Schatten-von Neumann Classes ) on the Torus for $$0

We provide a sharp sufficient condition on the symbol of a pseudo-differential operator on the torus $$\mathbb {T}^n$$ in order to guarantee its membership in the Schatten-von Neumann ideals $$\mathcal {S}_{p}(L^2(\mathbb {T}^n))$$ for lower orders of $$p,$$ namely when $$0