ON SHARP STRICHARTZ INEQUALITIES IN LOW DIMENSIONS DIRK HUNDERTMARK AND VADIM ZHARNITSKY Abstract. Recently Foschi gave a proof of a sharp Strichartz inequality in one and two dimensions. In this note, a new representation in terms of an orthogonal projection operator is obtained for the space time norm of solutions of the free Schr¨odinger equation in dimension one and two. As a consequence, the sharp Strichartz inequality follows from the elementary property that orthogonal projections do not increase the norm. 1. Introduction The solution u to the free Schr¨ odinger equation i∂ t u = −Δu on R d (1.1) with initial condition u(0) = f ∈ L 2 (R 2 ) is given by u(t, x) = (e itΔ f )(x) (1.2) where e itΔ is defined, for example, by the spectral theorem. Since e itΔ is a unitary group on L 2 (R d ), one immediately sees that u ∈ L ∞ t (L 2 (R d )). But in fact, due to the dispersive nature of the free Schr¨ odinger equation, the solution u, as a function of space-time, is even L p -bounded, ‖u‖ L p (R×R d ) ≤ S d ‖f ‖ L 2 (R d ) (1.3) where p = p(d)=2+ 4 d . This was first shown by Strichartz [10] who followed the L p restriction proof of Stein-Tomas. Later simplified proofs were given by Ginibre and Velo [4], see also [2, 11]. The sharp value of S d , i.e., the quantity S d = sup f =0, ‖u‖ L p (R×R d ) ‖f ‖ L 2 (R d ) (1.4) has been unknown until very recently. In fact, even the existence of maximizers for (1.4), that is, functions f ∗ = 0 such that one has equality in (1.4), S d = ‖e itΔ f ∗ ‖ L p (R d+1 ) ‖f ∗ ‖ L 2 (R) (1.5) Date : November 30, 2005. Revised February 13, 2006. Supported in part by NSF grants DMS–0400940 (D.H.) and DMS-0505216 (V.Z.). c 2006 by the authors. Faithful reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes. 1