The Boundedness for k-th Order Commutators of Fractional Integral Operators with Variable Kernels

Commutators of fractional integral operators play an important role in harmonic analysis due to their close connections with function regularity and partial differential equations. In this paper, we study higher-order commutators of fractional integral operators with rough variable kernels. Compared with first-order commutators, the higher-order setting involves more complicated interactions between the oscillation of the underlying BMO function and the fractional integral structure, which requires new ideas and techniques. We establish boundedness properties for these higher-order commutators under sharp conditions on the angular integrability of the variable kernels. Our results extend the existing boundedness theory of first-order commutators to higher-order cases and demonstrate the applicability of the developed techniques to a broader class of fractional integral operators with rough variable kernels.