On the Strichartz estimates for orthonormal systems of initial data with regularity

The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the form ‖∑jλ_j|e^itΔf_j|²‖_{L^p tLx ^q} ≲‖λ‖_ℓ^α for orthonormal systems (f_j)_j of initial data in L², firstly in work of Frank–Lewin–Lieb–Seiringer and later by Frank–Sabin. The primary objective is identifying the largest possible α as a function of p and q, and in contrast to the classical case, for such estimates the critical case turns out to be [Formula presented]. We consider the case of orthonormal systems (f_j)_j in the homogeneous Sobolev spaces H˙^s for [Formula presented] and we establish the sharp value of α as a function of p, q and s, except possibly an endpoint in certain cases. Furthermore, at the critical case [Formula presented] for general s, we show the veracity of the desired estimates when α=p if we consider frequency localised estimates, and the failure of the (non-localised) estimates when α=p; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.