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Journal of Operator Theory

Volume 82, Issue 2, Fall 2019  pp. 383-443.

Operator-valued local Hardy spaces

Authors:  Runlian Xia (1), Xiao Xiong (2)
Author institution:(1) Laboratoire de Math{\'e}matiques, Universit{\'e} de Franche-Comt{\'e}, 25030 Besan\c{c}on Cedex, France, \textit{and} Instituto de Ciencias Matem{\'a}ticas, 28049 Madrid, Spain
(2) Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China \textit{and} Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Summary: This paper gives a systematic study of operator-valued local\break Hardy spaces, which are localizations of the Hardy spaces defined by Mei. We prove the $\mathrm h_1$-$\mathrm{bmo}$ duality and the $\mathrm h_p$-$\mathrm h_q$ duality for any conjugate pair $(p,q)$ when $p\in(1, \infty)$. We show that $\mathrm h_1(\mathbb{R}^d, \mathcal M)$ and $\mathrm{bmo}(\mathbb{R}^d, \mathcal M)$ are also good endpoints of $L_p(L_\infty(\mathbb{R}^d) \overline{\otimes} \mathcal M)$ for interpolation. We obtain the local version of Calder\'on--Zygmund theory, and then deduce that the Poisson kernel in our definition of the local Hardy norms can be replaced by any reasonable test function. Finally, we establish the atomic decomposition of the local Hardy space $\mathrm h_1^\mathrm c(\mathbb{R}^d,\mathcal M)$.

Keywords: noncommutative $L_p$-spaces, operator-valued Hardy spaces, operator-valued $\mathrm{bmo}$ spaces, duality, interpolation, Calder\'on--Zygmund theory, characterization, atomic decomposition

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