2007/06/07 by Isroil A. Ikromov, Ikromov, Isroil A., Michael Kempe +3
Mathematics · #35D05 #35D10 #35G05 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #math.CA #msc:35D05 #msc:35D10 #msc:35G05
paper · pdf · doi:10.48550/arxiv.0706.1006
104 pages, 2 figures
arxiv created 2007/06/07 · openalex publication_date 2007/06/07 · arxiv updated 2009/12/01 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We study the boundedness problem for maximal operators \M associated to smooth hypersurfaces S 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 Lp(\RR3) if and only if p>h(S), where h(S) denotes the so-called height of the surface S. For non-analytic finite type 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 \M, and a conjecture by Iosevich and Sawyer which relates the Lp-boundedness of \M to an integrability condition on S for the distance function to tangential hyperplanes, in dimension three. In particular, we also give ess. 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. Arnol'd in our context.