Bounded composition operators on weighted function spaces in the unit disk

We introduce a general class of weighted spaces $\mathscr{H}(\beta)$ of holomorphic functions in the unit disk $\mathbb{D}$, which contains several classical spaces, such as Hardy space, Bergman space, Dirichlet space. We characterize boundedness of composition operators $C_{\varphi}$ induced by affine and monomial symbols $\varphi$ on these spaces $\mathscr{H}(\beta)$. We also establish a sufficient condition under which the operator $C_{\varphi}$ induced by the symbol $\varphi$ with relatively compact image $\varphi(\mathbb{D})$ in $\mathbb{D}$ is bounded on $\mathscr{H}(\beta)$. Note that in the setting of $\mathscr{H}(\beta)$, the characterizations of boundedness of composition operators $C_{\varphi}$ depend closely not only on functional properties of the symbols $\varphi$ but also on the behavior of the weight sequence $\beta$.