[Paper Review] Changes of variables in modulation and Wiener amalgam spaces

Michael Ruzhansky, Mitsuru Sugimoto+2 more|arXiv (Cornell University)|Mar 25, 2008

Mathematical Analysis and Transform Methods18 references15 citations

TL;DR

This paper investigates the invariance of modulation and Wiener amalgam spaces under global and local changes of variables, establishing Beurling–Helson type theorems and proving local boundedness of canonical transforms and Fourier integral operators. It shows that localizations of $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $ coincide, unifying the analysis of these spaces and enabling new continuity results for transforms and operators.

ABSTRACT

In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier Lebesgue spaces. Finally, counterparts of these results are discussed for spaces on the torus as well as for weighted spaces.

Motivation & Objective

• To investigate the invariance properties of modulation and Wiener amalgam spaces under global and local changes of variables.
• To establish local and global Beurling–Helson type theorems for changes of variables and canonical transforms on these function spaces.
• To analyze the continuity properties of canonical transforms and Fourier integral operators on $ \mathscr{F}L^q $, $ M^{p,q} $, and $ W^{p,q} $ spaces.
• To extend results to weighted spaces and function spaces on the torus, including counterparts for homogeneous changes of variables.
• To unify the analysis of $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $ via their local equivalence, simplifying proofs of invariance properties.

Proposed method

• Utilizes the equivalence of localizations of $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $ spaces to unify the analysis of invariance under changes of variables.
• Applies the theory of Fourier integral operators and Gabor theory to study continuity of homogeneous changes of variables with possible singularities.
• Employs the pullback of changes of variables on the Fourier transform side to define canonical transforms and analyze their boundedness.
• Uses convolution identities and modulation/Wiener amalgam space norms via short-time Fourier transforms with window functions $ \varphi_j $.
• Applies Hahn-Banach theorem to extend multiplication and convolution maps from $ \mathscr{S} $ to modulation and Wiener amalgam spaces.
• Establishes continuity via integral formulas (A.8) and (A.9) involving cross-window conditions for $ \varphi_0, \dots, \varphi_N $, ensuring independence of window choice.

Experimental results

Research questions

• RQ1 Under what conditions is a $ C^1 $ change of variables on $ \mathbb{R}^n $ bounded on modulation and Wiener amalgam spaces?
• RQ2 What is the relationship between localizations of $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $, and how does this equivalence simplify the study of invariance?
• RQ3 When is a canonical transform induced by a change of variables locally bounded on $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $?
• RQ4 Can the decay condition on amplitudes of Fourier integral operators be removed for canonical transforms in $ \mathscr{F}L^q $ spaces?
• RQ5 How do homogeneous changes of variables with singularities affect boundedness in modulation and Wiener amalgam spaces?

Key findings

• Localizations of $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $ spaces coincide, providing a unifying framework for studying invariance under changes of variables.
• The paper establishes a Beurling–Helson type theorem for Wiener amalgam spaces $ W^{p,q} $, showing that only affine maps preserve the space under $ C^1 $ changes.
• If the pullback by a change of variables $ \psi $ is bounded on $ L^q $, then the corresponding canonical transform $ I_\psi $ is locally continuous on $ M^{p,q} $, $ W^{p,q} $, and $ \mathscr{F}L^q $.
• For canonical transforms with amplitudes in $ S^0_{0,0} $ or $ M^\infty,1 $, local boundedness holds on $ \mathscr{F}L^q $ without requiring decay conditions.
• Homogeneous changes of variables with singularities on sets of different dimensions are shown to be continuous on modulation and Wiener amalgam spaces using Gabor theory.
• Multiplication and convolution maps between modulation and Wiener amalgam spaces are continuously extended via integral formulas (A.8) and (A.9), independent of window functions under appropriate weight conditions.