This is the first book to present a complete characterization of
Stein-Tomas type Fourier restriction estimates for large classes of
smooth hypersurfaces in three dimensions, including all real-analytic
hypersurfaces. The range of Lebesgue spaces for which these estimates
are valid is described in terms of Newton polyhedra associated to the
given surface.
Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of
Fourier restriction and some relations between the decay of the Fourier
transform of the surface measure and Stein-Tomas type restriction
estimates. Varchenko's ideas relating Fourier decay to associated Newton
polyhedra are briefly explained, particularly the concept of adapted
coordinates and the notion of height. It turns out that these classical
tools essentially suffice already to treat the case where there exist
linear adapted coordinates, and thus Ikromov and Müller concentrate on
the remaining case. Here the notion of r-height is introduced, which
proves to be the right new concept. They then describe decomposition
techniques and related stopping time algorithms that allow to partition
the given surface into various pieces, which can eventually be handled
by means of oscillatory integral estimates. Different interpolation
techniques are presented and used, from complex to more recent real
methods by Bak and Seeger.
Fourier restriction plays an important role in several fields, in
particular in real and harmonic analysis, number theory, and PDEs. This
book will interest graduate students and researchers working in such
fields.