A restriction theorem for flat manifolds of codimension two (Q1913494)

The main result of the paper is the following theorem: Let \(M= \{(x, x_{n+ 1}, x_{n+ 2})\in \mathbb{R}^{n+ 2}; x_{n+ 1}= \varphi_1(x), x_{n+ 2}= \varphi(x)\}\), \(n\geq 2\), where \(\varphi_j\in C^\infty_0(\mathbb{R}^n\backslash \{0\})\), \(\varphi_1\) is homogeneous of degree 1, and \(\varphi_2\) is homogeneous of degree \(m\geq 2\). Let \(\Sigma_j= \{x; \varphi_j(x)= 1\}\). Assume also that \(\varphi_2\) only vanishes at the origin and that \(\Sigma_2\) has everywhere nonvanishing Gaussian curvature. Let \[ F(\xi, \lambda_1, \lambda_2)= \int_{\mathbb{R}^n} e^{i(\langle\xi, x\rangle+ \lambda_1 \varphi_1(x)+ \lambda_2 \varphi_2(x))} \chi(x) dx, \] where \(\chi\in C^\infty_0(\mathbb{R}^n)\). (a) Suppose that the restriction of \(\varphi_1\) to the set where \(\varphi_2= 1\) is constant. Then \[ |F(\xi, \lambda_1, \lambda_2)|\leq C(|\xi|+ |\lambda_1|+ |\lambda_2|)^{- n/m}\tag{\(*\)} \] when \(m\geq 2n\). (b) Let \(M|_{\{x_{n+ 2}= 1\}}\) denote the restriction of \(M\) to \(\{x_{n+ 2}= 1\}\). If \(M|_{\{x_{n+ 2}= 1\}}\) (viewed as a submanifold of codimension 2 of \(\{x_{n+ 2}= 1\})\) satisfies a strong curvature condition (stated in the paper), then \((*)\) holds for \(m\geq 2\).