Stability of sharp Fourier restriction

AbstractWe solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

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Fourier restriction to polynomial curves I: a geometric inequality

American Journal of Mathematics ◽

10.1353/ajm.0.0127 ◽

2010 ◽

Vol 132 (4) ◽

pp. 1031-1076 ◽

Cited By ~ 11

Author(s):

Spyridon Dendrinos ◽

James Wright

Keyword(s):

Geometric Inequality ◽

Polynomial Curves ◽

Fourier Restriction

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Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller

Advances in Analysis ◽

10.23943/princeton/9780691159416.003.0013 ◽

2014 ◽

Keyword(s):

Harmonic Analysis ◽

Finite Type ◽

Maximal Function ◽

Oscillatory Integrals ◽

The Other ◽

Riemannian Surface ◽

Fourier Restriction ◽

Newton Polyhedra ◽

The Given ◽

Finite Type Hypersurfaces

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

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