Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces

Abstract

Let s w p be the set of all a ∈ l such that a w (x, D) is Schatten p-operator on L 2. Then we prove the following: \(S(m,g)\subseteq s_p^w\) iff \(m\in L^p\). Furthermore, \(L^p\cap S(m,g)\subseteq s_p^w\) when \(h_g^{N/2}m\in L^p\). Consequently, \(S^r_{\rho, \delta}\cap L^\infty \subseteq s^w_\infty\) when \(0\le \delta <\rho \le 1\); if \(g(z,\zeta )=\sum \lambda _j(z_j^2+\zeta _j^2)\), then \(G(z,\zeta )=\sum \lambda _j^\alpha (z_j^2+\zeta _j^2)\) is symplectically invariantly defined. Moreover, if \(-1\le \alpha \le 1\) and \( g\le g^{\sigma}\) is slowly varying (and σ-temperate), then the same is true for G; a generalization of sharp Garding's inequality.