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

By Isroil A. Ikromov, M. Kempe, D. Müller. 2010

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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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9127496e5a61b88ee350215f918a7084b78f0384

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Citations in the corpus

Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (2016, result): “There is a body of remarkable results on sharp decay estimates for oscillatory integrals with phase functions of two variables; in particular we mention the works by Varchenko [25], Arnold, Gusĕın-Zade, and Varchenko [1], Karpushkin [16], Magyar [17], Ikromov, Kempe, and Müller [13], Ikromov and M”
A multi-dimensional resolution of singularities with applications to analysis (2010, methodology): “Acknowledgements The third author thanks Professor Detlef Müller for valuable comments and Professor Kalle Karu for mentioning a reference for Lemma 2.4.”