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

Pierre Portal

TL;DR

The paper develops a unifying framework in which function spaces for harmonic analysis and PDE can be built as phase-space retracts from wave packet decompositions, enabling a PDE-driven choice of decomposition tailored to the operator at hand. By surveying modulation-type, directional FIO, operator-defined Hardy, Gaussian, and Schrödinger-adapted decompositions, it demonstrates how to obtain boundedness, embedding, and operator-invariance results that reproduce or improve classical mapping properties while offering sharper tools for nonlinear PDE via fixed-point arguments. A key contribution is the introduction of a new Schrödinger-adapted wave packet construction for operators of the form $\Delta-V$ with $V\in RH_q$, $q>d/2$, illustrating how localization in both position and momentum yields a robust framework for future PDE analysis. Together, these developments provide a practical user-guide for selecting phase-space decompositions to construct function spaces that align with a given PDE’s dynamics and regularity structure, enabling sharper estimates and broader applicability of harmonic analysis techniques.

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 Schrödinger operators of the form $Δ- V$ for $V \geq 0$, as a sneak peek into upcoming joint work with Dorothee Frey, Andrew Morris, and Adam Sikora.

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

TL;DR

The paper develops a unifying framework in which function spaces for harmonic analysis and PDE can be built as phase-space retracts from wave packet decompositions, enabling a PDE-driven choice of decomposition tailored to the operator at hand. By surveying modulation-type, directional FIO, operator-defined Hardy, Gaussian, and Schrödinger-adapted decompositions, it demonstrates how to obtain boundedness, embedding, and operator-invariance results that reproduce or improve classical mapping properties while offering sharper tools for nonlinear PDE via fixed-point arguments. A key contribution is the introduction of a new Schrödinger-adapted wave packet construction for operators of the form with , , illustrating how localization in both position and momentum yields a robust framework for future PDE analysis. Together, these developments provide a practical user-guide for selecting phase-space decompositions to construct function spaces that align with a given PDE’s dynamics and regularity structure, enabling sharper estimates and broader applicability of harmonic analysis techniques.

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 Schrödinger operators of the form for , as a sneak peek into upcoming joint work with Dorothee Frey, Andrew Morris, and Adam Sikora.
Paper Structure (6 sections, 1 theorem, 56 equations)

This paper contains 6 sections, 1 theorem, 56 equations.

Table of Contents

  1. Introduction
  2. Localisation: balls of fixed radius in momentum.
  3. Localisation: directional slices of dyadic annuli in momentum.
  4. Localisation: adapted energy levels in momentum.
  5. Localisation: position dependent critical energy levels in momentum.
  6. Localisation: critical ball in position, dual energy level in momentum.

Key Result

Theorem 6.1

Assume that, for all $\sigma>0$, $\psi(\sigma^{2}L)$ has an integral kernel $K_{\sigma^{2}}$ such that: $\exists g, h \in L^{1}((0,+\infty)) \quad \forall x, y \in \mathbb{R}^{d} \quad \forall j \in \mathbb{Z}^{d} \quad \forall \sigma \geqslant \rho\left(x_{j}\right)$ Then $\mathcal{W}\mathcal{W}^{*}$ extends to a bounded linear operator on $\ell^{p}\left(\mathbb{Z}^{d} ; L^{p}\left(\left(\mathbb{

Theorems & Definitions (2)

  • Theorem 6.1
  • proof