Quarklet Characterizations for Triebel-Lizorkin Spaces

TL;DR

This paper establishes a new quarklet-based characterization of one-dimensional Triebel-Lizorkin and Triebel-Lizorkin-Morrey spaces, providing equivalent quasi-norms and decompositions using biorthogonal spline wavelets.

Contribution

It introduces a novel quarklet decomposition for Triebel-Lizorkin spaces using biorthogonal spline wavelets and develops associated sequence spaces, extending to Triebel-Lizorkin-Morrey spaces.

Findings

Established quarklet characterizations for Triebel-Lizorkin spaces.

Derived new equivalent quasi-norms based on quarklet decompositions.

Extended the framework to Triebel-Lizorkin-Morrey spaces.

Abstract

In this paper we prove that under some conditions on the parameters the one-dimensional Triebel-Lizorkin spaces $ F^{s}_{r,q}(\mathbb{R}) $ can be described in terms of quarklets. So for functions from Triebel-Lizorkin spaces we obtain a quarkonial decomposition as well as a new equivalent quasi-norm. For that purpose we use quarklets that are constructed out of biorthogonal compactly supported Cohen-Daubechies-Feauveau spline wavelets, where the primal generator is a cardinal B-spline. Moreover we introduce some sequence spaces apposite to our quarklet system and study their properties. Finally we also obtain a quarklet characterization for the Triebel-Lizorkin-Morrey spaces $ \mathcal{E}^{s}_{u,r,q}(\mathbb{R}) $.