In this paper we prove an O'Neil inequality for the convolution operator (G-convolution) associated with the Gegenbauer differential operator G λ. By using an O'Neil inequality for rearrangements we obtain a pointwise rearrangement estimate of the G-convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the G-fractional maximal and G-fractional integral operators from the spaces L p,λ to L q,λ and from the spaces L 1,λ to the weak spaces W L p,λ .

On the real line, the Dunkl operators∀ν ≥ − 1 2 are differential-difference operators associated with the reflection group Z 2 on R, and on the R d the Dunkl operators { D k,j } d j=1 are the differential-difference operators associated with the reflection group Z d 2 on R d . In this paper, in the setting R we show that b ∈ BMO(R,dm ν ) if and only if the maximal commutator M b,ν is bounded on Orlicz spaces L Φ (R,dm ν ). Also in the setting R d we show that b ∈ BMO(R d ,h 2 k (x)dx) if and only if the maximal commutator M b,k is bounded on Orlicz spaces L Φ (R d ,h 2 k (x)dx).