Hardy!Sobolev spaces on strongly Lipschitz domains of Rn

Let Omega be a strongly Lipschitz domain of R-n. The Hardy spaces H-r(1)(Omega) and H-z(1)(Omega) have been introduced by Miyachi (Studia Math. 95(3) (1990) 205), Jonsson et al. (Studia Math. 80(2) (1984) 141) and Chang et al. (J. Funct. Anal. 114 (1993) 286). We first investigate spaces of functions in L-1 (Omega) whose gradients belong to H-r(1)(Omega) or H-z(1)(Omega), which we call Hardy-Sobolev spaces, following Strichartz (Coll. Math. 60-61(1) (1990) 129). Secondly, if L = -div Adel is a uniformly elliptic second-order divergence operator on Omega with measurable complex coefficients subject to the Dirichlet or the Neumann boundary condition, we compare the norms of L(1/2)f and delf in suitable Hardy spaces on Omega, depending on the boundary condition, under the assumption that the heat kernel of L satisfies suitable estimates. (C) 2004 Elsevier Inc. All rights reserved.