Precise smoothing effect in the exterior of balls

We are interested in this article in investigating the smoothing effect properties of the solutions of the Schrödinger equation. Since the work by Craig, Kapeller and Strausss [13], Kato [16], Constantin and Saut [12] establishing the smoothing property, many works have dealt with the understanding of this effect. In particular the work by Doi [14] and Burq [5, 6] shows that it is closely related to the infinite speed of propagation for the solutions of Schrödinger equation. Roughly speaking, if one considers a wave packet with wave length λ, it is known that it propagates with speed λ and the wave will stay in any bounded domain only for a time of order 1/λ. As a consequence, taking the L in time norm will lead to an improvement of 1/λ with respect to taking an L∞ norm, leading to a gain of 1/2 derivatives. This heuristic argument can be transformed into a proof of the smoothing effect either by direct calculations (in the case of the free Schrödinger equation) or by means of resolvent estimates (see [3] for the case of a perturbation by a potential or [7] for the boundary value problem). In view of this simple heuristics, it is natural to ask whether one can refine (and improve) such smoothing type estimates if one considers smaller space domains (whose size will shrink as the wave length increases). A very natural context in which one can test this heuristics is the case of the exterior of a convex body (or more generally the exterior of several convex bodies), in which case natural candidates for the λ dependent domains are λ−α neighborhoods of the boundary. This is the main aim of this paper. To keep the paper at a rather basic technical level, we choose to consider only balls, for which direct calculations (with Bessel functions) can be performed. Our first result reads as follows: