Trace Ideals for Pseudo-differential Operators and Their Commutators with Symbols in Α-modulation Spaces

The fact that symbols in the modulation space M1,1 generate pseudo-differential operators of the trace class was first mentioned by Feichtinger and the proof was given by Gröchenig [12]. In this paper, we show that the same is true if we replace M1,1 by more general α-modulation spaces which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corresponds to that of Gröchenig, and the one with α = 1 is a new result which states the trace property of the operators with symbols in the Besov space. As an application, we also discuss the trace property of the commutator [σ(X,D), a], where a(x) is a Lipschitz function and σ belongs to an α-modulation space.