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Morrey-Herz space; variable exponent; sublinear operator; commutator
We introduce a new type of variable exponent function spaces $M\dot K^{\alpha (\cdot ),\lambda }_{q,p(\cdot )}(\mathbb R^n)$ of Morrey-Herz type where the two main indices are variable exponents, and give some propositions of the introduced spaces. Under the assumption that the exponents $\alpha $ and $p$ are subject to the log-decay continuity both at the origin and at infinity, we prove the boundedness of a wide class of sublinear operators satisfying a proper size condition which include maximal, potential and Calderón-Zygmund operators and their commutators of BMO function on these Morrey-Herz type spaces by applying the properties of variable exponent and BMO norms.
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