The Unified Theory for the Necessity of Bounded Commutators and Applications

• Published in: The Journal of geometric analysis Vol. 30; no. 4; pp. 3995 - 4035
• Main Authors: Guo, Weichao, Lian, Jiali, Wu, Huoxiong
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2020; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Banach spaces; Commutators; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Function space; Geometry; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; 42B25; BMO; 42B20; Singular integrals; Lipschitz function spaces; Multilinear operators; Fractional integrals; Commutators; weights
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-019-00226-y

This paper gives unified criterions on the necessity of bounded commutators in linear and multilinear settings. Our results relax the restriction of Banach spaces in previous results to quasi-Banach spaces and extend
B
M
O
(
R
n
)
to the general
B
M
O
μ
, which includes
B
M
O
(
R
n
)
,
Lip
β
(
R
n
)
, and their weighted versions. Moreover, the conditions of kernels are also essentially weakened. As applications, some necessary conditions for bounded commutators, which are new in the endpoint case, and several new characterizations of
BMO
spaces, Lipschitz spaces, and their weighted versions via boundedness of commutators in various function spaces are deduced.