Higher-order commutators of fractional Hardy-type operators in grand variable Herz–Morrey spaces

In this paper, we investigate the boundedness properties of commutator associated with fractional Hardy-type operators in the framework of grand variable Herz–Morrey spaces. These spaces provide a flexible functional setting that simultaneously captures variable integrability, local Morrey-type behavior, and global Herz-type decay. Our main result establishes a Bounded Mean Oscillation (BMO) type estimate for commutator of fractional Hardy-type operators acting on grand variable Herz–Morrey spaces. The proof relies on the use of generalized Hölder and Minkowski inequalities together with a careful decomposition technique. In particular, the key summation involved in the analysis is split into several components, each of which is estimated under appropriate conditions on the parameters and the variable exponent functions. By combining these estimates in a systematic manner, we obtain the desired boundedness results for the commutator. The obtained results extend several known boundedness results for Hardy-type operators and their commutator to a more general functional framework involving both variable exponents and grand-type structures. Consequently, this work contributes to the development of harmonic analysis in generalized function spaces and provides new tools for the study of operators arising in problems with nonstandard growth conditions.