Scaling invariant smoothing estimates for the Schrödinger flow in dimension three

(1) sup λ>λ0, ε>0 ∥(−△+ V − (λ + iε) )−1∥∥ L2,σ(Rd)→L2,−σ(Rd) < ∞ provided that λ0 > 0, (1 + |x|)1+|V (x)| ∈ L∞ and σ > 12 . Here L(R) = {(1 + |x|)−σ f : f ∈ L(R)} is the usual weighted L2. The bound (1) is obtained from the same estimate for V = 0 by means of the resolvent identity. This bound for the free resolvent is related to the so called trace lemma, which refers to the statement that for every f ∈ L 12+ there is a restriction of f̂ to any (compact) hypersurface, and this restriction belongs to L2 relative to surface measure. Note that this fact does not require any curvature properties of the hypersurface in fact, it is proved by reduction to flat surfaces. Another fundamental restriction theorem is the Stein-Tomas theorem, see [Ste]. It requires the hypersurfaces S ⊂ R with d ≥ 2 to have non vanishing Gaussian curvature, and states that