Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates (Q5950063)

Let \(n\geq 2\) be a fixed integer. A function \(\varphi:{\mathbb R}^{n+1}\to H\) is called a red wave if it takes values in a finite dimensional complex Hilbert space \(H\), and its space-time Fourier transform \(\widehat \varphi\) is an \(L^2\) measure on the set \[ \{(\xi,|\xi|):\angle(\xi,e_1)\leq\tfrac \pi 8,\;2^k\leq|\xi|\leq 2^{k+1}\} \] for some integer \(k\), where \(e_1\) is a fixed basis vector. Similarly, a function \(\psi:{\mathbb R}^{n+1}\to H'\) is called a blue wave if it takes values in a finite dimensional complex Hilbert space \(H'\) and \(\widehat\psi\) is an \(L^2\) measure on \[ \{(\xi,-|\xi|):\angle(\xi,e_1)\leq\tfrac \pi 8,\;2^k\leq|\xi|\leq 2^{k+1}\} \] for some integer \(k\). In both cases, \(2^k\) is called the frequency of the waves \(\varphi\) and \(\psi\). Let \(E(\varphi)\) be the energy of \(\varphi\) defined by \[ E(\varphi)=\|\varphi(t)\|^2_2, \] where \(t\in{\mathbb R}\) is arbitrary. Let \(p_0=\frac {n+3}{n+1}\). In this paper, the author proves that if \(\varphi\) is a red wave of frequency 1 and \(\psi\) is a blue wave of frequency \(2^k\) for some \(k\geq 0\), then \[ \|\varphi\psi\|_p\leq C2^{k(\frac 1p-\frac 12+\varepsilon)}E(\varphi)^{1/2} E(\psi)^{1/2}\tag \(*\) \] for all \(2\geq p\geq p_0\) and \(\varepsilon>0\). In particular, if \(\varphi\) and \(\psi\) have frequency \(1\), then \[ \|\varphi\psi\|_p\leq CE(\varphi)^{1/2} E(\psi)^{1/2}\tag{\(**\)} \] for all \(2\geq p\geq p_0\). In these estimates, the constant \(C\) may depend on \(\varepsilon\) but is independent of \(H\) and \(H'\), and \(\varphi\psi:{\mathbb R}^{n+1}\to H\otimes H'\) denotes the tensor product \(\varphi\otimes\psi\) of \(\varphi\) and \(\psi\). The restriction \(p\geq p_0\) in estimates (\(\ast\)) and (\(\ast\ast\)) is sharp. Moreover, the estimate (\(\ast\ast\)) solves a conjecture of Machedon and Klainerman, and the case \(p>p_0\) was already obtained by \textit{T. Wolff} [Ann. Math. (2) 153, No. 3, 661-698 (2001)]. The estimate (\(\ast\)) is also sharp except for the \(\varepsilon\). The author conjectures that \(\varepsilon\) can be removed. The estimate (\(\ast\)) is necessary in order to develop some nearly-sharp \(L^p\) null form estimates which is obtained in the end of this paper. The main idea to show the estimate (\(\ast\)) is to localize the estimate (\(\ast\)) to a cube \(Q\) of side-length \(R\gg 2^k\), and obtain a bound independent of \(R\), which is obtained by induction on \(R\).