Hp Theory on a Smooth Domain in RN and Elliptic Boundary Value Problems

In this paper, we answer the following two questions. QUESTION 1. Let Ω be an (appropriate) domain in RN. What are the possible (natural) notions of Hp(Ω) that generalize the usual Hardy spaces Hp(RN)? We shall see that several versions are possible. We concentrate on the two most relevant here, which may be considered to be the "largest" version hpr(Ω), and the "smallest" version hpz(Ω).For the second question consider the Dirichlet and Neumann problems for Ω. The former consists in finding the u = G(ƒ) which solves Δu = ƒ in Ω, and u|∂λ 0. The latter consists in finding the u = G(ƒ) which solves Δu = ƒ in Ω, and (∂u/∂n→)|∂Ω = 0. QUESTION 2. In the context of the relevant Hp(Ω), can one obtain the boundedness of ƒ → (∂2G/∂xi∂xj)(ƒ), and ƒ → (∂G̃/∂xi ∂xj)(ƒ)?