Let M p;q denote the modulation space with parameters p; qA½1; N: If 1=p 1 þ 1=p 2 ¼ 1 þ 1=p 0 and 1=q 1 þ 1=q 2 ¼ 1=q 0 ; then it is proved that M p1;q1 Ã M p2;q2 CM p0;q0 : The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Cdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Cdo in the framework of modulation spaces. r

We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces. 9

We consider a broad family of test function spaces and their dual (distribution) space. The family includes Gelfand-Shilov spaces, a family of test function spaces introduced by S. Pilipović. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.2010 Mathematics Subject Classification. primary 46F05; 32A25; 32A36; secondary 35Q40; 30Gxx.

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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.

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