Global microlocal analysis on Rd with applications to hyperbolic partial differential equations and modulation spaces

This thesis treats different aspects of microlocal and time-frequency analysis, with particular emphasis on techniques involving multi-products of Fourier integral operators and one-parameter group properties for pseudodifferential operators. In the first part, we study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on Rd. We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. In the second part, we deduce lifting property for modulation spaces and construct explicit isomorpisms between them. To prove such results, we study one-parameter group properties for pseudo-differential operators with symbols in some Gevrey-Hörmander classes. Furthermore, we focus on some classes of pseudo-differential operators with symbols admitting anisotropic exponential growth at infinity. We deduce algebraic and invariance properties of these classes. Moreover, we prove mapping properties for these operators on Gelfand-Shilov spaces of type S and modulation spaces.