Weighted norm inequalities for vector-valued singular integrals on homogeneous spaces

Let $X$ be an homogeneous space and let $E$ be an UMD Banach space with a normalized unconditional basis $(e_j)_{j\geq 1}$. Given an operator $T$ from $L^{\infty}_c(X)$ in $L^1(X)$, we consider the vector-valued extension ${\widetilde T}$ of $T$ given by ${\widetilde T}(\sum_jf_je_j)=\sum_jT(f_j)e_j$. 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 conditions on the kernel of a singular integral operator to have the boundedness of the vector-valued extension of this operator on $L^p(X,Wd\mu;E)$ for $1