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Some recent progress on the restriction conjecture. (English) Zbl 1083.42008

Brandolini, Luca (ed.) et al., Fourier analysis and convexity. Boston, MA: Birkhäuser (ISBN 0-8176-3263-8/hbk). Applied and Numerical Harmonic Analysis, 217-243 (2004).
This is a nice survey paper on recent results on the restriction problem in harmonic analysis: for which sets \(S\subset\mathbb R^n\) and which \(p,\) \(1\leq p\leq2,\) can the Fourier transform of an \(L^p(\mathbb R^n)\) function be meaningfully restricted. Among infinitely many sets to consider, the author focus on sets \(S\) that are hypersurfaces or compact subsets of hypersurfaces. Three cases are model examples: sphere, paraboloid, and cone. They are hypersurfaces with curvature, enjoy a large group of symmetries, and are related via the Fourier transform to solutions to certain familiar PDEs. Several tools are considered in detail: the restriction to local estimates, the reduction to bilinear estimates, the introduction of wave packets, and the induction on scales. Since the submittance of this survey new results appeared in this area, mention, for example, D. Oberlin, “A restriction theorem for a \(k\)-surface in \(\mathbb R^n\)”. [Can. Math. Bull. 48, No. 2, 260–266 (2005; Zbl 1083.42007)].
For the entire collection see [Zbl 1057.43001].

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A85 Harmonic analysis on homogeneous spaces
53A05 Surfaces in Euclidean and related spaces

Citations:

Zbl 1083.42007
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Full Text: arXiv