Abstract. We consider the commutator operator $[B, \sigma(x,D)]$ of the multiplication operator by a function $B$ and a pseudo-differential operator $\sigma(x,D)$, and prove that $[B, \sigma(x,D)]$ is bounded on the local Hardy spaces $h^p({\msbm R}^n)$. We also show that our result is optimal.

AMS Subject Classification (1991): 42B20

Keyword(s): pseudo-differential operator, commutator, Hardy space, local Hardy space

Received January 13, 2009, and in revised form April 11, 2011. (Registered under 7/2009.)