Boundedness of anisotropic operators on vanishing Morrey type spaces

This paper investigates the boundedness of anisotropic almost quasi-linear integral operators of $(p,q)$-type in the framework of generalized Morrey spaces. Extending classical results, including the boundedness of the Riesz potentials in Lebesgue spaces and the Morrey space estimates of Chiarenza and Frasca, we develop a theory adapted to anisotropic settings characterized by non-uniform scaling. In particular, we consider generalized anisotropic Morrey spaces $\mathcal{M}^{p,\varphi}_{\mathbf a}(\mathbb{R}^n)$, defined in terms of suitable admissible functions $\varphi$ and anisotropy vector $\mathbf a$. We also establish corresponding boundedness results in anisotropic generalized vanishing Morrey spaces, which capture finer local behavior at small scales. Our results unify and extend the known isotropic theory, providing a flexible framework for the analysis of operators in anisotropic settings. As a consequence, they recover and generalize several classical boundedness properties.