A bilinear approach to cone multipliers. II: Applications (Q1976295)

Let \(S_1\) and \(S_2\) be two smooth compact hypersurfaces with boundary in \(\mathbb{R}^3\), with Lebesgue measure \(d\sigma_1\) and \(d\sigma_2\), respectively. If \(0< p,q\leq\infty\), one says that the bilinear adjoint restriction estimate \(R^*_{S_1,S_2}(p\times p\to q)\) holds if \[ \Biggl\|\prod^2_{t= 1} (f^\wedge_t d\sigma_t)\Biggr\|_{L^q(\mathbb{R}^3)}\leq C \prod^2_{t= 1}\|f_t\|_p, \] for all test functions \(f_1\), \(f_2\) supported on \(S_1\) and \(S_2\), respectively. This paper is a continuation of the previous paper [\textit{T. Tao} and \textit{A. Vargas}, Geom. Funct. Anal. 10, No. 1, 185-215 (2000; preceding review)]. In this previous paper, the authors gave some new linear and bilinear restriction estimates for the cone, sphere, and paraboloid in \(\mathbb{R}^3\), building upon and unifying previous work in this direction by Bourgain, Wolff, and others. In the present reviewed paper, the authors use these new estimates to give new progress on several open problems concerning the wave and Schrödinger equations in \(\mathbb{R}^{2+1}\), and convolution with curves in \(\mathbb{R}^3\).