A Unifying Approach for Certain Class of Maximal Functions

where U is the class of all h∈ L2(R+,r−1dr) with ‖h‖L2(R+,r−1dr) ≤ 1. The operator Ω was introduced by Chen and Lin [7]. They showed that Ω is bounded on Lp(Rn) for all p > 2n/(2n− 1) provided that Ω ∈ (Sn−1). Recently, we have been able to show that the Lp(Rn) boundedness of Ω still holds for all p ≥ 2 if the condition Ω ∈ (Sn−1) is replaced by the more natural and weaker condition Ω ∈ L(logL)1/2(Sn−1) [2]. Moreover, we showed that if the condition Ω∈ L(logL)1/2(Sn−1) is replaced by any condition in the form Ω ∈ L(logL)r(Sn−1) for some r < 1/2, then Ω might fail to be bounded on L2.