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

Abstract

In this paper we study the behavior of dilation operators $D_{\lambda }:f\mapsto f(\lambda \cdot )$ with $\lambda >1$ in the context of Triebel-Lizorkin-Morrey spaces ${\mathcal{E}}_{u,p,q}^s({\mathbb{R}}^d)$. For that purpose we prove upper and lower bounds for the operator (quasi-)norm $\Vert D_{\lambda }|\mathcal{L}({\mathcal{E}}_{u,p,q}^s({\mathbb{R}}^d))\Vert$. We show that for $s>{\sigma }_p$ the operator (quasi-)norm $\Vert D_{\lambda }|\mathcal{L}({\mathcal{E}}_{u,p,q}^s({\mathbb{R}}^d))\Vert$ up to constants behaves as ${\lambda }^{s-\frac{d}{u}}$. For the borderline case $s={\sigma }_p$ we observe a behavior of the form ${\lambda }^{{\sigma }_p-\frac{d}{u}}$, multiplied with logarithmic terms of $\lambda$ that also depend on the fine index $q$. For $s<{\sigma }_p$ and $p\ge 1$ we find the relation $\Vert D_{\lambda }|\mathcal{L}({\mathcal{E}}_{u,p,q}^s({\mathbb{R}}^d))\Vert \sim {\lambda }^{-\frac{d}{u}}$. The case $s<{\sigma }_p$ 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 ${\mathcal{E}}_{u,p,q}^s({\mathbb{R}}^d)$. Keywords: Dilation Operator, Morrey space, Triebel-Lizorkin-Morrey space, Fourier multiplier

33 pages; Dedicated to the 90th anniversary of Hans Triebel