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.