We obtain the factorization theorem for Hardy space via the variable exponent Lebesgue spaces. As an application, it is proved that if the commutator of Coifman, Rochberg and Weiss is bounded on the variable exponent Lebesgue spaces, then is a bounded mean oscillation (BMO) function.
A version of the John-Nirenberg inequality suitable for the functions with is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.
We study the mapping property of the commutator of Hardy-Littlewood maximal function on Triebel-Lizorkin spaces. Also, some new characterizations of the Lipschitz spaces are given.
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