Fourier Multipliers on Decomposition Spaces of Modulation and Triebel–Lizorkin Type

The family of anisotropic decomposition spaces of modulation and Triebel–Lizorkin type on \({\mathbb R}^n\) is a large family of smoothness spaces that include classical Besov, Triebel–Lizorkin, modulation and \(\alpha \)-modulation spaces. The decomposition space approach allows for a unified treatment of such smoothness spaces in both the isotropic and an anisotropic setting. We derive a boundedness result for Fourier multipliers on anisotropic decomposition spaces of modulation and Triebel–Lizorkin type. As an application, we obtain equivalent quasi-norm characterizations for this class of decomposition spaces.