Schatten–von Neumann properties in the Weyl calculus

Let Opt (a), for t ∈ R, be the pseudo-differential operator f (x) → (2π)−n a (1− t)x +ty, ξ f (y)ei x−y,ξ dy dξ and let Ip be the set of Schatten–von Neumann operators of order p ∈ [1,∞] on L2. We are especially concerned with the Weyl case (i.e. when t = 1/2). We prove that if m and g are appropriate metrics and weight functions respectively, hg is the Planck’s function, h k/2 g m ∈ Lp for some k 0 and a ∈ S(m,g), then Opt (a) ∈Ip, iff a ∈ Lp. Consequently, if 0 δ <ρ 1 and a ∈ Sr ρ,δ, then Opt (a) is bounded on L2, iff a ∈ L∞. © 2010 Elsevier Inc. All rights reserved.