We continue the work of X. Zhu, L. Hu and D. Qu (2022) by investigating the boundedness and essential norm of the Volterra integral operator $V_g$ and the associate integral operator $S_g$ from Morrey-type spaces ${\mathscr{H}}_K^p$ to Dirichlet–Morrey type spaces ${\mathcal{𝒟}}_{p-1}^{p,K}$.

There are the several different methods concerning studies of quadratic, quadratic fractional, linear and nonlinear integral equations of Volterra- and Fredholm-type in various spaces. Method of studies of such equations in locally convex spaces with approximation procedure and presented here is new, general, optimal and precise, and also different from those known in the literature. Examples provided in this paper illustrate chosen aspects of our method. The main tools used here are based on our special cases of very general and stronger periodic point, fixed point and approximation theorems concerning not necessarily continuous dynamic systems in not necessarily sequentially complete or separable gauge spaces.

We study compact intertwining relations for composition operators whose intertwining operators are Volterra-type operators between Morrey spaces and Bloch-type spaces in the unit disk. As a consequence, we find a new connection between Morrey spaces and Bloch-type spaces through these relations.

Let $T$ be a bounded operator on $L^p({ℝ}^n)$. Under the assumption that the kernel of $T$ satisfies some Hörmander-type estimates, we obtain a boundedness criterion for the multilinear commutators $T_{\overrightarrow{b}}$ on the weighted Lebesgue spaces $L^p(\omega )$ with $\overrightarrow{b}\in \text{BMO}({ℝ}^n)$ and $\omega$ belonging to the Muckenhoupt weight class $A_{p∕m^{\prime }}$. Further, for $\overrightarrow{b}\in \text{CMO}({ℝ}^n)$, the vanishing mean oscillation space, a criterion of $L^p$-weighted compactness of $T_{\overrightarrow{b}}$ is established. As applications, the weighted $L^p$-boundedness and $L^p$-compactness criteria can be applied to the $𝜃$-type Calderón–Zygmund operator and its commutators.