The closure of derivative tent spaces in the logarithmic Bloch-type norm

In this paper, the derivative tent space $DT_p^q(\alpha)$ is introduced. Then, we study $\mathcal{C}_{\mathcal{B}_{{\log}^\gamma}^\beta}(DT_p^q(\alpha)\cap\mathcal{B}_{{\log}^\gamma}^\beta)$, the closure of the derivative tent space $DT_p^q(\alpha)$ in the logarithmic Bloch-type space $\Blog$. As a byproduct, some new characterizations for $C_\mathcal{B}(\mathcal{D}^p_{\alpha} \cap \mathcal{B})$ and $C_{\mathcal{B}_{{\log}}}(\mathcal{D}^2_{\alpha}\cap\mathcal{B}_{{\log}})$ are obtained.