A bilinear extension inequality in two dimensions.
Let \[ (f d\sigma)\hat{\;} (x) =\int_{{\mathbb{S}}^1}e^{-ixy} f(y)\,d\sigma(y),\qquad x\in {\mathbb{R}}^2, \] where \(d\sigma\) is the surface measure on \({\mathbb{S}}^1\). Suppose that \(\widetilde{f}(x)= f(-x)\). In the paper it is shown that if \(f,g\in L^2({\mathbb{S}}^1)\) with dist(supp(\(f\)),supp(\(g\))) and dist(supp(\(\widetilde{f}\)),supp(\(g\))) bounded below, then the following bilinear estimate \[ \int_{{\mathbb{S}}^1}| (f d\sigma)\hat{\;} (Rx) (g \,d\sigma)\hat{\;}(Rx)| \,d\sigma(x) \leq\frac{c}{R^{5/6}}\| f\| _2\| g\| _2 \] holds for all \(R>0\). Moreover, the decay rate \(R^{-5/6}\) is optimal.
scientific article; zbMATH DE number 1970664