On a scale of anisotropic Sobolev spaces

TL;DR

This paper introduces a new family of anisotropic Sobolev spaces defined via Fourier multipliers, exploring their properties and applications to PDEs involving traveling wave solutions.

Contribution

It presents a novel three-parameter scale of anisotropic Sobolev spaces and analyzes their functional analytic properties, with applications to PDEs.

Findings

Spaces are well-defined and exhibit desirable functional properties.

Applications demonstrate relevance to traveling wave solutions in PDEs.

Provides a framework for further analysis of anisotropic PDE problems.

Abstract

We introduce a scale of anisotropic Sobolev spaces defined through a three-parameter family of Fourier multipliers and study their functional analytic properties. These spaces arise naturally in PDE when studying traveling wave solutions, and we give some simple applications of the spaces in this direction.