Operator-Valued Triebel–Lizorkin Spaces

  • Runlian XiaEmail author
  • Xiao Xiong


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.


Noncommutative \(L_p\)-spaces Operator-valued Triebel–Lizorkin spaces Operator-valued Hardy spaces Fourier multipliers Interpolation Characterizations Atomic decomposition 

Mathematics Subject Classification

Primary: 46L52 42B30 Secondary: 46L07 47L65 



The authors are greatly indebted to Professor Quanhua Xu for having suggested to them the subject of this paper, for many helpful discussions and very careful reading of this paper. The authors are partially supported by the National Natural Science Foundation of China (Grant No. 11301401).


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Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité Bourgogne Franche-ComtéBesançon CedexFrance
  2. 2.Instituto de Ciencias MatemáticasMadridSpain
  3. 3.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

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