Characterizations of elements with compact support in the dual spaces of $A_{p}(G)$-modules of $PM_{p}(G)$

For a locally compact group $G$ and $1 < p < \infty$, let $A_{p}(G)$ be the Figà-Talamanca-Herz algebra and let $PM_{p}(G)$ be its dual Banach space. For a Banach $A_{p}(G)$-module $X$ of $PM_{p}(G)$, we denote the norm closure of the subspace of the elements in $X^{*}$ with compact support by $A_{p,X}(G)$. We prove that an element $u$ of $X^{*}$ is in $A_{p,X}(G)$ if and only if for any $\epsilon > 0$, there exists a compact subset $K$ of $G$such that $\vert \langle u, f \rangle \vert < \epsilon$ for all $f\in X$ with $\Vert f\Vert \le 1$ and $supp \, (f)\subseteq G\sim K$. In particular, we have that an element $b$ of $W_{p}(G)$ is in $A_{p}(G)$ if and only if for any $\epsilon > 0$, there exists a compact subset $K$ of $G$ such that $\vert \langle u, f \rangle \vert < \epsilon$ for all $f\in L^{1}(G\sim K)$ with $\Vert f\Vert \le 1$. If $A_{p, X}(G)$ has an orthogonal complement $A_{p, X}^{s}(G)$ in $X^{*}$, we characterize $A_{p, X}^{s}(G)$ by the following condition: $u\in X^{*}$ is in $A_{p, X}^{s}(G)$ if and only if for any $\epsilon > 0$ and any compact subset $K$ of $G$, there exists some $f\in X$with $\Vert f\Vert \le 1$ and $supp\, (f)\subseteq G\sim K$ such that $\vert \langle u, f \rangle \vert > \Vert u\Vert - \epsilon$. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking $p=2$ and $X$ as some $C^{*}$-subalgebras of $PM_{p}(G)$.