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 to the classical Lebesgue space and from the variable dyadic martingale Hardy space to the variable Lebesgue space . Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from to and from the variable Hardy–Lorentz space to the variable Lorentz space . As a consequence, we can prove theorems about almost everywhere and norm convergence.
"Boundedness of Cesàro and Riesz means in variable dyadic Hardy spaces." Banach J. Math. Anal. 13 (3) 675 - 696, July 2019. https://doi.org/10.1215/17358787-2018-0037