Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra
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A01=Detlef Muller
A01=Isroil A. Ikromov
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Algorithm
Analytic function
Asymptotic analysis
Author_Detlef Muller
Author_Isroil A. Ikromov
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Big O notation
Borel measure
Category1=Non-Fiction
Category=PBKF
Category=PBM
Category=PBP
Change of variables
Coefficient
Combination
Convergent series
Convolution
Coordinate system
COP=United States
Corollary
Critical exponent
Degeneracy (mathematics)
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Derivative
Dimension
Direct proof
Dispersive partial differential equation
Division by zero
eq_isMigrated=2
eq_nobargain
Error term
Estimation
Existential quantification
Family of curves
Fibration
Fourier
Fourier inversion theorem
Fourier transform
Frequency domain
Frequency domain decomposition
Fubini's theorem
Hessian matrix
Hypersurface
Implicit function theorem
Inequality (mathematics)
Integer
Integration by parts
Interpolation theorem
Iterative method
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Line (geometry)
Line segment
Line-line intersection
Minkowski inequality
Monotonic function
Multiple integral
Natural number
Oscillatory integral
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Parameter
Partial derivative
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Rational number
Real number
Rectangle
Scientific notation
Series expansion
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Summation
Support (mathematics)
Taylor series
Tensor product
Theorem
Unit circle
Upper and lower bounds
Variable (mathematics)
Without loss of generality
Zero of a function
Product details
- ISBN 9780691170541
- Weight: 510g
- Dimensions: 152 x 235mm
- Publication Date: 24 May 2016
- Publisher: Princeton University Press
- Publication City/Country: US
- Product Form: Hardback
- Language: English
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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 Muller 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 Muller 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.
Isroil A. Ikromov is professor of mathematics at Samarkand State University in Uzbekistan. Detlef Muller is professor of mathematics at the University of Kiel in Germany.
