Estimates for maximal functions associated with hypersurfaces in ℝ^{3}and related problems of harmonic analysis

Isroil A. Ikromov Department of Mathematics, Samarkand State University Michael Kempe Mathematisches Seminar, C.A.-Universität Kiel Detlef Müller Mathematisches Seminar, C.A.-Universität Kiel

TBD mathscidoc:1701.332014

Acta Mathematica, 204, (2), 151-271, 2008.2
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.
Maximal operator; Hypersurface; Oscillatory integral; Newton diagram; Oscillation index; Fourier restriction theorem; Contact index
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@inproceedings{isroil2008estimates,
  title={Estimates for maximal functions associated with hypersurfaces in ℝ^{3}and related problems of harmonic analysis},
  author={Isroil A. Ikromov, Michael Kempe, and Detlef Müller},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203354946715723},
  booktitle={Acta Mathematica},
  volume={204},
  number={2},
  pages={151-271},
  year={2008},
}
Isroil A. Ikromov, Michael Kempe, and Detlef Müller. Estimates for maximal functions associated with hypersurfaces in ℝ^{3}and related problems of harmonic analysis. 2008. Vol. 204. In Acta Mathematica. pp.151-271. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203354946715723.
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