We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space $H_p$ to the classical Lebesgue space $L_p$ and from the variable dyadic martingale Hardy space $H_{p(\cdot )}$ to the variable Lebesgue space $L_{p(\cdot )}$. Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from $H_{p(\cdot )}$ to $L_{p(\cdot )}$ and from the variable Hardy–Lorentz space $H_{p(\cdot ),q}$ to the variable Lorentz space $L_{p(\cdot ),q}$. As a consequence, we can prove theorems about almost everywhere and norm convergence.