Karoline Johansson

Global wave-front sets of Banach, Fréchet and Modulation spacetypes, and pseudo-differential operators

We introduce global wave-front sets with respect to suitable Banach or Frechet spaces. An important special case appears when choosing these spaces as modulation spaces. We show that the standard p ...

Sharp convolution and multiplication estimates in weighted spaces

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hormander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipovic, 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 Equations248 (2010) 1972–2002].

MICRO-LOCAL ANALYSIS IN SOME SPACES OF ULTRADISTRIBUTIONS

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.

Global wave-front sets of intersection and union type

We show that a temperate distribution belongs to an ordered intersection or union of admissible Banach or Frechet 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.

Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems

We continue our analysis of the global wave-front sets we introduced on modulation spaces, here in relation with the corresponding class of Fourier integral operators. We obtain propagation results in terms of canonical transformations of the phase space, without requiring that the involved phase functions and amplitude are classical.

A Note on Wave-front Sets of Roumieu Type Ultradistributions

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.