Wavelets and Triebel type oscillation spaces

TL;DR

This paper uses wavelets to characterize Triebel-Lizorkin-Morrey spaces and establishes their properties, including semigroup characterization and operator continuity, with implications for Navier-Stokes equations.

Contribution

It introduces a wavelet-based characterization of Triebel-Lizorkin-Morrey spaces and proves the continuity of Calderón-Zygmund operators on these spaces.

Findings

Wavelet characterization of Triebel-Lizorkin-Morrey spaces

Fractional heat semigroup characterization of these spaces

Continuity of Calderón-Zygmund operators on the spaces

Abstract

We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$. Then we establish a characterization of $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calder\'on-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.