Continuity and Schatten–von Neumann Properties for Pseudo–Differential Operators and Toeplitz operators on Modulation Spaces

Let M (ω) be the modulation space with parameters p, q and weight function ω. We prove that if p1 = p2, q1 = q2, ω1 = ω0ω and ω2 = ω0, and ∂ a/ω0 ∈ L ∞ for all α, then the Ψdo at(x,D) : M p1,q1 (ω1) → M22 (ω2) is continuous. If instead a ∈ M p,q (ω) for appropriate p, q and ω, then we prove that the map here above is continuous, and if in addition pj = qj = 2, then we prove that at(x,D) is a Schatten-von Neumann operator of order p. We use these results to discuss continuity for Toeplitz operators. Mathematics Subject Classifications (2000): Primary: 47B10, 35S05, 47B35, 47B37; Secondary: 42B35, 46E35.