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 $\mathrm h_1^\mathrm c(\mathbb{R}^d,\mathcal M)$.
DOI: http://dx.doi.org/10.7900/jot.2018jun02.2191
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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