morrey Spaces. Acta Math. Appl. Sin. Engl. Ser. 39, 583–590 (2023). https://doi.org/10.1007/s10255-023-1077-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-023-1077-0