# Orlicz-Modulationsräume

• Orlicz modulation spaces

Schnackers, Catherine; Führ, Hartmut (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2014)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2014

Abstract

In this thesis we extend the definition of modulation spaces associated to mixed-norm Lebesgue spaces to Orlicz spaces and mixed-norm Orlicz spaces and analyse their properties. In particular we characterize embedding relations between these spaces and prove entropy inequalities of the short-time Fourier transform. The basic idea of modulation spaces is to impose a norm on the short-time Fourier transform of tempered distributions that leads to Banach spaces of distributions with a given time-frequency behavior. The modulation space M^{p,q} consists of all tempered distributions such that the short-time Fourier transform is a function in the mixed-norm Lebesgue space L^{p,q}. We extend this concept and examine modulation spaces associated to Orlicz spaces and mixed-norm Orlicz spaces. The classical modulation spaces can be understood as Coorbit spaces, but can also be directly defined as spaces of tempered distributions without explicit reference to Coorbit theory see [Gröchenig, 2001]. This double perspective is also developed systematically for Orlicz-modulation spaces in this work. Chapters 1 and 3 present the essential results from the time frequency analysis and the theory of Orlicz spaces. Definitions and properties of the Young functions Phi and the associated Orlicz spaces L^Phi are being discussed. The mixed Orlicz spaces L^{Phi_1 Phi_2} are introduced as spaces of vector valued functions. We then examine the properties required for the time frequency analysis and Coorbit theory of these spaces. Here the Delta_2 condition repeatedly plays a central role. Based on this, in chapter 4 we define the modulation spaces associated to Orlicz spaces and mixed-norm Orlicz spaces. These spaces consist of tempered distributions, whose short-time Fourier transform is a function contained in the Orlicz space or in the mixed-norm Orlicz space respectively. We show that the Orlicz modulation spaces are well-defined Banach spaces as long as the Young function and its complementary function are continuous. A corresponding statement also applies to mixed-norm Orlicz modulation spaces assuming that the Young functions are continuous and satisfy the Delta_2 condition and that the complementary functions are also continuous. In our proofs we will therefore use results from Coorbit theory, presented in chapter 2. Besides we will also give alternative proofs solely by means of functional analytical methods, similar to the reasoning in [Gröchenig, 2001]. In both approaches, we need the Delta_2 condition, whose role we therefore discuss in detail in chapter 5. In section 4.5 we give a characterization of embeddings of Orlicz modulation spaces by using relations of Young functions. Furthermore we derive conditions under which an embedding of Fourier-Lebesgue spaces into the modulation space with respect to the Young function Phi(x)=|x|^p ln(|x|+1) is valid. We finally deal with entropy in chapter 6. We derive sufficient conditions under which the discretization of the entropy approximates the continuous entropy of the short-time Fourier transform. Additionally we prove new estimates for the continuous entropy of the short-time Fourier transform under the assumption that the function lies in a suitable modulation space.