I have some difficulty in reading a paper of Jerison and Kenig: the inhomogeneous Dirichlet problem in Lipschitz domains.

For convenience, I explain some notations. $ \Omega$ is a bounded Lipschitz domain in $ \mathbb{R}^n$ and $ \delta(x)$ is a distance function from $ x$ to $ \partial \Omega$ . $ L^2_{1/2}$ is a Bessel potential space of order $ 1/2$ .

Define $ \Gamma (x) = \{ y\in \mathbb{R}^n : |x-y| \leq C \delta(x) \}$ where $ C>1$ a suitable constant. $ v^*$ denote the nontangential maximal function. Define $ $ S(v)(z) = \left(\int_{\Gamma (z)} \delta(x)^{2-n} |\nabla v(x)|^2 dx \right)^{\frac{1}{2}},$ $ the Lusin area integral.

Here is a corollary in Jerison and Kenig’s paper.

Corollary 5.5. Suppose $ v$ is a harmonic function in $ \Omega$ . Then the following are equivalent:

1. $ v^*$ is in $ L^2(\partial \Omega)$
2. $ v\in L^2_{1/2} (\Omega)$
3. $ S(v) \in L^2(\partial \Omega)$

In each case, there exists a function $ g\in L^2(\partial\Omega)$ such that $ v$ tends to $ g$ nontangentially a.e. on $ \partial \Omega$ .

I have a two questions on this corollary.

1. If we assume $ S(v)\in L^2(\partial \Omega)$ , using a result of Dahlberg, I can prove $ v\in L^2 (\partial\Omega)$ . Can I prove $ v\in L^2(\Omega)$ in this case?
2. According to some literature, the finiteness of $ S(v)$ is equivalent to the nontangential convergence of $ v$ . I checked the case when a domain is upper half plane. (Stein, Singular integral) But I cannot find an integrability result. How can I find such $ g \in L^2(\partial \Omega)$ so that $ v$ converges to $ g$ nontangentially a.e.? I’m not sure when $ \Omega =\mathbb{R}^n_+$ .

Thanks for in advance.