A Sharp Bilinear Estimate for the Klein–Gordon Equation in ℝ1+1

Abstract

Abstract. We prove a sharp bilinear estimate for the one dimensional KleinGordon equation. The proof involves an unlikely combination of five trigonometric identities. We also prove new estimates for the restriction of the Fourier transform to the hyperbola, where the pullback measure is not assumed to be compactly supported.

Scattering for the mass-critical nonlinear Klein-Gordon equations in three and higher dimensions

“…To do this, we prove a refined Strichartz estimates by utilising a bilinear restriction estimate proved recently by Candy and Herr [4], then follow the argument in [31], we can establish the inverse Strichartz estimate and therefore give the linear profile decomposition after applying the inverse Strichartz estimate inductively. For one dimensional case, the linear profile decomposition was proven in [24], but we give a shorter proof using a bilinear restriction estimate obtained by similar arguments in [56].…”

“…For the one-dimensional case, that is when d = 1, we use a different argument to show Theorem 4.7. Motivated by the argument in [56], we have for any f ∈ L 2 (R),…”

Bilinear identities involving thek-plane transform and Fourier extension operators

“…holds for some kernel K S1,S2 and such that the constant C is best possible; in many cases, extremizers for the above kinds of inequalities have also been characterized. This has been mostly studied for paraboloids [10], cones [7], spheres [20,11] and hyperboloids [27,23], with the corresponding natural interpretations in PDE. It should be noted that the bilinear estimates (1.3) and (1.4) also hold when E 2 f 2 is replaced by its complex conjugate E 2 f 2 .…”

A Morawetz Inequality for Gravity-Capillary Water Waves at Low Bond Number

“…In fact, the general study of Morawetz estimates has had a long history, which is too extensive to try to describe here. For further recent references we refer the reader to [43] for the wave equation, [42] for the Schrödinger equation, [49] for fractional dispersive equations. These estimates are very robust and also hold for nonlinear problems, which make them useful in the study of the Cauchy problem for nonlinear equations (see, e.g.…”