Maximal averages over surfaces

Let Mf(x) = sup(t>0)\f*delta(t)(psi d sigma)(x)\ denote the maximal operator associated with surface measure d sigma on a smooth surface S. We prove that if S is convex and has finite order contact with its tangent lines, then M is bounded on L-p(R-n), p > 2, if and only if d(x, H)(-1) is an element of L-loc(1/p)(S) for all tangent planes H not passing through the origin. Let M'f(x) = sup(t>0)\f*delta(t)'(psi d sigma)(x)\ be the maximal operator associated with a nonisotropic dilation delta(t)' of surface measure d sigma. We prove that M' often behaves far better than M due to a rotational curvature in the time parameter t. (C) 1997 Academic Press.