Planchon’s space–time Schrödinger estimate (Conjecture 1.2)

Establish the global space–time bound for the free Schrödinger evolution on R^{n+1}: for all exponents 2 ≤ p,r < ∞ satisfying 1/p + 1/r ≤ 1/2, show that ∥e^{itΔ}f∥_{L_x^p L_t^r(R^{n+1})} ≤ C ∥f∥_{H^s(R^n)} with s = n/2 − n/p − 1/r, for all initial data f ∈ H^s(R^n).

Background

This conjecture is the space–time counterpart to the classical Strichartz estimates, formulated by Planchon. It seeks the optimal regularity index s indicated by scaling for mixed L_x^p L_t^r norms of the Schrödinger solution. The authors confirm the conjecture in dimension n = 2 and improve known results in higher dimensions, but the full conjecture remains unproven in general.