Weighted Integral Inequalities for the Hardy Type Operator and the Fractional Maximal Operator

Let w ( x ), u ( x ) and υ( x ) be weight functions. In this paper, under appropriate conditions on Young's functions Φ 1 , Φ 2 we characterize the inequalityΦ 2 − 1 ( ∫ 0 ∞Φ 2 ( T f ( x ) ) w ( x ) d x ) ⩽ Φ 1 − 1 ( ∫ 0 ∞Φ 1 ( C f ( x ) u ( x ) ) υ ( x ) d x ) for the Hardy‐type operator T defined in [1] and the inequalityΦ 2 − 1 ( ∫ R nΦ 2 ( M α,υ ( f υ ) ( x ) ) w ( x ) d v ( x ) ) ⩽ Φ 1 − 1 ( ∫ R nΦ 1 ( C f ( x ) ) υ ( x ) d x ) for the fractional maximal operator M α, υ; defined in [8], as well as the corresponding weak‐type inequalities.