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We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to high-low frequency interactions for solutions to partial differential equations, as well as to the linear restriction problem for surfaces with degenerate curvature. As a consequence, we obtain new bilinear restriction estimates for elliptic phases and wave/Klein-Gordon interactions in the full bilinear range, and give a refined Strichartz inequality for the Klein-Gordon equation. In addition, we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves.

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We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space 9 H n´1 2 pR n q for n " 2, 3. The proof proceeds by using the Fierz identities to rewrite the equation in a form where the null structure of the system is readily apparent. This null structure is then exploited via bilinear estimates in spaces based on the null frame spaces of Tataru.We hope that the spaces and estimates used here can be applied to other nonlinear Dirac equations in the scale invariant setting. Our work complements recent results of Bejenaru-Herr who proved a similar result for n " 3 in the massive case.

In previous work on the Maxwell-Klein-Gordon system first global existence and then decay estimates have been shown. Here we show that the Maxwell-Klein-Gordon system in the Lorenz gauge satisfies the weak null condition and give detailed asymptotics for the scalar field and the potential. These asymptotics have two parts, one wave like along outgoing light cones at null infinity, and one homogeneous inside the light cone at time like infinity. Here the charge plays a crucial role in imposing an oscillating factor in the asymptotic system for the field, and in the null asymptotics for the potential. The Maxwell-Klein-Gordon system, apart from being of interest in its own right, also provides a simpler semi-linear model of the quasi-linear Einstein's equations where similar asymptotic results have previously been obtained in wave coordinates.where J µ is the asymptotic source termHereAdditionally, we have the following bound on the difference when |x| < tRemark. The second term in the right of (1.6) corresponds to a solution of a homogeneous wave equation that has same kind of asymptotics at null infinity as given in Theorem 1.1, compare Lindblad-Schlue [14].