Operator-valued local Hardy spaces

Abstract

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 h1-bmo duality and the hp-hq duality for any conjugate pair (p,q) when p∈(1,∞). We show that h1(Rd,M) and bmo(Rd,M) are also good endpoints of Lp(L∞(Rd)¯¯¯¯⊗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 hc1(Rd,M).