Let X be a homogeneous space and let E be a UMD Banach space with a normalized unconditional basis ${\left(e_j\right)}_{j\ge 1}$. Given an operator T from $L_c^{\infty }\left(X\right)$ to L¹(X), we consider the vector-valued extension T̃ of T given by $T̃\left({\sum }_j{f}_j{e}_j\right)={\sum }_{j}T\left(f_j\right)e_j$. We prove a weighted integral inequality for the vector-valued extension of the Hardy-Littlewood maximal operator and a weighted Fefferman-Stein inequality between the vector-valued extensions of the Hardy-Littlewood and the sharp maximal operators, in the context of Orlicz spaces. We give sufficient...