Restriction and decay for flat hypersurfaces.
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 restriction theorem'', with bounds uniform in the choice of surface. These two dimensional arguments are detailed further in this paper. The authors investigate whether such a universal restriction theorem holds in higher dimensions. They first observe that the assumption of positive affine curvature is insufficient, but leave open the possibility that such a theorem might hold if one assumes convexity. Then they turn to the question of decay estimates, since these estimates are the most natural way to try to obtain restriction estimates. They first recall that the decay estimate for \(\beta\) would imply the restriction estimate for \(q=2\) and \(p = 2(1+(n-1)\beta) / (2 + (n-1)\beta)\). Unfortunately they give examples where the decay estimate fails (but the restriction estimate still holds), even if one assumes convexity, thus showing that the restriction problem does not automatically reduce to the decay problem. In the case of cylindrically symmetric surfaces, more precise necessary and sufficient conditions are given to determine when the restriction and decay properties hold.
scientific article; zbMATH DE number 1876876