Trace ideals for pseudo-differential operators and their commutators with symbols in α-modulation spaces

That symbols in the modulation space M1,1 generate pseudo-differential operators of the trace class was first stated by Feichtinger and proved by Gröchenig in [13]. In this paper, we show that the same is true if we replace M1,1 by the 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 discuss the trace property of the commutator [α (X, D), a], where; a(χ) is a Lipschitz function and σ(χ, ξ) belongs to an α-modulation space.