Acta Mathematica

, Volume 204, Issue 2, pp 151–271

Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis

  • Isroil A. Ikromov
  • Michael Kempe
  • Detlef Müller

DOI: 10.1007/s11511-010-0047-6

Cite this article as:
Ikromov, I.A., Kempe, M. & Müller, D. Acta Math (2010) 204: 151. doi:10.1007/s11511-010-0047-6


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 Lp-boundedness of the associated maximal operator \( \mathcal{M} \), and a conjecture by Iosevich and Sawyer which relates the Lp-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.


Maximal operatorHypersurfaceOscillatory integralNewton diagramOscillation indexFourier restriction theoremContact index

2000 Math. Subject Classification

Primary 35D05 35D10 35G05

Copyright information

© Institut Mittag-Leffler 2010

Authors and Affiliations

  • Isroil A. Ikromov
    • 1
  • Michael Kempe
    • 2
  • Detlef Müller
    • 2
  1. 1.Department of MathematicsSamarkand State UniversitySamarkandUzbekistan
  2. 2.Mathematisches SeminarC.A.-Universität KielKielGermany