Theorems on iterations of partial integrals in a space with mixed norm
In $\mathbb{R}_2$, we consider partial integrals acting on the first or second variable and obtain conditions for bounded action in spaces of continuous functions with respect to one of the variables with values in the Lebesgue class $L_p$ with respect to the other variable. We assume that these functions are defined in a finite rectangle $D\in\mathbb{R}_2$. We prove theorems on the boundedness of iterations of these partial integrals in the spaces of anisotropic functions $C(D_\alpha{(1)}; L_p(D_{\overline{\alpha}}{(1)}))$, where $\alpha$ and $\overline{\alpha}$ are indices complementing each other up to the double index $(1;2)$.