Generalized Integration Operators from Weak to Strong Spaces of Vector-valued Analytic Functions

For a fixed nonnegative integer $m$, an analytic map $\varphi$ and function $\psi$, the generalized integration operator $I^{(m)}_{\varphi,\psi}$ is defined by \[ I^{(m)}_{\varphi,\psi} f(z) = \int_0^z f^{(m)}(\varphi(\zeta)) \psi(\zeta) \, d\zeta \] for $X$-valued $f$, where $X$ Banach space. Some estimates norm of $I^{(m)}_{\varphi,\psi} \colon wA^p_{\alpha}(X) \to A^p_{\alpha}(X)$ are obtained. In particular, it shown that Volterra $J_b bounded if only A^2_{\alpha} A^2_{\alpha}$ in Schatten class $S_p(A^2_{\alpha})$ $2 \leq p \lt \infty$ $\alpha \gt -1$. corresponding results established Hardy spaces Fock spaces.