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Abstract. We define the Wodzicki Residue TR(A) for A belonging to a space of operators with double order, denoted L m 1 ,m 2 cl . Such operators are globally defined initially on R n and then, more generally, on a class of non-compact manifolds, namely, the manifolds with cylindrical ends. The definition is based on the analysis of the associate zeta function ζ(A, z). Using this approach, under suitable ellipticity assumptions, we also compute a two terms leading part of the Weyl formula for a positive selfadjoint operator A ∈ L m 1 ,m 2 cl in the case m 1 = m 2 .

We introduce global wave-front sets WF B (f ), f ∈ S ′ (R d ), 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 WF B (f ). In particular, we prove that microlocality and microellipticity hold for a class of globally defined pseudo-differential operators Op t (a), acting continuously on the involved spaces.2000 Mathematics Subject Classification. 35A18,35S30,42B05,35H10.

We study the composition of an arbitrary number of Fourier integral operators A j , j = 1, . . . , M, M ≥ 2, defined through symbols belonging to the so-called SG classes. We give conditions ensuring that the composition A 1 • · · · • A M of such operators still belongs to the same class. Through this, we are then able to show well-posedness in weighted Sobolev spaces for first order hyperbolic systems of partial differential equations in SG classes, by constructing the associated fundamental solutions. These results expand the existing theory for the study of the properties "at infinity" of the solutions to hyperbolic Cauchy problems on R n with polynomially bounded coefficients.

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L_p-Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index.Comment: 41 pages, 1 figur