Acta Mathematica

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

Isroil A. Ikromov, Michael Kempe, and Detlef Müller

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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.


We acknowledge the support for this work by the Deutsche Forschungsgemeinschaft.

Article information

Acta Math., Volume 204, Number 2 (2010), 151-271.

Received: 18 February 2008
Revised: 7 September 2009
First available in Project Euclid: 31 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35D05 35D10 35G05

Maximal operator Hypersurface Oscillatory integral Newton diagram Oscillation index Fourier restriction theorem Contact index

2010 © Institut Mittag-Leffler


Ikromov, Isroil A.; Kempe, Michael; Müller, Detlef. Estimates for maximal functions associated with hypersurfaces in ℝ 3 and related problems of harmonic analysis. Acta Math. 204 (2010), no. 2, 151--271. doi:10.1007/s11511-010-0047-6.

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