  The theory of tent spaces on $\mathbb{R}^n$ was introduced by Coifman, Meyer
and Stein, including atomic decomposition, duality theory and so on. Russ
generalized the atomic decomposition for tent spaces to the case of spaces of
homogeneous type $(X,d,\mu)$. The main purpose of this paper is to extend the
results of Coifman, Meyer, Stein and Russ to weighted version. More precisely,
we obtain a $q$-atomic decomposition for the weighted tent spaces
$T^p_{2,w}(X)$, where $0<p\leq 1, 1<q<\infty,$ and $w\in A_\infty$. As an
application, we give an atomic decomposition for weighted Hardy spaces
associated to nonnegative self-adjoint operators on $X$.
