Construction of an atomic decomposition for functions with compact support

Chang, Krantz and Stein [D.-C. Chang, S.G. Krantz, E.M. Stein, Hp theory on a smooth domain in Rn and elliptic boundary value problems, J. Funct. Anal. 114 (1993) 286–347] proved that if f ∈ Hp(Rn) and f vanishes outside Ω, then f has an atomic decomposition whose atoms are contained in Ω. The purpose of this paper is to give another proof for the case n/(n + 1) < p 1 and Ω a cube. Our argument provides a simple, direct construction of the desired atomic decomposition, and it works in a class of function spaces more general than the usual Hardy spaces.  2005 Elsevier Inc. All rights reserved.