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

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.

💡 Research Summary

The paper presents a unifying perspective on the construction of a wide variety of function spaces that appear in harmonic analysis and the study of partial differential equations (PDEs). The central idea is that any such space can be realized as a retract of a simple Banach space defined on phase space (position × momentum) via a wave‑packet decomposition. Concretely, one fixes a linear lifting operator
  W : L²(ℝᵈ) → L²(ℝᵈ×ℝᵈ)
which satisfies a normalization condition (typically ∫₀^∞ ψ(σ)² dσ/σ = 1) so that WW = Id on L². For a chosen Banach space Y on phase space, the norm ‖f‖ = ‖Wf‖_Y defines a function space on ℝᵈ. If the projection WW is bounded on Y, then all the interpolation, duality, and functional‑calculus properties of Y are inherited by the retract, providing a systematic way to transfer known results from the phase‑space setting to the original function space.

The paper surveys several concrete choices of wave‑packet decompositions, each tailored to a different class of operators:

1. Fixed‑radius (modulation) packets – defined by ψ(D + η). The associated modulation spaces M^{p,q}_s recover Sobolev, Hardy, Triebel‑Lizorkin, and Besov norms. They are well suited for Fourier integral operators (FIOs) such as exp(iΔ) and exp(i√{−Δ}), because the phase of exp(i|ξ|²) can be linearised on balls of fixed radius. However, for exp(i√{−Δ}) the modulation spaces are not optimal, prompting the need for a finer localisation.

2. Directional slices of dyadic annuli – using polar coordinates (ω,σ) and symbols ψ_{σ,ω}(D) that localise both frequency magnitude and direction. The resulting spaces L^{q}_ω(L^{p}x(L²_σ)), denoted L^{q,p}{W,s}, capture the geometry of the wave propagator exp(i√{−Δ}) and align with modern ℓ² decoupling inequalities. They provide the sharp mapping properties for wave equations, namely exp(i√{−Δ}) ∈ B(W^{s,p}, W^{−s,p}) for s > (d−1)/2·|1/p−1/2|.

3. Energy‑level (heat‑semigroup) packets – built from the functional calculus of a non‑negative self‑adjoint operator L (e.g. a divergence‑form elliptic operator). One lifts f by Wf(x,σ)=ψ(σ²L)f(x). Instead of the product space L^{p}_x(L²_σ), the authors employ tent spaces T^{p,2}, which average over spatial balls and are robust under off‑diagonal decay estimates. This yields Hardy spaces H^{p}_L that coincide with L^{p} when L=Δ, but remain meaningful for rough operators where the heat kernel lacks Calderón‑Zygmund regularity. The boundedness of WW* on T^{p,2} follows from standard square‑function estimates, and the resulting spaces support a bounded holomorphic functional calculus and Riesz transform estimates.

4. Gaussian‑measure adapted packets – for the Ornstein‑Uhlenbeck operator L_{OU} on (ℝᵈ,γ). Since the Gaussian measure is not translation invariant, the authors again use ψ(σ²L_{OU}) to define a wave‑packet lift. This leads to Besov and Triebel‑Lizorkin spaces adapted to the Gaussian setting, and, at the endpoint p=1, to the Hardy space h¹(γ) constructed via a localized doubling property of γ on balls of radius proportional to ρ(x)=min(1,|x|^{-1}).

5. A new Schrödinger‑adapted packet – designed for operators Δ−V with non‑negative potentials V belonging to the reverse Hölder class RH_q (q > d/2). The lift Wf(x,σ)=ψ(σ²(Δ−V))f(x) respects the spatial inhomogeneity of V and will be the foundation of forthcoming joint work with Dorothee Frey, Andrew Morris, and Adam Sikora. The authors anticipate that this decomposition will yield function spaces on which the Schrödinger propagator enjoys optimal mapping properties, extending the paradigm of phase‑space localisation to potentials with limited regularity.

Throughout the paper the authors pose three guiding questions for each construction: (1) Is WW* bounded on the chosen Y? (2) Do the operators of interest (propagators, multipliers, Riesz transforms) act invariantly on the resulting space? (3) How does the new space embed into classical function spaces? Positive answers to (1) give interpolation and duality; (2) ensures the space is suitable for fixed‑point arguments in nonlinear PDE; (3) allows comparison with known optimal results.

In summary, the article provides a practical “user guide”: by selecting an appropriate wave‑packet decomposition—whether based on translation, directional localisation, functional calculus, or Gaussian geometry—researchers can systematically construct function spaces that are tailor‑made for the operators appearing in their PDE problems. The survey unifies classical constructions (Littlewood‑Paley, modulation, tent spaces) with recent advances (decoupling‑based spaces, Gaussian harmonic analysis) and introduces a promising new Schrödinger‑adapted framework, thereby equipping analysts with a versatile toolbox for future investigations.