On the bilinear estimate of Ozawa and Rogers for the one-dimensional Klein-Gordon equation and some related lower Jacobian estimates

We give a natural convexity proof of an elementary inequality used by Ozawa and Rogers in proving their bilinear estimate for the one-dimensional Klein-Gordon equation. This robust approach also enables us to derive the optimality of Ozawa-Rogers estimate and establish a new bilinear estimate. Our estimate is in sharp analogy with the bilinear estimate of Bez and Rogers for the wave equation. We also present an alternative proof of some lower Jacobian estimates used crucially by Ozawa and Rogers in proving their Fourier restriction results on the whole hyperbola. Based on interpolation, our proof is purely non-trigonometric.