Pseudo-differential operators and logarithmic Schatten classes

Abstract

For a Hilbert space ${{\mathcal {H}}}$ and positive numbers p and q, let $S_{p,q}({{\mathcal {H}}})$ stand for the class of those compact operators A on ${{\mathcal {H}}}$ such that $\sum ^{\infty }_{j=1}s_{j}(A)^{p}|\log s_{j}(A)|^{q}<\infty$, where $s_{j}(A)$ is the jth singular value of A counted from above with multiplicity. We characterize those weights M such that for any symbol $a\in S(M;\Phi ,\Psi )$ the Weyl quantization of a belongs to $S_{p,q}(L^{2})$.