Bilinear space–time estimate for Schrödinger evolutions (Conjecture B.1)

Establish the bilinear space–time inequality for separated frequency supports: for f1,f2 supported in subsets of B(Ne,1) with dist(supp f1, supp f2) ≥ 2, show that ∥(e^{itΔ}f1)(e^{itΔ}f2)∥_{L_x^p L_t^r(R^{n+1})} ≤ C_{p,r} N^{κ(p,r)} ∥f1∥_{L^2(R^n)} ∥f2∥_{L^2(R^n)} if and only if (n+2)/p + 1/r ≤ (n+1)/2 and 2 ≤ p,r < ∞, where κ(p,r) is the exponent of N appearing in (B.1).

Background

The authors propose a bilinear space–time conjecture that would imply the global space–time Conjecture 1.2 and is closely linked to Fourier restriction problems. They also provide a necessary condition supporting the proposed sharp range of (p,r). Establishing this bilinear inequality in the full if-and-only-if range would resolve Conjecture 1.2 via their reduction.