Characterizations of Variable Martingale Hardy Spaces Via Maximal Functions

Abstract

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 Lp(⋅) if 1 < p− ≤ p+ ≤ ∞. Moreover, the space generated by the Lp(⋅)-norm (resp. the Lp(⋅), q-norm) of the maximal operator is equivalent to the Hardy space Hp(⋅) (resp. to the Hardy-Lorentz space Hp(⋅), q). As special cases, our maximal operator contains the usual dyadic maximal operator and four other maximal operators investigated in the literature.