Spherical Maximal Operators on Radial Functions

where dσ is the rotationally invariant measure on Sd−1, normalized such that σ(Sd−1) = 1. Stein [5] showed that limt→0Atf(x) = f(x) almost everywhere, provided f ∈ L(R), p > d/(d − 1) and d ≥ 3. Later Bourgain [1] extended this result to the case d = 2. If p ≤ d/(d − 1) then pointwise convergence fails. However if {tj}j=1 is a fixed sequence converging to 0 then pointwise convergence may hold for all f ∈ L even if p ≤ d/(d− 1), and p depends on geometric properties of the sequence {tj}. According to a theorem by Stein [4] pointwise convergence holds for all f ∈ L if the associated maximal operator MEf(x) = sup t∈E |Atf(x)|