OPERATOR-VALUED LOCAL HARDY SPACES

This paper gives a systematic study of operator-valued local Hardy spaces, which are localizations of the Hardy spaces defined by Mei. We prove the h(1)-bmo duality and the h(p)-N-q duality for any conjugate pair (p, q) when 1 < p < infinity. We show that h(1)(R-d, M) and bmo(R-d, M) are also good endpoints of L-p(L-infinity(R-d )(circle times) over barM ) for interpolation. We obtain the local version of Calderon-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 h(1)(c)(R-d , M).