The Unified Theory for the Necessity of Bounded Commutators and Applications

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.