Fractional sublinear operators with rough kernels and their commutators on new vanishing generalized Morrey spaces

This study focuses on a class of fractional sublinear operators, denoted as $ S_{\Omega, \alpha} $, and their commutators $ S_{\Omega, \alpha, b} $ with rough kernels. We establish the boundedness of these operators on newly emerging vanishing generalized Morrey spaces that are characterized by $ (V_\infty) $ or $ (V^*) $. The primary innovation of this paper lies in the novel approach of controlling $ S_{\Omega, \alpha} $ via the Riesz potential $ I_{\alpha} $ and the management of $ S_{\Omega, \alpha, b} $ through fractional maximal commutators with rough kernels $ M_{\Omega, \alpha-\varepsilon, b} $ and $ M_{\Omega, \alpha+\varepsilon, b} $ for some $ \varepsilon > 0 $.