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This paper deals with the introduction and study of a new, Orlicz type space associated with the Gegenbauer differential operator and briefly called here a  G -Orlicz space. We prove that such spaces are Banach function spaces. The necessary and sufficient conditions are established which ensure the validity of weak and strong type inequalities for fractional maximal functions and fractional integrals of  G -Orlicz spaces.

Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator