Endpoint and Interpolation Estimates for Higher-Order Commutators of Rough Fractional Maximal Operators with Variable Kernels on Variable Exponent Morrey Spaces

Abstract

This paper investigates the higher-order commutators generated by fractional maximal operators with rough, spatially dependent kernels in the framework of variable exponent Morrey spaces. Under minimal log-Hölder continuity assumptions on the variable exponent profiles and suitable geometric constraints on the Morrey-type weights, we establish comprehensive strong interior boundedness results between the appropriate spaces. We further analyze the critical endpoint boundary configurations where classical strong-type boundedness fails due to Luxemburg norm blow-up, proving that a corresponding sharp weak-type estimate remains structurally valid and revealing the underlying transitions between different boundedness regimes. In addition, an abstract real interpolation framework of Grafakos--Martell type is developed to bridge these endpoint and interior strong estimates, thereby recovering a full continuous scale of intermediate regularity properties. These results extend the classical harmonic analysis scheme to a broader nonhomogeneous setting and provide new insights into the continuous interplay between rough kernels, variable exponents, and local weighted growth conditions.