Using wave packet decompositions to construct function spaces: a user guide

TL;DR

This paper surveys the construction of function spaces in harmonic analysis of PDEs using wave packet decompositions, providing a conceptual framework and a new construction for Schrödinger operators.

Contribution

It offers a unified framework for constructing function spaces via wave packet decompositions and introduces a new method for Schrödinger operators.

Findings

Provides a user guide for choosing wave packet decompositions

Surveys recent constructions of function spaces in harmonic analysis

Introduces a new construction for Schrödinger operators

Abstract

We survey the construction of a range of function spaces used in harmonic analysis of PDE, including classical results as well as recent developments. We frame these constructions in a common conceptual framework, where these function spaces arise as retracts of simple function spaces over phase space, through a projection associated with a wave packet decomposition. Finding appropriate function spaces to study a given PDE then consists in choosing a relevant wave packet decomposition. We provide a user guide to making such choices, and constructing the corresponding function spaces. This is done mostly by surveying recent constructions, but we also include a new construction, adapted to Schr\"odinger operators of the form $\Delta - V$ for $V \geq 0$, as a sneak peek into upcoming joint work with Dorothee Frey, Andrew Morris, and Adam Sikora.