Averages over Surfaces with Infinitely Flat Points

Let S be a hypersurface in Rn , n ≥ 2, and dμ = ψ dσ, where ψ ∈ C∞0 (Rn) and σ denotes the surface area measure on S. Define the maximal function M associated to S and μ by [formula] It was shown by Stein that when S is the sphere in Rn, n ≥ 3, M (the spherical maximal function) is bounded on Lp(Rn) if and only if p > n/(n − 1). It has also been shown that if S is of finite type, i.e., the curvature vanishes to at most a finite order m at every point of S, then there exists some number pm < ∈ such that M is bounded on Lp(Rn) (n ≥ 3) for all p ∈ (pm, ∈]. On the other hand it is well known that if S is flat, that is, S contains a point at which the curvature vanishes to infinite order, then M may not be bounded on any Lp(Rn), p < ∞. We show that under some hypotheses the maximal functions M associated to flat surfaces S ⊂ R3 are bounded on certain Orlicz spaces LΦ(R3) defined naturally in terms of S.