Global wave-front sets of Banach, Fréchet and modulation space types, and pseudo-differential operators

We introduce global wave-front sets WFB(f), f∈S′(Rd), with respect to suitable Banach or Fréchet spaces B. An important special case is given by the modulation spaces B=M(ω,B), where ω is an appropriate weight function and B is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to WFB(f). In particular, we prove that micro-locality and micro-ellipticity hold for a class of globally defined pseudo-differential operators Opt(a), acting continuously on the involved spaces.