Boundedness and compactness of operators related to time-frequency analysis

Abstract

In this thesis, we study different aspects of operators related to time-frequency analysis. Every linear and continuous operator from the Schwartz class into its dual, the space of tempered distributions, can be written as an integral operator with kernel K, or also as an integral Fourier operator (in fact, pseudodifferential). Different conditions on the kernel or the symbol and the phase (in the FIOs case) allow to extend the operator to various spaces of functions and distributions. Below we detail the contents of the memory. At the first chapter we introduce the notation, the definitions of some spaces and the preliminary results that will be used throughout the thesis. The second chapter is devoted to the study of uncondicional multipliers. The main result, an improvement of Orlicz's theorem, shows that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained. The aim of the third chapter is to investigate compactness for Fourier integral operators when acting on weighted modulation spaces, using the matrix representation of Fourier integral operators with respect to a Gabor frame. As a consequence, we recover and improve some known results on compactness of pseudodifferential operators. At the fourth chapter we study conditions for the boundedness of Fourier integral operators with Hölder-continuous phase on Lebesgue spaces. We prove boundedness in L1 with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The continuity in L2 is studied as well by providing sufficient conditions and relevant counterexamples. At the last chapter we find some conditions for continuity of unimodular Fourier multipliers on modulation spaces. We find some results assuming that the second derivatives of the phase are bounded or, more generally, that its second derivatives belong to a particular Wiener amalgam space, in particular, its second derivatives could have strong oscillations at infinity.