Maximal Averages over Surfaces

LetMf(x)=supt>0|f*δt(ψdσ)(x)|denote the maximal operator associated with surface measuredσon a smooth surfaceS. We prove that ifSis convex and has finite order contact with its tangent lines, then M is bounded onLp(Rn),p>2, if and only ifd(x,H)−1∈L1/loc(S) for all tangent planesHnot passing through the origin. LetM′f(x)=supt>0|f*δ′t(ψdσ)(x)|be the maximal operator associated with a nonisotropic dilationδ′tof surface measuredσ. We prove that M′ often behaves far better than M due to a rotational curvature in the time parametert.