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

We introduce global wave-front sets WFB(f), f is an element of'(R-d), with respect to suitable Banach or Frechet spaces B. An important special case is given by the modulation spaces B = M(omega,B), where omega 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 Op(t)(alpha), acting continuously on the involved spaces. (C) 2013 Elsevier Inc. All rights reserved.