Fourier Restriction For Hypersurfaces In Three Dimensions And Newton Polyhedra (am-194) (annals Of Mathematics Studies)
by Isroil A. Ikromov /
2016 / English / EPUB
29.7 MB Download
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.
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.
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.
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.