The ``orthogonal'' expansion of Sobolev spaces in a sum of analytic and coanalytic subspaces (Q1594000)

The authors characterize the orthogonal complements of the set of holomorphic \(W^m_p(G)\) functions. Here \(W^m_p(G)\) is the usual Sobolev space of functions having \(m\) derivatives in \(L^p\) for \(m \in \mathbb{N}\cup \{0 \}\), and is the dual space of \(W^{-m}_{p{\prime}}(G)\) for \(m = -1, -2, \ldots\), and \(\frac 1p + \frac 1p' = 1\). The holomorphic \(W^m_p(G)\) functions are \[ \mathcal O^m_p (G) = \{f \in W^m_p(G) |\partial_{\overline {z}} f(z) = 0 \}, \] and the derivatives are ordinary derivatives if \(m\geq 1\), and distribution derivatives otherwise. We have denoted \(\partial_{\overline {z}}= \frac 12 (\partial_x + \partial_y)\), and, \[ \mathcal O^m_p (G)^{\perp} = \{f \in W^m_p(G) |\int \int_G f(z) \overline {\phi (z)} dx dy = 0, \phi \in \mathcal O^m_p (G) \}, \;m \geq 0, \] and the integral is replaced by duality if \(m < 0\). Two further spaces are needed. The first is \(V^{\infty}_0(G) = C^{\infty}(\overline G) \cap C_0(\overline G)\), the space of infinitely differentiable functions continuous up to the boundary \(\Gamma\) and equal to zero on the boundary. The second is its completion in the \(W^m_p(G)\) norm, which is denoted \(W^m_{p,0}(G)\). The main result is \[ \mathcal O^m_p(G)^{\perp} = \partial_z W^{m+1}_{p,0}(G), m \in\mathbb{Z}, p>1, \] i.e., a function \(q \in W^m_p (G)\) is orthogonal to \(\mathcal O^m_p(G)\) if and only if there is a unique \(r \in W^{m+1}_{p,0}(G)\) that is a solution of \( \partial_z r = q, r(z) |_{\Gamma} = 0\). Moreover, it follows that \[ W^m_p(G) = \mathcal O^m_p(G) \oplus \partial_z W^{m+1}_{p,0}(G). \] They also give a version valid for Clifford algebras.