On the bilinear estimate of Ozawa and Rogers for the one-dimensional Klein-Gordon equation

We give a natural proof of an elementary inequality used by Ozawa and Rogers in proving their bilinear estimate for the one-dimensional Klein-Gordon equation. This new approach also enables us to derive the sharpness of Ozawa-Rogers estimate and establish a new bilinear estimate. Our estimate is analogous to the bilinear estimate of Bez and Rogers for the wave equation.