AVERAGES OVER SURFACES WITH INFINITELY FLAT POINTS

Let S be a hypersurface in R '', n greater than or equal to 2, and d m u = psi d sigma, where psi epsilon C-0 (infinity) (R(n)) and sigma den otes the surface area measure on S. Define the maximal function M asso ciated to S and mu by [GRAPHICS] It was shown by Stein that when S is the sphere in R(n), n greater than or equal to 3, M (the spherical max imal function) is bounded on L(p)(R(n)) if and only if p > n/(n = l). It has also been shown that if S is of finite type, i.e., the curvatur e vanishes to at most a finite order m at every point of S, then there exists some number p(m) < infinity such that M is bounded on L(p)(R(n )) (n greater than or equal to 3) for all p epsilon (p(m), infinity]. On the other hand it is well known that if S is flat, that is, S conta ins a point at which the curvature vanishes to infinite order, then M may not be bounded on any L(p)(R(n)), p < infinity(.) We show that und er some hypotheses the maximal functions M associated to flat surfaces S subset of R(3) are bounded on certain Orlicz spaces L(Phi)(R(3)) de fined naturally in terms of S. (C) 1995 Academic Press, Inc.