Operator-valued Triebel-Lizorkin spaces

TL;DR

This paper develops Fourier multiplier theorems and characterizations for operator-valued Triebel-Lizorkin spaces, connecting them with Hardy spaces and establishing atomic decompositions, advancing the analysis of pseudo-differential operators with operator-valued symbols.

Contribution

It introduces new Fourier multiplier theorems and atomic decompositions for operator-valued Triebel-Lizorkin spaces, linking them with Hardy spaces and enabling analysis of operator-valued pseudo-differential operators.

Findings

Established Fourier multiplier theorems for square functions.

Connected operator-valued Triebel-Lizorkin spaces with Hardy spaces.

Proved atomic decompositions for these spaces.

Abstract

This paper is devoted to the study of operator-valued Triebel-Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel-Lizorkin spaces on $\mathbb{R}^d$. As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood-Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel-Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.