Inhomogeneous Strichartz estimates for Schrödinger's equation

Foschi and Vilela in their independent works [3] and [13] showed that the range of (1/r,1/r˜) for which the inhomogeneous Strichartz estimate ‖∫0tei(t−s)ΔF(⋅,s)ds‖LtqLxr≲‖F‖Ltq˜′Lxr˜′ holds for some q  , q˜ is contained in the closed pentagon with vertices A, B  , B′B′, P  , P′P′ except for the points P  , P′P′ (see Fig. 1). We obtain the estimate for the corner points P  , P′P′.