Schatten–von Neumann properties in the Weyl calculus

Let Opt(a)Opt(a), for t∈Rt∈R, be the pseudo-differential operatorf(x)↦(2π)−n∫∫a((1−t)x+ty,ξ)f(y)ei〈x−y,ξ〉dydξ and let IpIp be the set of Schatten–von Neumann operators of order p∈[1,∞]p∈[1,∞] on L2L2. We are especially concerned with the Weyl case (i.e. when t=1/2t=1/2). We prove that if m and g   are appropriate metrics and weight functions respectively, hghg is the Planck's function, hgk/2m∈Lp for some k⩾0k⩾0 and a∈S(m,g)a∈S(m,g), then Opt(a)∈IpOpt(a)∈Ip, iff a∈Lpa∈Lp. Consequently, if 0⩽δ<ρ⩽10⩽δ<ρ⩽1 and a∈Sρ,δr, then Opt(a)Opt(a) is bounded on L2L2, iff a∈L∞a∈L∞.