Differences weighted composition operators acting between kind of weighted Bergman-type spaces and the Bers-type space -I-

Let $\mathcal{O}(\mathbb{D})$ denote the class of all analytic or holomorphic functions on the open unit disk $\mathbb{D}$ of $\mathbb{C}$. Let $\phi$ and $\psi$ are an analytic self-maps of $\mathbb{D}$ and $u,v\in \mathcal{O}(\mathbb{D}).$ The difference of two weighted composition operators is defined by

$$T_{\phi ,\psi }f(z):=(W_{\phi ,u}f-W_{\psi ,v}f)(z)=u(z)(f\circ \phi )(z)-v(z)(f\circ \psi )(z),f\in \mathcal{O}(\mathbb{D})\text{and}z\in \mathbb{D}.$$

The boundedness and compactness of the differences of two weighted composition operators from ${\mathcal{H}}_{\alpha }^{\mathrm{\infty }}(\mathbb{D})$ spaces into ${\mathcal{N}}_K(\mathbb{D})$ spaces (resp. from ${\mathcal{N}}_K(\mathbb{D})$ into ${\mathcal{H}}_{\alpha }^{\mathrm{\infty }}(\mathbb{D})$) are investigate in this paper.