Problems in harmonic analysis related to hypersurfaces

• Applicant: Professor Dr. Detlef Müller
• Subject Area: Mathematics
• Term: from 2013 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 237750060

• Final Report Year: 2021

Final Report Abstract

This research project within the field of Euclidean Harmonic Analysis primarily aimed at further developing the following important branches of Fourier Analysis: a) The theory of maximal functions associated to a given hypersurface in three-dimensional Euclidean space, which are defined by means of the averages of a given function over all dilates of the given surface. b) The theory of Fourier restriction to general smooth hypersurfaces in three-dimensional space. Both branches had been initiated by E.M. Stein (Princeton), actually a) by his study of the case of the Euclidean unit sphere, i.e., the so-called spherical maximal function. This had been initially motivated by Stein’s quest for dimension free estimates for the classical centered Hardy-Littlewood maximal operator, which plays, for instance, a fundamental role in Lebesgue’s differentiation theory. On a). In joint work with Prof. I. Ikromov (Samarkand, Usbekistan), with whom I have a long standing cooperation and who has regularly visited us in Kiel, as well as Prof. S. Dendrinos (Cork, Irland) and Dr. S. Buschenhenke (Kiel), we have succeeded within this research project to clarify the boundedness properties of such maximal operators on Lebesgue spaces for almost all classes of hypersurfaces. The question remains open at this stage essentially only for a small exceptional class of surfaces exhibiting singularities of type A (in the sense of V. Arnold). On b) Within this subproject too, we have been able achieve major progress. In joint work of mine with Ikromov, we have been able to essentially completely answer the question which range of so-called Fourier restriction estimates of Stein-Tomas type can hold for any given finite type hypersurface. Such estimates are of great importance also to the theory of dispersive differential equations. Our results have been further developed to mixed-norms “Strichartz estimates” in the outstanding doctoral thesis of L. Palle. As a second highlight I would like to mention joint articles with Dr. S. Buschenhenke and Prof. A. Vargas (Madrid), in which we have been able to achieve some breakthroughs in establishing Fourier restriction estimates for hyperbolic (i.e., saddle type) surfaces which go beyond Stein-Tomas type estimates. Here we had essentially entered new territory, since before, with the exception of a special case, essentially only elliptic surfaces (such as spheres) had been treated. What came as a surprise to us is that the so-called “bilinear method”, which had been introduced by Fields medalist J. Bourgain, has turned to be rather unsuited for the study of hyperbolic surfaces, since it proved to be non-robust under deformations of the given surface. In contrast, the new “polynomial partitioning” method, which had only recently been successfully applied to the study of Fourier restriction estimates for elliptic surfaces, turned out to be robust even in the setting of hyperbolic surfaces, besides being more powerful.

Publications

• Fourier restriction for hypersurfaces in three dimensions and Newton polyhedra; Annals of Mathematics Studies 194, Princeton University Press, Princeton and Oxford 2016; 260 pp.
  I. A. Ikromov, D. Müller
  (See online at https://doi.org/10.1515/9781400881246)
• A Fourier restriction theorem for a two-dimensional surface of finite type; Analysis & PDE 10 No. 4 (2017), 817–891
  S. Buschenhenke, D. Müller, A. Vargas
  (See online at https://doi.org/10.2140/apde.2017.10.817)
• Estimates for maximal functions associated to hypersurfaces in R3 with height h < 2 : Part I; Trans. Amer. Math. Soc.; Vol. 372 No. 2 (2019), 1363–1406
  S. Buschenhenke, S. Dendrinos, I. A. Ikromov, D. Müller
  (See online at https://doi.org/10.1090/tran/7633)
• A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning
  S. Buschenhenke, D. Müller, A. Vargas
  (See online at https://doi.org/10.48550/arXiv.2003.01619)
• A Fourier restriction theorem for a perturbed hyperboloid; Proceedings London Math. Soc., Vol. 120 No. 3 (2020), 124–154
  S. Buschenhenke, D. Müller, A. Vargas
  (See online at https://doi.org/10.1112/plms.12286)
• Fourier restriction for smooth hyperbolic 2-surfaces; Preprint 2020
  S. Buschenhenke, D. Müller, A. Vargas
  (See online at https://doi.org/10.48550/arXiv.2010.10449)
• Mixed norm Strichartz-type estimates for hypersurfaces in three dimensions; Math. Z. 297 (2021), no. 3-4, 1529–1599
  L. Palle
  (See online at https://doi.org/10.1007/s00209-020-02568-8)
• On Fourier restriction for finite-type perturbations of the hyperboloid paraboloid; in Geometric Aspects of Harmonic Analysis (conference proceedings) Springer INdAM Series Vol. 45, pp. 193–222, 2021; ISBN 978-3-030-72057-5, ISBN 978-3-030-72058-2 (eBook)
  S. Buschenhenke, D. Müller, A. Vargas
  (See online at https://doi.org/10.1007/978-3-030-72058-2_5)
• Partitions of flat one-variate functions and a Fourier restriction theorem for related perturbations of the hyperbolic paraboloid; J. Geom. Anal. 31 (2021), no. 7, 6941–6986
  S. Buschenhenke, D. Müller, A. Vargas
  (See online at https://doi.org/10.1007/s12220-020-00587-9)
• Strichartz estimates for mixed homogeneous surfaces in three dimensions
  L. Palle
  (See online at https://doi.org/10.48550/arXiv.2004.07751)