Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system

Multi-scale bilinear restriction estimates for general phases

“…The second consequence is an extension of (1.1) from free waves e itΦ(−i∇) f , to more general functions belonging to certain adapted function spaces U 2 Φ , see Corollary 1.6 below. This corresponds to a multi-scale version of recent work of the author and Herr [4]. The new bilinear bounds we obtain here, are applied in [5,6] to obtain conditional scattering results for the Dirac-Klein-Gordon system.…”

“…The key assumption (A1) is essentially equivalent to the conditions appearing previously in the literature [14, 1,4], except that we require a normalised version to correctly determine the dependence of the constant on the phases Φ j . To gain a better geometric understanding of (A1), we observe that letting ξ ′ → ξ and taking η = h − ξ, since ∇Φ j (ξ) − ∇Φ k (h − ξ) is normal to Σ j (h), (A1) implies the local condition…”

“…The proof of Theorem 1.2 follows the strategy used by Tao in the proof of the endpoint bilinear restriction estimate for the cone [21]. In particular, we replace the combinatorial Kakeya type arguments used by Wolff in the seminal paper [27] and further exploited in, for instance, [22,14,4,20], with energy estimates across transverse hypersurfaces. A similar argument was used by Bejenaru [1] to obtain a bilinear restriction estimate with q = r for general phases at unit scale.…”

Null Structures and Degenerate Dispersion Relations in Two Space Dimensions

“…The right hand side of (3.26) is evaluated in U 2 −based spaces, which, as we shall see later, is insufficient for our fixed point argument. However, we can use the following interpolation result from [HHK09] to transfer from U p −based estimates to V p −based estimates, provided we have an additional U q −based estimate for some q > p. Another way of transferring from U p to V p can be found in [CH16], which is not easy to implement here. Proposition 6.…”

On the radius of spatial analyticity for solutions of the Dirac–Klein–Gordon equations in two space dimensions

“…In particle physics, DKG arises as a model for forces between nucleons, mediated by mesons; see [3]. The well-posedness of the Cauchy problem in space dimensions d ≤ 3 with data in the family of Sobolev spaces H s (R d ) = W s,2 (R d ) has been extensively studied; see [9,12,11,16,21,28,2,7] and the references therein.…”