Counterexamples to Strichartz estimates for the wave equation in domains

Let Ω be the upper half plane {(x, y) ∈ R, x > 0, y ∈ R}. Define the Laplacian on Ω to be ∆D = ∂ 2 x + (1 + x)∂ 2 y , together with Dirichlet boundary conditions on ∂Ω: one may easily see that Ω, with the metric inherited from ∆D, is a strictly convex domain. We shall prove that, in such a domain Ω, Strichartz estimates for the wave equation suffer losses when compared to the usual case Ω = R, at least for a subset of the usual range of indices. Our construction is microlocal in nature; in [7] we prove that the same result holds true for any regular domain Ω ⊂ R, d = 2, 3, 4, provided there exists a point in T ∗∂Ω where the boundary is microlocally strictly convex. Definition 1.1. Let q, r ≥ 2, (q, r, α) 6= (2,∞, 1). A pair (q, r) is called α-admissible if