We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko’s notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates.

Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the L^p-boundedness of the associated maximal operator $ \mathcal{M} $, and a conjecture by Iosevich and Sawyer which relates the L^p-boundedness of $ \mathcal{M} $ to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3.

In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an $ {L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S) $ Fourier restriction theorem for S.

Let $ E \subset \mathbb{C} $ be a compact set, $ g:\mathbb{C} \to \mathbb{C} $ be a K-quasiconformal map, and let 0 < t < 2. Let $ {\mathcal{H}^t} $ denote t-dimensional Hausdorff measure. Then $ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $

This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in [2] and answers in the positive a conjecture of K. Astala in op. cit.