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Estimates for maximal functions associated with hypersurfaces in \(\mathbb R^3\) and related problems of harmonic analysis. (English) Zbl 1223.42011

From the authors’ abstract: 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 \(L^p(\mathbb R^3)\) 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 \(h(\varphi)\) for functions \(\varphi\) such that \(S\) can be locally represented as the graph of \(\varphi\) after rotation of coordinates. Several consequences of this result are discussed. In particular, we verify a conjecture of 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 \({\mathcal M}\) 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 \(L^p(\mathbb R^3) - L^2(S)\) Fourier restriction theorem for \(S\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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