On uniformly continuous surjections between $C_p$-spaces over metrizable spaces

Let $X$ be metrizable, $Y$ be perfectly normal and suppose that there exists a uniformly continuous surjection $T: C_{p}(X) \to C_{p}(Y)$ (resp., $T: C_{p}^*(X) \to C_{p}^*(Y)$), where $C_{p}(X)$ (resp., $C_{p}^*(X)$) denotes the space of all real-valued continuous (resp., continuous and bounded) functions on $X$ endowed with the pointwise convergence topology. We show that if additionally $T$ is an inversely bounded mapping and $X$ has some dimensional-like property $\mathcal P$, then so does $Y$. For example, this is true if $\mathcal P$ is one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. Also, we consider other properties $\mathcal P$: of being a scattered, or a strongly $\sigma$-scattered space, or being a $\Delta_1$-space (see [17]). Our results strengthen and extend several results from [6], [13], [17].