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The aim of this paper is to give some constructions results of averagingoperators on Hom-Lie algebras. The homogeneous averaging operators on$q$-deformed Witt and $q$-deformed $W(2,2)$ Hom-algebras are classified. Asapplications, the induced Hom-Leibniz algebra structures are obtained and theirmultiplicativity conditions are also given.

Averaging operators on $q$-deformed Witt and $q$-deformed $W(2,2)$ algebras