Martingale Inequalities in Variable Exponent Hardy spaces with $0<p^{-}\le p^{+}<\mathrm{\infty }$

We investigate the properties of the variable Lebesgue spaces with quasi-norm on a probability space, and give the atomic decompositions suited to the variable exponent martingale Hardy spaces. Using the decompositions and the harmonic mean of a variable exponent, we obtain several continuous embedding relations between martingale Hardy spaces with small exponent. Finally, we extend these results to the cases $0<p^{-}\le p^{+}<\mathrm{\infty }\mathrm{.}$