Schatten–von Neumann properties in the Weyl calculus

AbstractLet Opt(a), for t∈R, be the pseudo-differential operatorf(x)↦(2π)−n∫∫a((1−t)x+ty,ξ)f(y)ei〈x−y,ξ〉dydξ 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, hgk/2m∈Lp for some k⩾0 and a∈S(m,g), then Opt(a)∈Ip, iff a∈Lp. Consequently, if 0⩽δ