Hardy Sobolev spaces on strongly Lipschitz domains of [formula omitted

Abstract

Let Ω be a strongly Lipschitz domain of R n . The Hardy spaces H r 1 ( Ω ) and H z 1 ( Ω ) 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 ( Ω ) whose gradients belong to H r 1 ( Ω ) or H z 1 ( Ω ) , which we call Hardy–Sobolev spaces, following Strichartz (Coll. Math. 60–61(1) (1990) 129). Secondly, if L = - ÷ A ∇ is a uniformly elliptic second-order divergence operator on Ω with measurable complex coefficients subject to the Dirichlet or the Neumann boundary condition, we compare the norms of L 1 / 2 f and ∇ f in suitable Hardy spaces on Ω , depending on the boundary condition, under the assumption that the heat kernel of L satisfies suitable estimates.