For \(\vec {p} = (p_1, p_2, \dots , p_n)\) with \(0<p_i<\infty , 0<q< \infty \) , and \(\alpha \in \mathbb {R}\) , we consider the inhomogeneous weighted mixed-norm Besov spaces \(B^{\alpha , q}_{\vec {p}} (w, \mathbb {R}^n)\) , where the weight w is a function of \(x_n\) only. We denote the unweighted space, where \(w=1\) , as \(B^{\alpha , q}_{\vec {p}} (\mathbb {R}^n)\) . Under the assumption that w belongs to the Muckenhoupt class \(A_{\max (1, p_n)}(\mathbb {R})\) , we prove characterizations of the quasi-norm of \(f \in B^{\alpha , q}_{\vec {p}} (w)\) in terms of an associated quasi-norm on the sequence of coefficients of f in various Littlewood-Paley and wavelet decompositions. We apply these results to show that if \(\gamma >0, \vec {p}= (p^{\, \prime }, p_{n+1}) \in (0, \infty )^{n+1}\) , and \(w \in A_{\max (1, p_{n+1})}(\mathbb {R})\) , the trace operator \(\begin{aligned}T: B^{\alpha , q}_{\vec {p}} (w, \mathbb {R}^{n+1}) \rightarrow B^{\alpha - \gamma /p_{n+1}, q}_{\vec {p}^{\, \prime }} (\mathbb {R}^n),\end{aligned}\) is bounded if and only if the weight w satisfies \(\begin{aligned} w([0, 2^{-j}]) \ge C 2^{-j \gamma }\,\,\, \text{ for } \text{ all } \,\,\, j \in \mathbb {N},\end{aligned}\) for a certain range of \(\alpha , \vec {p}, q\) , and \(\gamma \) . Note that the loss \(\gamma /p_{n+1}\) in the smoothness index from the domain to the co-domain of T is independent of \(p_1, \dots , p_n, q\) , and \(\alpha \) . If also there exists \(C>0\) such that \(w([0, 2^{-j}]) \le C 2^{-j \gamma }\) for all \(j \in \mathbb {N}\) , then there exists a linear, bounded extension operator \(E: B^{\alpha - \gamma /p_{n+1}, q}_{\vec {p}^{\, \prime }} (\mathbb {R}^n) \rightarrow B^{\alpha , q}_{\vec {p}} (w, \mathbb {R}^{n+1})\) such that \(T\circ E\) is the identity on \(B^{\alpha - \gamma /p_{n+1}, q}_{\vec {p}^{\, \prime }} (\mathbb {R}^n)\) .
