Characterizations of Variable Martingale Hardy Spaces Via Maximal Functions

We introduce a new type of dyadic maximal operators and prove that under the log-Hölder continuity condition of the variable exponent p (⋅), it is bounded on L p (⋅) if 1 < p − ≤ p + ≤ ∞. Moreover, the space generated by the L p (⋅) -norm (resp. the L p (⋅), q -norm) of the maximal operator is equivalent to the Hardy space H p (⋅) (resp. to the Hardy-Lorentz space H p (⋅), q ). As special cases, our maximal operator contains the usual dyadic maximal operator and four other maximal operators investigated in the literature.