Weighted differentiation composition operators from $\mathcal{Q}_K(p,q)$ spaces to classical weighted spaces

For $-2 q \infty$ and $0 p \infty$, the $\mathcal{Q}_K(p,q)$ space is the space of all analytic functions on the open unit disk $\mathbb{D}$ satisfying $$\sup_{a\in\mathbb{D}}\int_\mathbb{D}|f (z)|^p(1-|z|^2)^q K(g(z,a))dA(z) \infty,$$ where $g(z,a)=\log\frac{1}{|\sigma_a(z)|}$ is the Green s function on $\mathbb{D}$ and $K:[0,\infty)\rightarrow[0,\infty)$, is a right-continuous and non-decreasing function. The boundedness and compactness of the weighted differentiation composition operators from $\mathcal{Q}_K(p,q)$ spaces and $\mathcal{Q}_{K,0}(p,q)$ spaces into the classical weighted spaces and the little classical weighted spaces are characterized.