Restriction and decay for flat hypersurfaces.
type: Item
instance of: scholarly article
title: Restriction and decay for flat hypersurfaces.
zbMATH Open document ID: 1043.42007
DOI: 10.5565/PUBLMAT_46202_04
author: Anthony Carbery
author: S. N. Ziesler
published in: Publicacions Matemàtiques
publication date: 2002-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/41458
review text: Let \(S = \{ (t, \gamma(t)): t \in \Omega \}\) be a hypersurface in \(\mathbb R^n\), where \(\Omega\) is a bounded open subset of \(\mathbb R^{n-1}\) and \(\gamma: \Omega \to \mathbb R\) is a smooth function. Assume we have a non-negative affine curvature, thus \(\det(Hess(\gamma)) \geq 0\) on \(\Omega\). The authors consider two questions: the restriction problem, which asks for which pairs \((p,q)\) of exponents one has an estimate of the form \[ \left(\int_\Omega | \hat f(t,\gamma(t))| ^q \psi(t) \det(Hess(\gamma))^{q/p'(n-1)}\;dt\right)^{1/q} \leq C_{p,q} \| f \| _{L^p(\mathbb R^n)} \] for all functions \(f\), and the decay problem, which asks for which \(\beta\) one has an estimate of the form \[ \left| \int_\Omega e^{i \xi \cdot (t,\gamma(t))} \psi(t) \det(Hess(\gamma))^{\beta+i\alpha}\;dt\right| \leq C (1 + | \alpha| )^N / | \xi_n| ^{\beta(n-1)} \] for all \(\alpha \in \mathbb R\) and some \(N > 0\). Here \(\psi\) is a cutoff to \(\Omega\) which is either smooth or rough. The powers of the affine curvature \(\det(Hess(\gamma))\) are mandated by affine invariance considerations. In the case when \(\gamma\) is uniformly elliptic, the restriction estimate is known for all \(1 \leq p \leq 2n+2/(n+3)\) and \(q \leq (n-1)/(n+1)p'\) (this is the famous Tomas-Stein restriction theorem); further recent improvements to the exponents are known. Also, the decay estimate is known to be true for \(\beta=1/2\) in this case. When however \(\gamma\) is allowed to be flat, it is known from Knapp-type examples that the above estimates fail for certain choices of exponents unless one places an appropriate damping power of the affine curvature as was done above. In [Stud. Math. 51, 169--182 (1974; Zbl 0291.42003)] \textit{P. Sjölin} showed in two dimensions that one can recover the full range of \(p\) and \(q\) for arbitrary convex curves (i.e., \(1 \leq p \leq 4/3\) and \(q = p'/3\)) if one introduces the above damping factor, thus obtaining what might be termed an ``universal r…
Mathematics Subject Classification ID: 42B20
zbMATH DE Number: 1876876
zbMATH Keywords: Fourier transform
zbMATH Keywords: restriction theorems
zbMATH Keywords: decay estimates
reviewed by: Terence Tao
MaRDI profile type: Publication
OpenAlex ID: W2611741147
description: scientific article; zbMATH DE number 1876876