[Paper Review] Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization

Hartmut Führ, Azita Mayeli|arXiv (Cornell University)|Jul 23, 2010

Mathematical Analysis and Transform Methods6 references4 citations

TL;DR

This paper establishes wavelet characterizations of homogeneous Besov spaces on stratified Lie groups using both continuous and discrete wavelet systems. It proves that different sub-Laplacians yield equivalent Besov spaces and constructs universal Banach frames for all $\dot{B}_{p,q}^{s}$ with $1\leq p,q<\infty$, $s\in\mathbb{R}$, ensuring unconditional convergence and norm equivalence via wavelet coefficients.

ABSTRACT

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces ${\dot B}_{p,q}^{s}$, with $1 \le p,q &lt; \infty$ and $s \in \mathbb{R}$.

Motivation & Objective

• To extend wavelet characterizations of Besov spaces from Euclidean spaces to stratified Lie groups.
• To define homogeneous Besov spaces $\dot{B}_{p,q}^{s}$ on stratified Lie groups using a Littlewood-Paley-type decomposition based on the spectral measure of a sub-Laplacian.
• To prove that the scale of Besov spaces is independent of the choice of sub-Laplacian, ensuring consistency with group structure.
• To establish continuous and discrete wavelet characterizations of $\dot{B}_{p,q}^{s}$, including frame and atomic decomposition properties.
• To construct universal Banach frames for all $\dot{B}_{p,q}^{s}$ spaces via sufficiently dense regular sampling sets on the group.

Proposed method

• Define homogeneous Besov spaces $\dot{B}_{p,q}^{s}$ via a Littlewood-Paley decomposition using the spectral calculus of a sub-Laplacian on a stratified Lie group $G$.
• Use a generalized $\phi$-function framework to prove that different sub-Laplacians yield equivalent Besov spaces, ensuring group-intrinsic consistency.
• Characterize $\dot{B}_{p,q}^{s}$ via the continuous wavelet transform with a wide class of admissible wavelets, including the heat semigroup as a special case.
• Introduce the coefficient space $\dot{b}_{p,q}^{s}$ to link wavelet coefficients to Besov norms, enabling discrete characterization.
• Employ oscillation estimates to bridge continuous and discrete decompositions, ensuring stability under sampling.
• Prove that for sufficiently dense regular sampling sets $\Gamma$, the discrete wavelet system $\{\psi_{j,\gamma}\}_{j\in\mathbb{Z}, \gamma\in\Gamma}$ forms a universal Banach frame for all $\dot{B}_{p,q}^{s}$, with norm equivalence $\|f\|_{\dot{B}_{p,q}^{s}} \asymp \|\{r_{j,\gamma}\}\|_{\dot{b}_{p,q}^{s}}$.

Experimental results

Research questions

• RQ1 Can homogeneous Besov spaces on stratified Lie groups be consistently defined via Littlewood-Paley decompositions based on sub-Laplacians, independent of the specific choice of sub-Laplacian?
• RQ2 How can continuous wavelet transforms be used to characterize homogeneous Besov spaces on stratified Lie groups?
• RQ3 What conditions ensure that discrete wavelet systems on stratified Lie groups form Banach frames for all $\dot{B}_{p,q}^{s}$ spaces?
• RQ4 Can oscillation estimates be used to bridge continuous and discrete wavelet decompositions in the context of homogeneous Besov spaces?
• RQ5 Is there a universal wavelet frame that works across all $\dot{B}_{p,q}^{s}$ spaces for $1\leq p,q<\infty$ and $s\in\mathbb{R}$?

Key findings

• The scale of homogeneous Besov spaces $\dot{B}_{p,q}^{s}$ on a stratified Lie group is independent of the choice of sub-Laplacian, ensuring intrinsic consistency with the group structure.
• Continuous wavelet transforms with a wide class of admissible wavelets provide a valid characterization of $\dot{B}_{p,q}^{s}$, including the heat semigroup as a special case.
• The wavelet coefficient sequence of any $f\in\dot{B}_{p,q}^{s}$ lies in the coefficient space $\dot{b}_{p,q}^{s}$, establishing a discrete characterization.
• For sufficiently dense regular sampling sets $\Gamma$, the discrete wavelet system $\{\psi_{j,\gamma}\}_{j\in\mathbb{Z}, \gamma\in\Gamma}$ forms a universal Banach frame for all $\dot{B}_{p,q}^{s}$ with $1\leq p,q<\infty$ and $s\in\mathbb{R}$.
• The wavelet synthesis formula $f = \sum_{j,\gamma} 2^{-jQ} \langle f, \tilde{\psi}_{j,\gamma} \rangle \psi_{j,\gamma}$ converges unconditionally in $\dot{B}_{p,q}^{s}$, with coefficients satisfying $\|\{r_{j,\gamma}\}\|_{\dot{b}_{p,q}^{s}} \asymp \|f\|_{\dot{B}_{p,q}^{s}}$.
• The tightness of the wavelet frame converges to one as the sampling density increases, particularly when measured with respect to the Besov norm from the same window $\psi$.