Restriction theorems for a surface with negative curvature
The author gives the following: Let \(S\) be the surface in \(\mathbb R^3\), defined by \(S=\{(\xi,\eta,\tau); \tau=\xi\eta, | \xi| , | \eta| \leq 1\}\). Then for any two compact subsets of \(S\),\, \(S_1,S_2\) satisfying \(| \xi_1-\xi_2| , | \eta_1-\eta_2| \geq 1\) for all \((\xi_1,\eta_1, \xi_1\eta_1)\in S_1\) and \((\xi_2,\eta_2, \xi_2\eta_2)\in S_2\), it holds \[ \| (f_1d\sigma_1)\hat{\,\,}\,(f_2d\sigma_2)\hat{\,\,}\| _{L^q(\mathbb R^3)}\leq C\| f_1\| _{L^p(S_1)}\| f_2\| _{L^p(S_2)} \] for any \(q>5/3\), \(\frac{5}{2q}+\frac1p<3\), where \(d\sigma_j\) is the surface Lebesgue measure of \(S_j\) and \[ (f_jd\sigma_j)\hat{\,\,}(x)=\int_{S_j}f(y)e^{2\pi ix\cdot y}\,d\sigma_j(y),\quad (j=1,2). \] Similar bilinear restriction theorems hold for elliptic type surfaces under the hypothesis that \(S_1\) and \(S_2\) are separated compact subsets of \(S\), in the general dimension [\textit{T. Tao, A. Vargas} and \textit{L. Vega}, J. Am. Math. Soc. 11, No. 4, 967--1000 (1998; Zbl 0924.42008)], and [\textit{T. Tao} and \textit{A. Vargas}, Geom. Funct. Anal. 10, No.~1, 185--215 (2000; Zbl 0949.42012)]. As in the elliptic type surface, she deduces the linear restriction theorem from the above result: For the above \(S\), it holds \(\| (fd\sigma)\hat{\,\,}\| _{L^r(\mathbb R^3)}\leq C\| f\| _{L^p(S)}\) for any \(r>10/3\), \(p'<r/2\).
scientific article; zbMATH DE number 2146796