# Fourier Multipliers on Anisotropic Mixed-Norm Spaces of Distributions

## TL;DR

This paper introduces a new Hormander type condition for Fourier multipliers on anisotropic mixed-norm distribution spaces, establishing boundedness, continuity, and norm equivalences in various anisotropic function spaces.

## Contribution

It develops a generalized Hormander condition for anisotropic mixed-norm spaces and proves boundedness and continuity of Fourier multipliers in these settings.

## Key findings

- Boundedness of Fourier multipliers on anisotropic Besov and Triebel-Lizorkin spaces.
- Continuity results for operators on mixed Sobolev and Lebesgue spaces.
- Establishment of lifting properties and equivalent norms in anisotropic distribution spaces.

## Abstract

A new general Hormander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multi- pliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well.