[Paper Review] Embeddings of decomposition spaces

Felix Voigtlaender|arXiv (Cornell University)|May 31, 2016

Mathematical Analysis and Transform Methods4 citations

TL;DR

This paper establishes a general, combinatorial framework for determining the existence of continuous embeddings between decomposition spaces—key function spaces in harmonic analysis such as Besov, α-modulation, and coorbit spaces. By reducing the problem to geometric and sequence-norm conditions on frequency coverings and weights, the framework enables embedding analysis without Fourier analysis, yielding sharp, verifiable criteria for embeddings between these spaces, including novel 'smoothness-for-integrability' trade-offs.

ABSTRACT

Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two decomposition spaces, is there an embedding between the two? A decomposition space $\mathcal{D}(\mathcal{Q}, L^p, Y)$ can be described using : a covering $\mathcal{Q}=(Q_{i})_{i\in I}$ of the frequency domain, an exponent $p$ and a sequence space $Y\subset\mathbb{C}^{I}$. Given these, the decomp. space norm of a distribution $g$ is $\| g\| _{\mathcal{D}(\mathcal{Q}, L^p, Y)}=\left\| \left(\left\| \mathcal{F}^{-1}\left(φ_{i}\widehat{g} ight) ight\| _{L^{p}} ight)_{i\in I} ight\| _{Y}$, where $(φ_{i})_{i\in I}$ is a suitable partition of unity for $\mathcal{Q}$. We establish readily verifiable criteria which ensure an embedding $\mathcal{D}(\mathcal{Q}, L^{p_1}, Y)\hookrightarrow\mathcal{D}(\mathcal{P}, L^{p_2}, Z)$, mostly concentrating on the case, $Y=\ell_{w}^{q_{1}}(I)$ and $Z=\ell_{v}^{q_{2}}(J)$. The relevant sufficient conditions are $p_{1}\leq p_{2}$, and finiteness of a norm of the form \[ \left\| \left(\left\| (α_{i}\,β_j \cdot v_{j}/w_{i})_{i\in I_{j}} ight\| _{\ell^{t}} ight)_{j\in J} ight\| _{\ell^{s}}

Motivation & Objective

• To develop a unified, accessible framework for determining continuous embeddings between decomposition spaces, which are central in harmonic analysis.
• To reduce the analysis of embeddings—previously requiring deep Fourier analysis—to purely geometric and combinatorial conditions on frequency coverings and sequence weights.
• To provide sharp, verifiable criteria for embeddings between decomposition spaces, including cases where smoothness and integrability are traded.
• To characterize when two decomposition spaces coincide, showing that their underlying coverings and weights must be equivalent under non-trivial conditions.
• To demonstrate the framework's power by recovering and improving known results on embeddings in α-modulation and Besov spaces.

Proposed method

• Define decomposition spaces D(Q, L^p, Y) via a frequency covering Q, L^p norm per tile, and sequence space Y on the index set I.
• Use a nested sequence norm involving the intersection sets I_j = {i ∈ I : Q_i ∩ P_j ≠ ∅} to characterize embedding conditions.
• Establish sufficient conditions for embedding D(Q, L^{p_1}, ℓ^{q_1}_w) ↪ D(P, L^{p_2}, ℓ^{q_2}_v) via finiteness of the norm ‖(α_i β_j v_j / w_i)_i∈I_j‖_{ℓ^t_j∈J}^{ℓ^s}.
• Prove sharpness of criteria under mild assumptions on coverings Q and P, showing the conditions are both necessary and sufficient in many cases.
• Apply the framework to α-modulation and Besov spaces, recovering and improving known embedding theorems.
• Use a rigidity result to show that D(Q, L^p, ℓ^q_w) = D(P, L^p, ℓ^q_v) only if p = p', q = q', and Q, P, w, v are equivalent in a suitable sense.

Experimental results

Research questions

• RQ1 Under what conditions does a continuous embedding exist between two decomposition spaces D(Q, L^{p_1}, ℓ^{q_1}_w) and D(P, L^{p_2}, ℓ^{q_2}_v)?
• RQ2 Can the existence of such embeddings be determined without Fourier analysis, relying only on the geometric structure of the coverings Q and P?
• RQ3 What is the precise relationship between the exponents p_1, p_2, q_1, q_2 and the weights w, v that determine embedding existence?
• RQ4 When do two decomposition spaces with different coverings or weights coincide?
• RQ5 Can the framework improve or generalize known embedding results for α-modulation and Besov spaces?

Key findings

• The embedding D(Q, L^{p_1}, ℓ^{q_1}_w) ↪ D(P, L^{p_2}, ℓ^{q_2}_v) exists if and only if p_1 ≤ p_2 and the nested sequence norm ‖(α_i β_j v_j / w_i)_i∈I_j‖_{ℓ^t_j∈J}^{ℓ^s} is finite, under suitable assumptions on Q and P.
• The framework allows for 'smoothness-for-integrability' trade-offs, as seen in the case p_1 ≠ p_2, generalizing classical Sobolev embeddings.
• For p_1 = p_2, the necessary and sufficient conditions for embedding are fully characterized in terms of the nested norm and the weight behavior.
• The framework recovers and improves all known embedding results for α-modulation spaces, showing that previous results can be strengthened.
• The rigidity result shows that two decomposition spaces D(Q, L^p, ℓ^q_w) and D(P, L^p, ℓ^q_v) can only be equal if their coverings Q and P, and weights w and v, are equivalent in a precise sense.
• The theory is sharp: for relatively moderate coverings and most parameter ranges, the criteria provide a complete characterization of embedding existence.