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We give a natural convexity proof of an elementary inequality used by Ozawaand Rogers in proving their bilinear estimate for the one-dimensionalKlein-Gordon equation. This robust approach also enables us to derive theoptimality of Ozawa-Rogers estimate and establish a new bilinear estimate. Ourestimate is in sharp analogy with the bilinear estimate of Bez and Rogers forthe wave equation. We also present an alternative proof of some lower Jacobianestimates used crucially by Ozawa and Rogers in proving their Fourierrestriction results on the whole hyperbola. Based on interpolation, our proofis purely non-trigonometric.

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