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

For several decades now, one of the central objects of investigation in Euclidean harmonic analysis is the question in which form geometric properties of submanifolds and subvarieties of Euclidean space are influencing various problems in Fourier analysis and real analysis. Particular instances of this are Fourier restriction estimates, which have been formulated and popularized by E.M. Stein (Princeton) and which, in a special case, date back even to an article by A. Zygmund from 1974. The corresponding dual "Strichartz" estimates have turned out to be of fundamental importance in the modern theory of partial differential equations. Other, related questions are for instance the question of uniform estimates of the Fourier transform of surface carried measures on submanifolds, or the boundedness of associated maximal operators on Lebesgue spaces, such as Stein's spherical maximal operator. Substantial progress on these questions has been achieved in the first place for the "generic" case of hypersurfaces with non-vanishing Gaussian curvature, through deep work by many leading analysts, including E.M. Stein, C. Fefferman, J. Bourgain, T. Wolff und T. Tao and, most recently, Bourgain and L. Guth. These have revealed numerous interesting connections with problems in other fields, such as geometric measure theory, combinatorial arithmetics and topology. Nevertheless, many of these questions remain wide open in higher dimensions.Moreover, except for special classes of hypersurfaces, very little was known for surfaces on which the Gaussian curvature may vanish. In a series of joint articles with I. Ikromov (Samarkand), we have been able to make substantial progress also for such hypersurfaces in three dimensions. Among other results, we have derived sharp L^p-estimates for the maximal functions associated to such hypersurfaces for p>2, as well as sharp Stein-Tomas type restriction estimates for a large class of smooth hypersurfaces, which includes all analytic hypersurfaces. The follow-up project is aiming at rounding off the corresponding studies for two-dimensional hypersurfaces. Among other things, we want to estimate the corresponding maximal functions on L^p also for the remaining cases of ''height'' <2 in the sense of Varchenko and study Strichartz estimates for hypersurfaces. Moreover, in joint work with A. Vargas (Madrid) and S. Buschenhenke (Birmingham), we plan to study L^p-L^q - Fourier restriction for ''hyperbolic'' surfaces of negative Gaussian curvature, for which very little known to date, by means of the so-called bilinear method (to which Mrs. Vargas has made fundamental contributions).

• DFG Programme: Research Grants