On the boundedness of dilation operators in the context of Triebel-Lizorkin-Morrey spaces

TL;DR

This work addresses the boundedness of dilation operators on Triebel-Lizorkin-Morrey spaces $\mathcal{E}^{s}_{u,p,q}(\mathbb{R}^d)$ by deriving sharp $\lambda$-dependent bounds for the operator norm of $D_\lambda$. Employing Fourier-analytic techniques, dyadic decompositions, and Morrey-space Fourier multiplier results, the authors obtain precise asymptotics: for $\lambda\ge 2$, $\|D_\lambda\|_{\mathcal{L}(\mathcal{E}^{s}_{u,p,q})} \sim \lambda^{s- d/u}$ when $s>\sigma_p$, and a borderline behavior $\sim \lambda^{\sigma_p- d/u}$ with log corrections (dependent on $p$ and $q$) when $s=\sigma_p$, while for $s<\sigma_p$ one has $\sim \lambda^{- d/u}$ (with refinements for $p<1$). The paper also provides a refined set of tools, including an advanced Fourier multiplier theorem for band-limited Morrey spaces and new equivalent (quasi-)norms for $\mathcal{E}^{s}_{u,p,q}$, enriching the functional-analytic toolkit for these spaces. The results have potential applications in regularity analysis for PDEs and refined localization techniques, where dilation-invariant characterizations are crucial. Overall, the work sharpens the understanding of scale interactions in Morrey-type Triebel-Lizorkin spaces and extends classical boundedness results to a broad parameter range.

Abstract

In this paper we study the behavior of dilation operators $ D_λ\colon f \mapsto f(λ\,\cdot) $ with $ λ> 1 $ in the context of Triebel-Lizorkin-Morrey spaces $\mathcal{E}^{s}_{u,p,q}(\mathbb{R}^d)$. For that purpose we prove upper and lower bounds for the operator (quasi-)norm $\| D_λ\,|\, \mathcal{L}\big(\mathcal{E}^s_{u,p,q}(\mathbb{R}^d)\big) \| $. We show that for $s>σ_p $ the operator (quasi-)norm $\| D_λ\,|\, \mathcal{L}\big(\mathcal{E}^s_{u,p,q}(\mathbb{R}^d)\big) \| $ up to constants behaves as $λ^{s - \frac{d}{u}} $. For the borderline case $ s = σ_{p} $ we observe a behavior of the form $λ^{σ_p- \frac{d}{u}}$, multiplied with logarithmic terms of $λ$ that also depend on the fine index $q$. For $s < σ_{p}$ and $p \geq 1$ we find the relation $\| D_λ\,|\, \mathcal{L}\big(\mathcal{E}^s_{u,p,q}(\mathbb{R}^d)\big) \| \sim λ^{ - \frac{d}{u}}$. The case $s < σ_{p}$ and $p < 1$ is investigated as well. Our proofs are mainly based on the Fourier analytic approach to Triebel-Lizorkin-Morrey spaces. As byproducts we show an advanced Fourier multiplier theorem for band-limited functions in the context of Morrey spaces and derive some new equivalent (quasi-)norms and characterizations of $\mathcal{E}^{s}_{u,p,q}(\mathbb{R}^d)$. Keywords: Dilation Operator, Morrey space, Triebel-Lizorkin-Morrey space, Fourier multiplier

Theorems & Definitions (40)

• Theorem 1.1
• Corollary 1
• Definition 1: Morrey space $\mathcal{M}^{u}_{p}({\mathbb{R}}^d)$
• Lemma 1
• proof
• Proposition 1: TangXu
• Theorem 2.1
• proof
• Proposition 2: IzuSawTan2015
• proof