A b s t r a c t . Our first aim in this paper is to prove the boundedness of some sublinear operators on Herz spaces with variable exponent. As an application, we give characterizations and unconditional bases of the spaces in terms of wavelets.

Our aim in the present paper is to prove the boundedness of vectorvalued commutators on Herz spaces with variable exponent. In order to obtain the result, we clarify a relation between variable exponent and BMO norms.

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Our first aim in this paper is to prove boundedness of commutators with fractional integrals on Lebesgue spaces with variable exponent. We additionally obtain the boundedness on Herz spaces with variable exponent applying some properties of variable exponent and BMO norms.

Our goal is to obtain the John-Nirenberg inequality for ball Banach function spaces X, provided that the Hardy-Littlewood maximal operator M is bounded on the associate space X by using the extrapolation. As an application we characterize BMO, the bounded mean oscillation, via the norm of X.