Abstract: Based on the properties of Hardy operators and BMO functions, together with the definition of HerzMorrey spaces with variable exponent, and by using the Hlder inequalities, we established the boundedness of the higher order commutators generated by fractional Hardy operators with variable index and BMO functions on the HerzMorrey spaces with variable exponent. The result generalized the boundedness of the higher order commutators of the classical fractional Hardy operators to the higher order commutators of fractional Hardy operators with variable index. When the variable exponent β(x) was a constant, the fractional Hardy operators  were the classical fractional Hardy operators.

Key words: Herz-Morrey space, fractional Hardy operator, BMO space, commutator, variable index