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

Abstract

The classical Strichartz estimates for the free Schrodinger propagator have recently been substantially generalised to estimates of the form parallel to Sigma(j )lambda(j) vertical bar e(it Delta )f(j )vertical bar(2)parallel to(Ltp Lxq )less than or similar to parallel to lambda parallel to(l alpha) for orthonormal systems (f(j))(j) of initial data in L-2, firstly in work of Frank-Lewin-Lieb-Seiringer and later by Frank-Sabin. The primary objective is identifying the largest possible a as a function of p and q, and in contrast to the classical case, for such estimates the critical case turns out to be (p, q) = (d+1/d, d+1/d-1). We consider the case of orthonormal systems (f(j))(j) in the homogeneous Sobolev spaces (H) over dot(s) for s is an element of (0, d/2) and we establish the sharp value of alpha as a function of p, q and s, except possibly an endpoint in certain cases. Furthermore, at the critical case (p, q) = (d+1/d-2s, d(d+1)/(d-1)(d-2s)) for general s, we show the veracity of the desired estimates when alpha = p if we consider frequency localised estimates, and the failure of the (non-localised) estimates when alpha = p; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting. (C) 2019 Elsevier Inc. All rights reserved.