Hardy spaces and the Dirichlet problem on Lipschitz domains (Q1122684)

Let D be a Lipschitz domain in \(R^ n\) (n\(\geq 3)\) and consider the ``Hardy space'' \(H^ p(D,d\sigma)=\{u\); \(\Delta u=0\) in D, \(u(P_ 0)=0\) and \(Nu\in L^ p(\partial D,d\sigma)\}\), where \(P_ 0\in D\) is a fixed point, \(d\sigma\) is the surface measure on \(\partial D\) and Du is the non-tangential maximal function of u. The main concern of this paper is to provide an atomic decomposition of functions in \(H^ p(D,d\sigma)\) and to describe the dual of this space in case \(1\leq p<2\). The main result, to whose proof a good part of the paper is devoted, is the following pairing between functions in \(H^ 1(D,d\sigma)\) and BMO- functions: \[ | \int_{\partial D}u(Q)f(Q) d\omega (Q)| \leq C \| Nu\|_{L^ 1(d\sigma)} \| f\|_{BMO} \] for \(u\in {\mathcal L}(\bar D)\) and \(f\in BMO_{\sigma}(\omega)\), where \(\omega\) is the harmonic measure evaluated at \(P_ 0\), \({\mathcal L}(\bar D)=\{u\); Lipschitz on \(\bar D,\) \(\Delta u=0\) in D and \(u(P_ 0)=0\}\) and \[ BMO_{\sigma}(\omega)=\{f;\quad \sup_{\Delta}\frac{1}{\sigma (\Delta)}\int_{\Delta}| f-f_{\Delta}| \quad d\omega \quad +\int_{\partial D}| f | \quad d\omega <\infty \} \] with \(f_{\Delta}=(1/\omega (\Delta))\int_{\Delta}f d\omega\), and \(\Delta\) denotes the surface ball.