We illustrate the composition properties for an extended family of SG Fourier integral operators. We prove continuity results on modulation spaces, and study mapping properties of global wave-front sets for such operators. These extend classical results to more general situations. For example, there are no requirements on homogeneity for the phase functions. Finally, we apply our results to the study of the propagation of singularities, in the context of modulation spaces, for the solutions to the Cauchy problems for the corresponding linear hyperbolic operators. 

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl.1 (2010) 341–376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations 248 (2010) 1972–2002].

We show that a temperate distribution belongs to an ordered intersection or union of admissible Banach or Fréchet spaces if and only if the corresponding global wave-front set of union or intersection type is empty. We also discuss the situation where intersections and unions of sequences of spaces with two indices are involved. A main situation where the present theory applies is given by sequences of weighted, general modulation spaces.

We study wave-front sets in weighted Fourier–Lebesgue spaces and corresponding spaces of ultradistributions. We give a comparison of these sets with the well-known wave-front sets of Roumieu type ultradistributions. Then we study convolution relations in the framework of ultradistributions. Finally, we introduce modulation spaces and corresponding wave-front sets, and establish invariance properties of such wave-front sets.

We introduce global wave-front sets with respect to suitable Banach or Fréchet spaces. An important special case appears when choosing these spaces as modulation spaces. We show that the standard properties for known notions of wave-front set extend to global wave-front sets. In particular, we prove that microlocality and microellipticity hold for a class of globally defined pseudo-differential operators, acting continuously on the involved spaces.

Let ω, ω ₀ be appropriate weight functions and ${\fancyscript{B}}$ be an invariant BF-space. We introduce the wave-front set ${{\rm WF}_{\mathcal{B}}(f)}$ with respect to the weighted Fourier Banach space ${\mathcal{B}=\fancyscript{F} \fancyscript{B}(\omega )}$ . We prove that the usual mapping properties for pseudo-differential operators Op _t (a) with symbols a in ${S^{(\omega_0)}_{\rho, 0}}$ hold for such wave-front sets. In particular we prove ${{\rm WF}_{\mathcal C}({\rm Op}_t (a) f)\subseteq {\rm WF}_{\mathcal{B}}(f)}$ and ${{\rm WF}_{\mathcal{B}}(f) \subseteq {\rm WF} _{\mathcal C}({\rm Op}_t (a) f)\bigcup {\rm Char} (a)}$ . Here ${\mathcal{C}=\fancyscript{F} \fancyscript{B}(\omega /\omega_0)}$ and Char(a) is the set of characteristic points of a.

We introduce discrete wave-front sets of sup type and prove that they coincide with the Hörmander wave-front set of a distribution. To that end we recall the notion of admissible lattice pairs and wave-front sets in Fourier-Lebesgue spaces.

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of "continuous type". This implies that the coefficients of a Gabor frame expansion of $f$ are parameter dependent, and describe the wave-front set of $f$.

We extend some results from [14] and [19], concerning wave-front sets of Fourier–Lebesgue and modulation space types, to a broader class of spaces of ultradistributions. We relate these wave-front sets one to another and to the usual wave-front sets of ultradistributions.

Furthermore, we give a description of discrete wave-front sets by intro- ducing the notion of discretely regular points, and prove that these wave-front sets coincide with corresponding wave-front sets in [19]. Some of these inves- tigations are based on the properties of the Gabor frames.

Let B be a translation invariant Banach function space (BF-space). In this paper we prove that every temperate distribution f can be associated with a function F analytic in the convex tube Ω = {z in C^d; | Im z| < 1 } such that the wave-front set of f of Fourier BF-space types in intersection with R^d ×S^d-1 consists of the points (x, ξ) such that F does not belong to the Fourier BF-space at x − i ξ.